mesfunc5.miz

begin

 reconsider jj = 1 as Real;

 theorem :: MESFUNC5:1



 Th1: for x,y be R_eal holds |.(x - y).| = |.(y - x).|

 proof

 let x,y be R_eal;

 |.(y - x).| = |.( - (x - y)).| by XXREAL_3: 26;

 hence thesis by EXTREAL1: 29;

 end;

 theorem :: MESFUNC5:2



 Th2: for x,y be R_eal holds (y - x) <= |.(x - y).|

 proof

 let x,y be R_eal;

 ( - |.(x - y).|) <= (x - y) by EXTREAL1: 20;

 then ( - (x - y)) <= |.(x - y).| by XXREAL_3: 60;

 hence thesis by XXREAL_3: 26;

 end;

 theorem :: MESFUNC5:3



 Th3: for x,y be R_eal, e be Real st |.(x - y).| < e & not (x = +infty & y = +infty or x = -infty & y = -infty ) holds x <> +infty & x <> -infty & y <> +infty & y <> -infty

 proof

 let x,y be R_eal, e be Real;

 assume

 A1: |.(x - y).| < e;

 (y - x) <= |.(x - y).| by Th2;

 then

 A2: (y - x) < e by A1, XXREAL_0: 2;

 (x - y) <= |.(x - y).| by EXTREAL1: 20;

 then (x - y) < e by A1, XXREAL_0: 2;

 hence thesis by A2, XXREAL_3: 54;

 end;

 theorem :: MESFUNC5:4



 Th4: for n be Nat, p be ExtReal st 0 <= p & p < n holds ex k be Nat st 1 <= k & k <= ((2 |^ n) * n) & ((k - 1) / (2 |^ n)) <= p & p < (k / (2 |^ n))

 proof

 let n be Nat;

 let p be ExtReal;

 assume that

 A1: 0 <= p and

 A2: p < n;

 0 in REAL by XREAL_0:def 1;

 then

 reconsider p1 = p as Element of REAL by A1, A2, XXREAL_0: 46;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 set k = [\((p1 * (2 |^ n)) + 1)/];



 A3: (((p1 * (2 |^ n)) + 1) - 1) = (p1 * (2 |^ n));

 then

 A4: 0 < k by A1, INT_1:def 6;

 then

 reconsider k as Element of NAT by INT_1: 3;



 A5: (p1 * (2 |^ n)) < k by A3, INT_1:def 6;



 A6: 0 < (2 |^ n) by PREPOWER: 6;

 A7:

 now

 (p1 * (2 |^ n)) < ((2 |^ n) * n) by A2, A6, XREAL_1: 68;

 then

 A8: ((p1 * (2 |^ n)) + 1) < (((2 |^ n) * n) + 1) by XREAL_1: 6;

 reconsider N = ((2 |^ n) * n) as Integer;

 assume

 A9: k > ((2 |^ n) * n);



 A10: [\N/] = N by INT_1: 25;

 k <= ((p1 * (2 |^ n)) + 1) by INT_1:def 6;

 then ((2 |^ n) * n) < ((p1 * (2 |^ n)) + 1) by A9, XXREAL_0: 2;

 hence contradiction by A9, A8, A10, INT_1: 67;

 end;

 take k;

 k <= ((p1 * (2 |^ n)) + 1) by INT_1:def 6;

 then

 A11: (k - 1) <= (p1 * (2 |^ n)) by XREAL_1: 20;

 ( 0 + 1) <= k by A4, NAT_1: 13;

 hence thesis by A6, A7, A5, A11, XREAL_1: 79, XREAL_1: 81;

 end;

 theorem :: MESFUNC5:5



 Th5: for n,k be Nat, p be ExtReal st k <= ((2 |^ n) * n) & n <= p holds (k / (2 |^ n)) <= p

 proof

 let n,k be Nat;

 let p be ExtReal;

 assume that

 A1: k <= ((2 |^ n) * n) and

 A2: n <= p;

 assume p < (k / (2 |^ n));

 then n < (k / (2 |^ n)) by A2, XXREAL_0: 2;

 hence contradiction by A1, PREPOWER: 6, XREAL_1: 79;

 end;

 theorem :: MESFUNC5:6



 Th6: for x,y,k be ExtReal st 0 <= k holds (k * ( max (x,y))) = ( max ((k * x),(k * y))) & (k * ( min (x,y))) = ( min ((k * x),(k * y)))

 proof

 let x,y,k be ExtReal;

 assume

 A1: 0 <= k;

 now

 per cases by XXREAL_0: 16;

 suppose

 A2: ( max (x,y)) = x;

 then y <= x by XXREAL_0:def 10;

 then (k * y) <= (k * x) by A1, XXREAL_3: 71;

 hence (k * ( max (x,y))) = ( max ((k * x),(k * y))) by A2, XXREAL_0:def 10;

 end;

 suppose

 A3: ( max (x,y)) = y;

 then x <= y by XXREAL_0:def 10;

 then (k * x) <= (k * y) by A1, XXREAL_3: 71;

 hence (k * ( max (x,y))) = ( max ((k * x),(k * y))) by A3, XXREAL_0:def 10;

 end;

 end;

 hence (k * ( max (x,y))) = ( max ((k * x),(k * y)));

 per cases by XXREAL_0: 15;

 suppose

 A4: ( min (x,y)) = x;

 then x <= y by XXREAL_0:def 9;

 then (k * x) <= (k * y) by A1, XXREAL_3: 71;

 hence thesis by A4, XXREAL_0:def 9;

 end;

 suppose

 A5: ( min (x,y)) = y;

 then y <= x by XXREAL_0:def 9;

 then (k * y) <= (k * x) by A1, XXREAL_3: 71;

 hence thesis by A5, XXREAL_0:def 9;

 end;

 end;

 theorem :: MESFUNC5:7

 for x,y,k be R_eal st k <= 0 holds (k * ( min (x,y))) = ( max ((k * x),(k * y))) & (k * ( max (x,y))) = ( min ((k * x),(k * y)))

 proof

 let x,y,k be R_eal;

 assume

 A1: k <= 0 ;

 hereby

 per cases by XXREAL_0: 16;

 suppose ( max (x,y)) = x;

 then

 A2: y <= x by XXREAL_0:def 10;

 then (k * x) <= (k * y) by A1, XXREAL_3: 101;

 then ( max ((k * x),(k * y))) = (k * y) by XXREAL_0:def 10;

 hence (k * ( min (x,y))) = ( max ((k * x),(k * y))) by A2, XXREAL_0:def 9;

 end;

 suppose ( max (x,y)) = y;

 then

 A3: x <= y by XXREAL_0:def 10;

 then (k * y) <= (k * x) by A1, XXREAL_3: 101;

 then ( max ((k * x),(k * y))) = (k * x) by XXREAL_0:def 10;

 hence (k * ( min (x,y))) = ( max ((k * x),(k * y))) by A3, XXREAL_0:def 9;

 end;

 end;

 per cases by XXREAL_0: 15;

 suppose ( min (x,y)) = x;

 then

 A4: x <= y by XXREAL_0:def 9;

 then (k * y) <= (k * x) by A1, XXREAL_3: 101;

 then ( min ((k * x),(k * y))) = (k * y) by XXREAL_0:def 9;

 hence thesis by A4, XXREAL_0:def 10;

 end;

 suppose ( min (x,y)) = y;

 then

 A5: y <= x by XXREAL_0:def 9;

 then (k * x) <= (k * y) by A1, XXREAL_3: 101;

 then ( min ((k * y),(k * x))) = (k * x) by XXREAL_0:def 9;

 hence thesis by A5, XXREAL_0:def 10;

 end;

 end;

 begin

 definition

 let IT be set;

 :: MESFUNC5:def1

 attr IT is nonpositive means for x be R_eal holds x in IT implies x <= 0 ;

 end

 definition

 let R be Relation;

 :: MESFUNC5:def2

 attr R is nonpositive means ( rng R) is nonpositive;

 end

 theorem :: MESFUNC5:8



 Th8: for X be set, F be PartFunc of X, ExtREAL holds F is nonpositive iff for n be object holds (F . n) <= 0.

 proof

 let X be set, F be PartFunc of X, ExtREAL ;

 hereby

 assume F is nonpositive;

 then

 A1: ( rng F) is nonpositive;

 let n be object;

 per cases ;

 suppose n in ( dom F);

 then (F . n) in ( rng F) by FUNCT_1:def 3;

 hence (F . n) <= 0. by A1;

 end;

 suppose not n in ( dom F);

 hence (F . n) <= 0. by FUNCT_1:def 2;

 end;

 end;

 assume

 A2: for n be object holds (F . n) <= 0. ;

 let y be R_eal;

 assume y in ( rng F);

 then ex x be object st x in ( dom F) & y = (F . x) by FUNCT_1:def 3;

 hence thesis by A2;

 end;

 theorem :: MESFUNC5:9



 Th9: for X be set, F be PartFunc of X, ExtREAL st for n be set st n in ( dom F) holds (F . n) <= 0. holds F is nonpositive

 proof

 let X be set, F be PartFunc of X, ExtREAL such that

 A1: for n be set st n in ( dom F) holds (F . n) <= 0. ;

 let y be R_eal;

 assume y in ( rng F);

 then ex x be object st x in ( dom F) & y = (F . x) by FUNCT_1:def 3;

 hence thesis by A1;

 end;

 definition

 let R be Relation;

 :: MESFUNC5:def3

 attr R is without-infty means

 : Def3: not -infty in ( rng R);

 :: MESFUNC5:def4

 attr R is without+infty means not +infty in ( rng R);

 end

 definition

 let X be non empty set, f be PartFunc of X, ExtREAL ;

 :: original: without-infty

 redefine

 :: MESFUNC5:def5

 attr f is without-infty means

 : Def5: for x be object holds -infty < (f . x);

 compatibility

 proof

 hereby

 assume f is without-infty;

 then

 A1: not -infty in ( rng f);

 hereby

 let x be object;

 per cases ;

 suppose x in ( dom f);

 then (f . x) <> -infty by A1, FUNCT_1:def 3;

 hence -infty < (f . x) by XXREAL_0: 6;

 end;

 suppose not x in ( dom f);

 hence -infty < (f . x) by FUNCT_1:def 2;

 end;

 end;

 end;

 assume

 A2: for x be object holds -infty < (f . x);

 now

 assume -infty in ( rng f);

 then ex x be object st x in ( dom f) & (f . x) = -infty by FUNCT_1:def 3;

 hence contradiction by A2;

 end;

 hence thesis;

 end;

 :: original: without+infty

 redefine

 :: MESFUNC5:def6

 attr f is without+infty means for x be object holds (f . x) < +infty ;

 compatibility

 proof

 hereby

 assume f is without+infty;

 then

 A3: not +infty in ( rng f);

 hereby

 let x be object;

 per cases ;

 suppose x in ( dom f);

 then (f . x) <> +infty by A3, FUNCT_1:def 3;

 hence (f . x) < +infty by XXREAL_0: 4;

 end;

 suppose not x in ( dom f);

 hence (f . x) < +infty by FUNCT_1:def 2;

 end;

 end;

 end;

 assume

 A4: for x be object holds (f . x) < +infty ;

 now

 assume +infty in ( rng f);

 then ex x be object st x in ( dom f) & (f . x) = +infty by FUNCT_1:def 3;

 hence contradiction by A4;

 end;

 hence thesis;

 end;

 end

 theorem :: MESFUNC5:10



 Th10: for X be non empty set, f be PartFunc of X, ExtREAL holds (for x be set st x in ( dom f) holds -infty < (f . x)) iff f is without-infty

 proof

 let X be non empty set, f be PartFunc of X, ExtREAL ;

 hereby

 assume

 A1: for x be set st x in ( dom f) holds -infty < (f . x);

 now

 let x be object;

 per cases ;

 suppose x in ( dom f);

 hence -infty < (f . x) by A1;

 end;

 suppose not x in ( dom f);

 hence -infty < (f . x) by FUNCT_1:def 2;

 end;

 end;

 hence f is without-infty;

 end;

 assume f is without-infty;

 hence thesis;

 end;

 theorem :: MESFUNC5:11



 Th11: for X be non empty set, f be PartFunc of X, ExtREAL holds (for x be set st x in ( dom f) holds (f . x) < +infty ) iff f is without+infty

 proof

 let X be non empty set, f be PartFunc of X, ExtREAL ;

 hereby

 assume

 A1: for x be set st x in ( dom f) holds (f . x) < +infty ;

 now

 let x be object;

 per cases ;

 suppose x in ( dom f);

 hence (f . x) < +infty by A1;

 end;

 suppose not x in ( dom f);

 hence (f . x) < +infty by FUNCT_1:def 2;

 end;

 end;

 hence f is without+infty;

 end;

 assume f is without+infty;

 hence thesis;

 end;

 theorem :: MESFUNC5:12



 Th12: for X be non empty set, f be PartFunc of X, ExtREAL st f is nonnegative holds f is without-infty by SUPINF_2: 51;

 theorem :: MESFUNC5:13



 Th13: for X be non empty set, f be PartFunc of X, ExtREAL st f is nonpositive holds f is without+infty by Th8;

 registration

 let X be non empty set;

 cluster nonnegative -> without-infty for PartFunc of X, ExtREAL ;

 coherence by Th12;

 cluster nonpositive -> without+infty for PartFunc of X, ExtREAL ;

 coherence by Th13;

 end

 theorem :: MESFUNC5:14



 Th14: for X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL st f is_simple_func_in S holds f is without+infty & f is without-infty

 proof

 let X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL ;

 assume

 A1: f is_simple_func_in S;

 hereby

 assume not f is without+infty;

 then +infty in ( rng f);

 then (f " { +infty }) <> {} by FUNCT_1: 72;

 then

 consider x be object such that

 A2: x in (f " { +infty }) by XBOOLE_0:def 1;



 A3: f is real-valued by A1, MESFUNC2:def 4;

 (f . x) in { +infty } by A2, FUNCT_1:def 7;

 hence contradiction by A3, TARSKI:def 1;

 end;

 hereby

 assume not f is without-infty;

 then -infty in ( rng f);

 then (f " { -infty }) <> {} by FUNCT_1: 72;

 then

 consider x be object such that

 A4: x in (f " { -infty }) by XBOOLE_0:def 1;



 A5: f is real-valued by A1, MESFUNC2:def 4;

 (f . x) in { -infty } by A4, FUNCT_1:def 7;

 hence contradiction by A5, TARSKI:def 1;

 end;

 end;

 theorem :: MESFUNC5:15



 Th15: for X be non empty set, Y be set, f be PartFunc of X, ExtREAL st f is nonnegative holds (f | Y) is nonnegative

 proof

 let X be non empty set, Y be set, f be PartFunc of X, ExtREAL ;

 assume

 A1: f is nonnegative;

 now

 let x be object;

 assume

 A2: x in ( dom (f | Y));

 then ((f | Y) . x) = (f . x) by FUNCT_1: 47;

 hence 0 <= ((f | Y) . x) by A1, A2, SUPINF_2: 39;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:16



 Th16: for X be non empty set, f,g be PartFunc of X, ExtREAL st f is without-infty & g is without-infty holds ( dom (f + g)) = (( dom f) /\ ( dom g))

 proof

 let X be non empty set;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is without-infty and

 A2: g is without-infty;

 not -infty in ( rng g) by A2;

 then

 A3: (g " { -infty }) = {} by FUNCT_1: 72;

 not -infty in ( rng f) by A1;

 then (f " { -infty }) = {} by FUNCT_1: 72;

 then (((f " { +infty }) /\ (g " { -infty })) \/ ((f " { -infty }) /\ (g " { +infty }))) = {} by A3;

 then ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ {} ) by MESFUNC1:def 3;

 hence thesis;

 end;

 theorem :: MESFUNC5:17

 for X be non empty set, f,g be PartFunc of X, ExtREAL st f is without-infty & g is without+infty holds ( dom (f - g)) = (( dom f) /\ ( dom g))

 proof

 let X be non empty set;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is without-infty and

 A2: g is without+infty;

 not +infty in ( rng g) by A2;

 then

 A3: (g " { +infty }) = {} by FUNCT_1: 72;

 not -infty in ( rng f) by A1;

 then (f " { -infty }) = {} by FUNCT_1: 72;

 then (((f " { +infty }) /\ (g " { +infty })) \/ ((f " { -infty }) /\ (g " { -infty }))) = {} by A3;

 then ( dom (f - g)) = ((( dom f) /\ ( dom g)) \ {} ) by MESFUNC1:def 4;

 hence thesis;

 end;

 theorem :: MESFUNC5:18



 Th18: for X be non empty set, S be SigmaField of X, f,g be PartFunc of X, ExtREAL , F be Function of RAT , S, r be Real, A be Element of S st f is without-infty & g is without-infty & (for p be Rational holds (F . p) = ((A /\ ( less_dom (f,p))) /\ (A /\ ( less_dom (g,(r - p qua Complex)))))) holds (A /\ ( less_dom ((f + g),r))) = ( union ( rng F))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let f,g be PartFunc of X, ExtREAL ;

 let F be Function of RAT , S;

 let r be Real;

 let A be Element of S;

 assume that

 A1: f is without-infty and

 A2: g is without-infty and

 A3: for p be Rational holds (F . p) = ((A /\ ( less_dom (f,p))) /\ (A /\ ( less_dom (g,(r - p qua Complex)))));



 A4: ( dom (f + g)) = (( dom f) /\ ( dom g)) by A1, A2, Th16;



 A5: ( union ( rng F)) c= (A /\ ( less_dom ((f + g),r)))

 proof

 let x be object;

 assume x in ( union ( rng F));

 then

 consider Y be set such that

 A6: x in Y and

 A7: Y in ( rng F) by TARSKI:def 4;

 consider p be object such that

 A8: p in ( dom F) and

 A9: Y = (F . p) by A7, FUNCT_1:def 3;

 reconsider p as Rational by A8;



 A10: x in ((A /\ ( less_dom (f,p))) /\ (A /\ ( less_dom (g,(r - p))))) by A3, A6, A9;

 then

 A11: x in (A /\ ( less_dom (f,p))) by XBOOLE_0:def 4;

 then

 A12: x in A by XBOOLE_0:def 4;



 A13: x in ( less_dom (f,p)) by A11, XBOOLE_0:def 4;

 x in (A /\ ( less_dom (g,(r - p)))) by A10, XBOOLE_0:def 4;

 then

 A14: x in ( less_dom (g,(r - p))) by XBOOLE_0:def 4;

 reconsider x as Element of X by A10;

 (f . x) +infty by XXREAL_0: 4;



 A16: -infty < (f . x) by A1;



 A17: -infty < (g . x) by A2;



 A18: (g . x) < (r - p) by A14, MESFUNC1:def 11;

 then (g . x) <> +infty by XXREAL_0: 4;

 then

 reconsider f1 = (f . x), g1 = (g . x) as Element of REAL by A16, A17, A15, XXREAL_0: 14;



 A19: p < (r - g1) by A18, XREAL_1: 12;

 f1 < p by A13, MESFUNC1:def 11;

 then f1 < (r - g1) by A19, XXREAL_0: 2;

 then

 A20: (f1 + g1) < r by XREAL_1: 20;



 A21: x in ( dom g) by A14, MESFUNC1:def 11;

 x in ( dom f) by A13, MESFUNC1:def 11;

 then

 A22: x in ( dom (f + g)) by A4, A21, XBOOLE_0:def 4;



 then ((f + g) . x) = ((f . x) + (g . x)) by MESFUNC1:def 3

 .= (f1 + g1) by SUPINF_2: 1;

 then x in ( less_dom ((f + g),r)) by A20, A22, MESFUNC1:def 11;

 hence thesis by A12, XBOOLE_0:def 4;

 end;

 (A /\ ( less_dom ((f + g),r))) c= ( union ( rng F))

 proof

 let x be object;

 assume

 A23: x in (A /\ ( less_dom ((f + g),r)));

 then

 A24: x in A by XBOOLE_0:def 4;



 A25: x in ( less_dom ((f + g),r)) by A23, XBOOLE_0:def 4;

 then

 A26: x in ( dom (f + g)) by MESFUNC1:def 11;

 then

 A27: x in ( dom f) by A4, XBOOLE_0:def 4;



 A28: ((f + g) . x) < r by A25, MESFUNC1:def 11;



 A29: x in ( dom g) by A4, A26, XBOOLE_0:def 4;

 reconsider x as Element of X by A23;



 A30: -infty < (f . x) by A1;



 A31: ((f . x) + (g . x)) < r by A26, A28, MESFUNC1:def 3;

 then

 A32: (g . x) <> +infty by A30, XXREAL_3: 52;



 A33: -infty < (g . x) by A2;

 then (f . x) <> +infty by A31, XXREAL_3: 52;

 then

 reconsider f1 = (f . x), g1 = (g . x) as Element of REAL by A30, A33, A32, XXREAL_0: 14;

 (f . x) < (r - (g . x)) by A31, A30, A33, XXREAL_3: 52;

 then

 consider p be Rational such that

 A34: f1 < p and

 A35: p < (r - g1) by RAT_1: 7;

 not (r - p) <= g1 by A35, XREAL_1: 12;

 then x in ( less_dom (g,(r - p))) by A29, MESFUNC1:def 11;

 then

 A36: x in (A /\ ( less_dom (g,(r - p)))) by A24, XBOOLE_0:def 4;

 p in RAT by RAT_1:def 2;

 then p in ( dom F) by FUNCT_2:def 1;

 then

 A37: (F . p) in ( rng F) by FUNCT_1:def 3;

 x in ( less_dom (f,p)) by A27, A34, MESFUNC1:def 11;

 then x in (A /\ ( less_dom (f,p))) by A24, XBOOLE_0:def 4;

 then x in ((A /\ ( less_dom (f,p))) /\ (A /\ ( less_dom (g,(r - p))))) by A36, XBOOLE_0:def 4;

 then x in (F . p) by A3;

 hence thesis by A37, TARSKI:def 4;

 end;

 hence thesis by A5;

 end;

 definition

 let X be non empty set;

 let f be PartFunc of X, REAL ;

 :: MESFUNC5:def7

 func R_EAL f -> PartFunc of X, ExtREAL equals f;

 coherence by NUMBERS: 31, RELSET_1: 7;

 end

 theorem :: MESFUNC5:19



 Th19: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is nonnegative & g is nonnegative holds (f + g) is nonnegative

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is nonnegative and

 A2: g is nonnegative;

 for x be object st x in ( dom (f + g)) holds 0 <= ((f + g) . x)

 proof

 let x be object;

 assume

 A3: x in ( dom (f + g));

 0 <= (f . x) by A1, SUPINF_2: 51;

 then

 A4: (g . x) <= ((f . x) + (g . x)) by XXREAL_3: 39;

 0 <= (g . x) by A2, SUPINF_2: 51;

 hence thesis by A3, A4, MESFUNC1:def 3;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:20



 Th20: for X be non empty set, f be PartFunc of X, ExtREAL , c be Real st f is nonnegative holds ( 0 <= c implies (c (#) f) is nonnegative) & (c <= 0 implies (c (#) f) is nonpositive)

 proof

 let X be non empty set;

 let f be PartFunc of X, ExtREAL ;

 let c be Real;

 set g = (c (#) f);

 assume

 A1: f is nonnegative;

 hereby

 set g = (c (#) f);

 assume

 A2: 0 <= c;

 for x be object st x in ( dom g) holds 0 <= (g . x)

 proof

 let x be object;

 0 <= (f . x) by A1, SUPINF_2: 51;

 then

 A3: 0 <= (c * (f . x)) by A2;

 assume x in ( dom g);

 hence thesis by A3, MESFUNC1:def 6;

 end;

 hence (c (#) f) is nonnegative by SUPINF_2: 52;

 end;

 assume

 A4: c <= 0 ;

 now

 let x be set;

 0 <= (f . x) by A1, SUPINF_2: 51;

 then

 A5: (c * (f . x)) <= 0 by A4;

 assume x in ( dom g);

 hence (g . x) <= 0 by A5, MESFUNC1:def 6;

 end;

 hence thesis by Th9;

 end;

 theorem :: MESFUNC5:21



 Th21: for X be non empty set, f,g be PartFunc of X, ExtREAL st (for x be set st x in (( dom f) /\ ( dom g)) holds (g . x) <= (f . x) & -infty < (g . x) & (f . x) < +infty ) holds (f - g) is nonnegative

 proof

 let X be non empty set, f,g be PartFunc of X, ExtREAL ;

 assume

 A1: for x be set st x in (( dom f) /\ ( dom g)) holds (g . x) <= (f . x) & -infty < (g . x) & (f . x) < +infty ;

 now

 let x be object;

 assume

 A2: x in ( dom (f - g));

 ( dom (f - g)) = ((( dom f) /\ ( dom g)) \ (((f " { +infty }) /\ (g " { +infty })) \/ ((f " { -infty }) /\ (g " { -infty })))) by MESFUNC1:def 4;

 then ( dom (f - g)) c= (( dom f) /\ ( dom g)) by XBOOLE_1: 36;

 then 0 <= ((f . x) - (g . x)) by A1, A2, XXREAL_3: 40;

 hence 0 <= ((f - g) . x) by A2, MESFUNC1:def 4;

 end;

 hence thesis by SUPINF_2: 52;

 end;



 Lm1: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL holds ( max+ f) is nonnegative & ( max- f) is nonnegative & |.f.| is nonnegative

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;



 A1: for x be object st x in ( dom ( max- f)) holds 0 <= (( max- f) . x) by MESFUNC2: 13;

 for x be object st x in ( dom ( max+ f)) holds 0 <= (( max+ f) . x) by MESFUNC2: 12;

 hence ( max+ f) is nonnegative & ( max- f) is nonnegative by A1, SUPINF_2: 52;

 now

 let x be object;

 assume x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 hence 0 <= ( |.f.| . x) by EXTREAL1: 14;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:22



 Th22: for X be non empty set, f,g be PartFunc of X, ExtREAL st f is nonnegative & g is nonnegative holds ( dom (f + g)) = (( dom f) /\ ( dom g)) & (f + g) is nonnegative

 proof

 let X be non empty set, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is nonnegative and

 A2: g is nonnegative;

 thus

 A3: ( dom (f + g)) = (( dom f) /\ ( dom g)) by A1, A2, Th16;

 now

 let x be object;

 assume

 A4: x in (( dom f) /\ ( dom g));



 A5: 0 <= (g . x) by A2, SUPINF_2: 51;

 0 <= (f . x) by A1, SUPINF_2: 51;

 then 0 <= ((f . x) + (g . x)) by A5;

 hence 0 <= ((f + g) . x) by A3, A4, MESFUNC1:def 3;

 end;

 hence thesis by A3, SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:23



 Th23: for X be non empty set, f,g,h be PartFunc of X, ExtREAL st f is nonnegative & g is nonnegative & h is nonnegative holds ( dom ((f + g) + h)) = ((( dom f) /\ ( dom g)) /\ ( dom h)) & ((f + g) + h) is nonnegative & for x be set st x in ((( dom f) /\ ( dom g)) /\ ( dom h)) holds (((f + g) + h) . x) = (((f . x) + (g . x)) + (h . x))

 proof

 let X be non empty set;

 let f,g,h be PartFunc of X, ExtREAL ;

 assume that

 A1: f is nonnegative and

 A2: g is nonnegative and

 A3: h is nonnegative;



 A4: (f + g) is nonnegative by A1, A2, Th22;

 then

 A5: ( dom ((f + g) + h)) = (( dom (f + g)) /\ ( dom h)) by A3, Th22;

 hence ( dom ((f + g) + h)) = ((( dom f) /\ ( dom g)) /\ ( dom h)) by A1, A2, Th22;

 thus ((f + g) + h) is nonnegative by A3, A4, Th22;

 hereby

 let x be set;

 assume x in ((( dom f) /\ ( dom g)) /\ ( dom h));

 then

 A6: x in (( dom (f + g)) /\ ( dom h)) by A1, A2, Th22;

 then

 A7: x in ( dom (f + g)) by XBOOLE_0:def 4;



 thus (((f + g) + h) . x) = (((f + g) . x) + (h . x)) by A5, A6, MESFUNC1:def 3

 .= (((f . x) + (g . x)) + (h . x)) by A7, MESFUNC1:def 3;

 end;

 end;

 theorem :: MESFUNC5:24



 Th24: for X be non empty set, f,g be PartFunc of X, ExtREAL st f is without-infty & g is without-infty holds ( dom (( max+ (f + g)) + ( max- f))) = (( dom f) /\ ( dom g)) & ( dom (( max- (f + g)) + ( max+ f))) = (( dom f) /\ ( dom g)) & ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = (( dom f) /\ ( dom g)) & ( dom ((( max- (f + g)) + ( max+ f)) + ( max+ g))) = (( dom f) /\ ( dom g)) & (( max+ (f + g)) + ( max- f)) is nonnegative & (( max- (f + g)) + ( max+ f)) is nonnegative

 proof

 let X be non empty set;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is without-infty and

 A2: g is without-infty;



 A3: ( dom (f + g)) = (( dom f) /\ ( dom g)) by A1, A2, Th16;

 then

 A4: ( dom ( max- (f + g))) = (( dom f) /\ ( dom g)) by MESFUNC2:def 3;



 A5: for x be set holds (x in ( dom ( max- f)) implies -infty < (( max- f) . x)) & (x in ( dom ( max+ f)) implies -infty < (( max+ f) . x)) & (x in ( dom ( max+ g)) implies -infty < (( max+ g) . x)) & (x in ( dom ( max- g)) implies -infty < (( max- g) . x)) by MESFUNC2: 12, MESFUNC2: 13;

 then

 A6: ( max+ f) is without-infty by Th10;



 A7: ( max- f) is without-infty by A5, Th10;



 A8: for x be set holds (x in ( dom ( max+ (f + g))) implies -infty < (( max+ (f + g)) . x)) & (x in ( dom ( max- (f + g))) implies -infty < (( max- (f + g)) . x)) by MESFUNC2: 12, MESFUNC2: 13;

 then ( max+ (f + g)) is without-infty by Th10;

 then

 A9: ( dom (( max+ (f + g)) + ( max- f))) = (( dom ( max+ (f + g))) /\ ( dom ( max- f))) by A7, Th16;

 ( max- (f + g)) is without-infty by A8, Th10;

 then

 A10: ( dom (( max- (f + g)) + ( max+ f))) = (( dom ( max- (f + g))) /\ ( dom ( max+ f))) by A6, Th16;



 A11: ( max- g) is without-infty by A5, Th10;



 A12: ( dom ( max- f)) = ( dom f) by MESFUNC2:def 3;



 A13: ( max+ g) is without-infty by A5, Th10;



 A14: ( dom ( max- g)) = ( dom g) by MESFUNC2:def 3;



 A15: ( dom ( max+ f)) = ( dom f) by MESFUNC2:def 2;

 then

 A16: ( dom (( max- (f + g)) + ( max+ f))) = (( dom g) /\ (( dom f) /\ ( dom f))) by A4, A10, XBOOLE_1: 16;

 ( dom ( max+ (f + g))) = (( dom f) /\ ( dom g)) by A3, MESFUNC2:def 2;

 then

 A17: ( dom (( max+ (f + g)) + ( max- f))) = (( dom g) /\ (( dom f) /\ ( dom f))) by A12, A9, XBOOLE_1: 16;

 hence ( dom (( max+ (f + g)) + ( max- f))) = (( dom f) /\ ( dom g)) & ( dom (( max- (f + g)) + ( max+ f))) = (( dom f) /\ ( dom g)) by A4, A15, A10, XBOOLE_1: 16;



 A18: ( dom ( max+ g)) = ( dom g) by MESFUNC2:def 2;



 A19: for x be object holds (x in ( dom (( max+ (f + g)) + ( max- f))) implies 0 <= ((( max+ (f + g)) + ( max- f)) . x)) & (x in ( dom (( max- (f + g)) + ( max+ f))) implies 0 <= ((( max- (f + g)) + ( max+ f)) . x))

 proof

 let x be object;

 hereby

 assume

 A20: x in ( dom (( max+ (f + g)) + ( max- f)));

 then

 A21: 0 <= (( max- f) . x) by MESFUNC2: 13;

 0 <= (( max+ (f + g)) . x) by A20, MESFUNC2: 12;

 then 0 <= ((( max+ (f + g)) . x) + (( max- f) . x)) by A21;

 hence 0 <= ((( max+ (f + g)) + ( max- f)) . x) by A20, MESFUNC1:def 3;

 end;

 assume

 A22: x in ( dom (( max- (f + g)) + ( max+ f)));

 then

 A23: 0 <= (( max+ f) . x) by MESFUNC2: 12;

 0 <= (( max- (f + g)) . x) by A22, MESFUNC2: 13;

 then 0 <= ((( max- (f + g)) . x) + (( max+ f) . x)) by A23;

 hence thesis by A22, MESFUNC1:def 3;

 end;

 then

 A24: for x be set holds (x in ( dom (( max+ (f + g)) + ( max- f))) implies -infty < ((( max+ (f + g)) + ( max- f)) . x)) & (x in ( dom (( max- (f + g)) + ( max+ f))) implies -infty < ((( max- (f + g)) + ( max+ f)) . x));

 then (( max+ (f + g)) + ( max- f)) is without-infty by Th10;



 then ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = ((( dom f) /\ ( dom g)) /\ ( dom g)) by A14, A11, A17, Th16

 .= (( dom f) /\ (( dom g) /\ ( dom g))) by XBOOLE_1: 16;

 hence ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = (( dom f) /\ ( dom g));

 (( max- (f + g)) + ( max+ f)) is without-infty by A24, Th10;

 then ( dom ((( max- (f + g)) + ( max+ f)) + ( max+ g))) = ((( dom f) /\ ( dom g)) /\ ( dom g)) by A18, A13, A16, Th16;

 then ( dom ((( max- (f + g)) + ( max+ f)) + ( max+ g))) = (( dom f) /\ (( dom g) /\ ( dom g))) by XBOOLE_1: 16;

 hence ( dom ((( max- (f + g)) + ( max+ f)) + ( max+ g))) = (( dom f) /\ ( dom g));

 thus thesis by A19, SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:25



 Th25: for X be non empty set, f,g be PartFunc of X, ExtREAL st f is without-infty & f is without+infty & g is without-infty & g is without+infty holds ((( max+ (f + g)) + ( max- f)) + ( max- g)) = ((( max- (f + g)) + ( max+ f)) + ( max+ g))

 proof

 let X be non empty set, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is without-infty and

 A2: f is without+infty and

 A3: g is without-infty and

 A4: g is without+infty;



 A5: ( dom ( max- (f + g))) = ( dom (f + g)) by MESFUNC2:def 3;

 for x be object st x in ( dom ( max- g)) holds 0 <= (( max- g) . x) by MESFUNC2: 13;

 then

 A6: ( max- g) is nonnegative by SUPINF_2: 52;

 for x be object st x in ( dom ( max+ g)) holds 0 <= (( max+ g) . x) by MESFUNC2: 12;

 then

 A7: ( max+ g) is nonnegative by SUPINF_2: 52;



 A8: ( dom ( max- f)) = ( dom f) by MESFUNC2:def 3;

 for x be object st x in ( dom ( max+ (f + g))) holds 0 <= (( max+ (f + g)) . x) by MESFUNC2: 12;

 then

 A9: ( max+ (f + g)) is nonnegative by SUPINF_2: 52;

 for x be object st x in ( dom ( max+ f)) holds 0 <= (( max+ f) . x) by MESFUNC2: 12;

 then

 A10: ( max+ f) is nonnegative by SUPINF_2: 52;



 A11: ( dom ( max+ f)) = ( dom f) by MESFUNC2:def 2;



 A12: ( dom ( max+ g)) = ( dom g) by MESFUNC2:def 2;



 A13: ( dom ( max- g)) = ( dom g) by MESFUNC2:def 3;

 for x be object st x in ( dom ( max- f)) holds 0 <= (( max- f) . x) by MESFUNC2: 13;

 then

 A14: ( max- f) is nonnegative by SUPINF_2: 52;



 A15: ( dom ( max+ (f + g))) = ( dom (f + g)) by MESFUNC2:def 2;

 then

 A16: ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = ((( dom (f + g)) /\ ( dom f)) /\ ( dom g)) by A8, A13, A9, A14, A6, Th23;

 then

 A17: ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = (( dom (f + g)) /\ (( dom f) /\ ( dom g))) by XBOOLE_1: 16;

 for x be object st x in ( dom ( max- (f + g))) holds 0 <= (( max- (f + g)) . x) by MESFUNC2: 13;

 then

 A18: ( max- (f + g)) is nonnegative by SUPINF_2: 52;



 A19: for x be object st x in ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) holds (((( max+ (f + g)) + ( max- f)) + ( max- g)) . x) = (((( max- (f + g)) + ( max+ f)) + ( max+ g)) . x)

 proof

 let x be object;

 assume

 A20: x in ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g)));

 then

 A21: x in ( dom g) by A16, XBOOLE_0:def 4;

 then

 A22: (( max+ g) . x) = ( max ((g . x), 0 )) by A12, MESFUNC2:def 2;



 A23: (g . x) <> +infty by A4;



 A24: ( dom (f + g)) = (( dom f) /\ ( dom g)) by A1, A3, Th16;



 then

 A25: (( max+ (f + g)) . x) = ( max (((f + g) . x), 0 )) by A15, A17, A20, MESFUNC2:def 2

 .= ( max (((f . x) + (g . x)), 0 )) by A17, A20, A24, MESFUNC1:def 3;



 A26: x in ( dom f) by A17, A20, A24, XBOOLE_0:def 4;

 then

 A27: (( max+ f) . x) = ( max ((f . x), 0 )) by A11, MESFUNC2:def 2;



 A28: (( max- (f + g)) . x) = ( max (( - ((f + g) . x)), 0 )) by A5, A17, A20, A24, MESFUNC2:def 3

 .= ( max (( - ((f . x) + (g . x))), 0 )) by A17, A20, A24, MESFUNC1:def 3;



 A29: (f . x) <> -infty by A1;

 then

 A30: ( - (f . x)) <> +infty by XXREAL_3: 23;



 A31: (f . x) <> +infty by A2;



 A32: (( max- f) . x) = ( max (( - (f . x)), 0 )) by A8, A26, MESFUNC2:def 3;



 A33: (( max- g) . x) = ( max (( - (g . x)), 0 )) by A13, A21, MESFUNC2:def 3;



 A34: (g . x) <> -infty by A3;

 then

 A35: ( - (g . x)) <> +infty by XXREAL_3: 23;

 A36:

 now

 per cases ;

 suppose

 A37: 0 <= (f . x);

 then

 A38: (( max- f) . x) = 0 by A32, XXREAL_0:def 10;

 per cases ;

 suppose

 A39: 0 <= (g . x);

 then (( max- g) . x) = 0 by A33, XXREAL_0:def 10;



 then

 A40: (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = ((((f . x) + (g . x)) + 0 ) + 0 ) by A25, A37, A38, A39, XXREAL_0:def 10

 .= (((f . x) + (g . x)) + 0 ) by XXREAL_3: 4

 .= ((f . x) + (g . x)) by XXREAL_3: 4;



 A41: (( max+ g) . x) = (g . x) by A22, A39, XXREAL_0:def 10;

 (( max- (f + g)) . x) = 0 by A28, A37, A39, XXREAL_0:def 10;

 then (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = (( 0 + (f . x)) + (g . x)) by A27, A37, A41, XXREAL_0:def 10;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A40, XXREAL_3: 4;

 end;

 suppose

 A42: (g . x) < 0 ;

 then

 A43: (( max+ g) . x) = 0 by A22, XXREAL_0:def 10;



 A44: (( max- g) . x) = ( - (g . x)) by A33, A42, XXREAL_0:def 10;

 per cases ;

 suppose

 A45: 0 <= ((f . x) + (g . x));

 then (( max- (f + g)) . x) = 0 by A28, XXREAL_0:def 10;

 then

 A46: (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = (( 0 + (f . x)) + 0 ) by A27, A37, A43, XXREAL_0:def 10;

 (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = ((((f . x) + (g . x)) + 0 ) + ( - (g . x))) by A25, A38, A44, A45, XXREAL_0:def 10

 .= (((f . x) + (g . x)) - (g . x)) by XXREAL_3: 4

 .= ((f . x) + ((g . x) - (g . x))) by A23, A34, XXREAL_3: 30

 .= ((f . x) + 0 ) by XXREAL_3: 7;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A46, XXREAL_3: 4;

 end;

 suppose

 A47: ((f . x) + (g . x)) < 0 ;

 then (( max+ (f + g)) . x) = 0 by A25, XXREAL_0:def 10;

 then (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (( 0 + 0 ) + ( - (g . x))) by A38, A44;

 then

 A48: (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = ( 0 + ( - (g . x)));

 (( max- (f + g)) . x) = ( - ((f . x) + (g . x))) by A28, A47, XXREAL_0:def 10;



 then (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = ((( - ((f . x) + (g . x))) + (f . x)) + 0 ) by A27, A37, A43, XXREAL_0:def 10

 .= (( - ((f . x) + (g . x))) + (f . x)) by XXREAL_3: 4

 .= ((( - (g . x)) - (f . x)) + (f . x)) by XXREAL_3: 25

 .= (( - (g . x)) + (( - (f . x)) + (f . x))) by A31, A30, A35, XXREAL_3: 29;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A48, XXREAL_3: 7;

 end;

 end;

 end;

 suppose

 A49: (f . x) < 0 ;

 then

 A50: (( max- f) . x) = ( - (f . x)) by A32, XXREAL_0:def 10;

 per cases ;

 suppose

 A51: 0 <= (g . x);

 then

 A52: (( max+ g) . x) = (g . x) by A22, XXREAL_0:def 10;



 A53: (( max- g) . x) = 0 by A33, A51, XXREAL_0:def 10;

 per cases ;

 suppose

 A54: 0 <= ((f . x) + (g . x));

 then

 A55: (( max- (f + g)) . x) = 0 by A28, XXREAL_0:def 10;

 (( max+ f) . x) = 0 by A27, A49, XXREAL_0:def 10;

 then

 A56: (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = (( 0 + 0 ) + (g . x)) by A52, A55;

 (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = ((((f . x) + (g . x)) + ( - (f . x))) + 0 ) by A25, A50, A53, A54, XXREAL_0:def 10

 .= (((f . x) + (g . x)) + ( - (f . x))) by XXREAL_3: 4

 .= ((g . x) + ((f . x) - (f . x))) by A31, A29, A23, A34, XXREAL_3: 29

 .= ((g . x) + 0 ) by XXREAL_3: 7;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A56;

 end;

 suppose

 A57: ((f . x) + (g . x)) < 0 ;

 then (( max- (f + g)) . x) = ( - ((f . x) + (g . x))) by A28, XXREAL_0:def 10;



 then

 A58: (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = ((( - ((f . x) + (g . x))) + 0 ) + (g . x)) by A27, A49, A52, XXREAL_0:def 10

 .= (( - ((f . x) + (g . x))) + (g . x)) by XXREAL_3: 4

 .= ((( - (f . x)) - (g . x)) + (g . x)) by XXREAL_3: 25

 .= (( - (f . x)) + (( - (g . x)) + (g . x))) by A23, A30, A35, XXREAL_3: 29;

 (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (( 0 + ( - (f . x))) + 0 ) by A25, A50, A53, A57, XXREAL_0:def 10

 .= ( 0 + ( - (f . x))) by XXREAL_3: 4;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A58, XXREAL_3: 7;

 end;

 end;

 suppose

 A59: (g . x) < 0 ;

 then (( max- g) . x) = ( - (g . x)) by A33, XXREAL_0:def 10;



 then

 A60: (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (( 0 + ( - (f . x))) + ( - (g . x))) by A25, A49, A50, A59, XXREAL_0:def 10

 .= (( - (f . x)) - (g . x)) by XXREAL_3: 4;



 A61: (( max+ g) . x) = 0 by A22, A59, XXREAL_0:def 10;

 (( max- (f + g)) . x) = ( - ((f . x) + (g . x))) by A28, A49, A59, XXREAL_0:def 10;



 then (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) = ((( - ((f . x) + (g . x))) + 0 ) + 0 ) by A27, A49, A61, XXREAL_0:def 10

 .= (( - ((f . x) + (g . x))) + 0 ) by XXREAL_3: 4

 .= ( - ((f . x) + (g . x))) by XXREAL_3: 4;

 hence (((( max+ (f + g)) . x) + (( max- f) . x)) + (( max- g) . x)) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A60, XXREAL_3: 25;

 end;

 end;

 end;



 A62: ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = ((( dom ( max+ (f + g))) /\ ( dom ( max- f))) /\ ( dom ( max- g))) by A9, A14, A6, Th23;

 (((( max- (f + g)) + ( max+ f)) + ( max+ g)) . x) = (((( max- (f + g)) . x) + (( max+ f) . x)) + (( max+ g) . x)) by A5, A11, A12, A18, A10, A7, A16, A20, Th23;

 hence thesis by A9, A14, A6, A20, A62, A36, Th23;

 end;

 ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = (( dom f) /\ ( dom g)) by A1, A3, Th24;

 then ( dom ((( max+ (f + g)) + ( max- f)) + ( max- g))) = ( dom ((( max- (f + g)) + ( max+ f)) + ( max+ g))) by A1, A3, Th24;

 hence thesis by A19, FUNCT_1: 2;

 end;

 theorem :: MESFUNC5:26



 Th26: for C be non empty set, f be PartFunc of C, ExtREAL , c be Real st 0 <= c holds ( max+ (c (#) f)) = (c (#) ( max+ f)) & ( max- (c (#) f)) = (c (#) ( max- f))

 proof

 let C be non empty set;

 let f be PartFunc of C, ExtREAL ;

 let c be Real;

 assume

 A1: 0 <= c;



 A2: ( dom ( max+ (c (#) f))) = ( dom (c (#) f)) by MESFUNC2:def 2

 .= ( dom f) by MESFUNC1:def 6

 .= ( dom ( max+ f)) by MESFUNC2:def 2

 .= ( dom (c (#) ( max+ f))) by MESFUNC1:def 6;

 for x be Element of C st x in ( dom ( max+ (c (#) f))) holds (( max+ (c (#) f)) . x) = ((c (#) ( max+ f)) . x)

 proof

 let x be Element of C;

 assume

 A3: x in ( dom ( max+ (c (#) f)));

 then

 A4: x in ( dom (c (#) f)) by MESFUNC2:def 2;

 then x in ( dom f) by MESFUNC1:def 6;

 then

 A5: x in ( dom ( max+ f)) by MESFUNC2:def 2;



 A6: (( max+ (c (#) f)) . x) = ( max (((c (#) f) . x), 0 )) by A3, MESFUNC2:def 2

 .= ( max ((c * (f . x)), 0 )) by A4, MESFUNC1:def 6;

 ((c (#) ( max+ f)) . x) = (c * (( max+ f) . x)) by A2, A3, MESFUNC1:def 6

 .= (c * ( max ((f . x), 0 ))) by A5, MESFUNC2:def 2

 .= ( max ((c * (f . x)),(c * 0 qua ExtReal))) by A1, Th6;

 hence thesis by A6;

 end;

 hence ( max+ (c (#) f)) = (c (#) ( max+ f)) by A2, PARTFUN1: 5;



 A7: ( dom ( max- (c (#) f))) = ( dom (c (#) f)) by MESFUNC2:def 3

 .= ( dom f) by MESFUNC1:def 6

 .= ( dom ( max- f)) by MESFUNC2:def 3

 .= ( dom (c (#) ( max- f))) by MESFUNC1:def 6;

 for x be Element of C st x in ( dom ( max- (c (#) f))) holds (( max- (c (#) f)) . x) = ((c (#) ( max- f)) . x)

 proof

 let x be Element of C;

 assume

 A8: x in ( dom ( max- (c (#) f)));

 then

 A9: x in ( dom (c (#) f)) by MESFUNC2:def 3;

 then x in ( dom f) by MESFUNC1:def 6;

 then

 A10: x in ( dom ( max- f)) by MESFUNC2:def 3;



 A11: (( max- (c (#) f)) . x) = ( max (( - ((c (#) f) . x)), 0 )) by A8, MESFUNC2:def 3

 .= ( max (( - (c * (f . x))), 0 )) by A9, MESFUNC1:def 6;

 ((c (#) ( max- f)) . x) = (c * (( max- f) . x)) by A7, A8, MESFUNC1:def 6

 .= (c * ( max (( - (f . x)), 0 ))) by A10, MESFUNC2:def 3

 .= ( max ((c * ( - (f . x))),(c * 0 qua ExtReal))) by A1, Th6

 .= ( max (( - (c * (f . x))),(c * 0 qua ExtReal))) by XXREAL_3: 92;

 hence thesis by A11;

 end;

 hence thesis by A7, PARTFUN1: 5;

 end;

 theorem :: MESFUNC5:27



 Th27: for C be non empty set, f be PartFunc of C, ExtREAL , c be Real st 0 <= c holds ( max+ (( - c) (#) f)) = (c (#) ( max- f)) & ( max- (( - c) (#) f)) = (c (#) ( max+ f))

 proof

 let C be non empty set;

 let f be PartFunc of C, ExtREAL ;

 let c be Real;

 assume

 A1: 0 <= c;



 A2: ( dom ( max+ (( - c) (#) f))) = ( dom (( - c) (#) f)) by MESFUNC2:def 2;

 then ( dom ( max+ (( - c) (#) f))) = ( dom f) by MESFUNC1:def 6;

 then

 A3: ( dom ( max+ (( - c) (#) f))) = ( dom ( max- f)) by MESFUNC2:def 3;

 then

 A4: ( dom ( max+ (( - c) (#) f))) = ( dom (c (#) ( max- f))) by MESFUNC1:def 6;

 for x be Element of C st x in ( dom ( max+ (( - c) (#) f))) holds (( max+ (( - c) (#) f)) . x) = ((c (#) ( max- f)) . x)

 proof

 let x be Element of C;

 assume

 A5: x in ( dom ( max+ (( - c) (#) f)));



 then

 A6: (( max+ (( - c) (#) f)) . x) = ( max (((( - c) (#) f) . x), 0 )) by MESFUNC2:def 2

 .= ( max ((( - c) * (f . x)), 0 )) by A2, A5, MESFUNC1:def 6

 .= ( max (( - (c * (f . x))), 0 )) by XXREAL_3: 92;

 ((c (#) ( max- f)) . x) = (c * (( max- f) . x)) by A4, A5, MESFUNC1:def 6

 .= (c * ( max (( - (f . x)), 0 ))) by A3, A5, MESFUNC2:def 3

 .= ( max ((c * ( - (f . x))),(c * 0 ))) by A1, Th6

 .= ( max (( - (c * (f . x))),(c * 0 qua ExtReal))) by XXREAL_3: 92;

 hence thesis by A6;

 end;

 hence ( max+ (( - c) (#) f)) = (c (#) ( max- f)) by A4, PARTFUN1: 5;



 A7: ( dom ( max- (( - c) (#) f))) = ( dom (( - c) (#) f)) by MESFUNC2:def 3;

 then ( dom ( max- (( - c) (#) f))) = ( dom f) by MESFUNC1:def 6;

 then

 A8: ( dom ( max- (( - c) (#) f))) = ( dom ( max+ f)) by MESFUNC2:def 2;

 then

 A9: ( dom ( max- (( - c) (#) f))) = ( dom (c (#) ( max+ f))) by MESFUNC1:def 6;

 for x be Element of C st x in ( dom ( max- (( - c) (#) f))) holds (( max- (( - c) (#) f)) . x) = ((c (#) ( max+ f)) . x)

 proof

 let x be Element of C;

 assume

 A10: x in ( dom ( max- (( - c) (#) f)));



 then

 A11: (( max- (( - c) (#) f)) . x) = ( max (( - ((( - c) (#) f) . x)), 0 )) by MESFUNC2:def 3

 .= ( max (( - (( - c) * (f . x))), 0 )) by A7, A10, MESFUNC1:def 6

 .= ( max ((( - ( - c)) * (f . x)), 0 )) by XXREAL_3: 92;

 ((c (#) ( max+ f)) . x) = (c * (( max+ f) . x)) by A9, A10, MESFUNC1:def 6

 .= (c * ( max ((f . x), 0 ))) by A8, A10, MESFUNC2:def 2

 .= ( max ((c * (f . x)),(c * 0 qua ExtReal))) by A1, Th6;

 hence thesis by A11;

 end;

 hence thesis by A9, PARTFUN1: 5;

 end;

 theorem :: MESFUNC5:28



 Th28: for X be non empty set, f be PartFunc of X, ExtREAL , A be set holds ( max+ (f | A)) = (( max+ f) | A) & ( max- (f | A)) = (( max- f) | A)

 proof

 let X be non empty set;

 let f be PartFunc of X, ExtREAL ;

 let A be set;



 A1: ( dom ( max+ (f | A))) = ( dom (f | A)) by MESFUNC2:def 2

 .= (( dom f) /\ A) by RELAT_1: 61

 .= (( dom ( max+ f)) /\ A) by MESFUNC2:def 2

 .= ( dom (( max+ f) | A)) by RELAT_1: 61;

 for x be Element of X st x in ( dom ( max+ (f | A))) holds (( max+ (f | A)) . x) = ((( max+ f) | A) . x)

 proof

 let x be Element of X;

 assume

 A2: x in ( dom ( max+ (f | A)));

 then

 A3: ((( max+ f) | A) . x) = (( max+ f) . x) by A1, FUNCT_1: 47;



 A4: x in (( dom ( max+ f)) /\ A) by A1, A2, RELAT_1: 61;

 then

 A5: x in ( dom ( max+ f)) by XBOOLE_0:def 4;



 A6: x in A by A4, XBOOLE_0:def 4;

 (( max+ (f | A)) . x) = ( max (((f | A) . x), 0 )) by A2, MESFUNC2:def 2

 .= ( max ((f . x), 0 )) by A6, FUNCT_1: 49;

 hence thesis by A5, A3, MESFUNC2:def 2;

 end;

 hence ( max+ (f | A)) = (( max+ f) | A) by A1, PARTFUN1: 5;



 A7: ( dom ( max- (f | A))) = ( dom (f | A)) by MESFUNC2:def 3

 .= (( dom f) /\ A) by RELAT_1: 61

 .= (( dom ( max- f)) /\ A) by MESFUNC2:def 3

 .= ( dom (( max- f) | A)) by RELAT_1: 61;

 for x be Element of X st x in ( dom ( max- (f | A))) holds (( max- (f | A)) . x) = ((( max- f) | A) . x)

 proof

 let x be Element of X;

 assume

 A8: x in ( dom ( max- (f | A)));

 then

 A9: ((( max- f) | A) . x) = (( max- f) . x) by A7, FUNCT_1: 47;



 A10: x in (( dom ( max- f)) /\ A) by A7, A8, RELAT_1: 61;

 then

 A11: x in ( dom ( max- f)) by XBOOLE_0:def 4;



 A12: x in A by A10, XBOOLE_0:def 4;

 (( max- (f | A)) . x) = ( max (( - ((f | A) . x)), 0 )) by A8, MESFUNC2:def 3

 .= ( max (( - (f . x)), 0 )) by A12, FUNCT_1: 49;

 hence thesis by A11, A9, MESFUNC2:def 3;

 end;

 hence thesis by A7, PARTFUN1: 5;

 end;

 theorem :: MESFUNC5:29



 Th29: for X be non empty set, f,g be PartFunc of X, ExtREAL , B be set st B c= ( dom (f + g)) holds ( dom ((f + g) | B)) = B & ( dom ((f | B) + (g | B))) = B & ((f + g) | B) = ((f | B) + (g | B))

 proof

 let X be non empty set, f,g be PartFunc of X, ExtREAL , B be set such that

 A1: B c= ( dom (f + g));

 for x be object st x in ( dom g) holds (g . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider gg = g as Function of ( dom g), ExtREAL by FUNCT_2: 3;

 for x be object st x in ( dom (g | B)) holds ((g | B) . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider gb = (g | B) as Function of ( dom (g | B)), ExtREAL by FUNCT_2: 3;

 now

 let x be object;

 assume

 A2: x in ((g " { +infty }) /\ B);

 then

 A3: x in B by XBOOLE_0:def 4;



 A4: x in (g " { +infty }) by A2, XBOOLE_0:def 4;

 then x in ( dom gg) by FUNCT_2: 38;

 then x in (( dom gg) /\ B) by A3, XBOOLE_0:def 4;

 then

 A5: x in ( dom (gg | B)) by RELAT_1: 61;

 (gg . x) in { +infty } by A4, FUNCT_2: 38;

 then (gb . x) in { +infty } by A5, FUNCT_1: 47;

 hence x in ((g | B) " { +infty }) by A5, FUNCT_2: 38;

 end;

 then

 A6: ((g " { +infty }) /\ B) c= ((g | B) " { +infty });

 now

 let x be object;

 assume

 A7: x in ((g | B) " { +infty });

 then

 A8: x in ( dom gb) by FUNCT_2: 38;

 then

 A9: x in (( dom g) /\ B) by RELAT_1: 61;

 then

 A10: x in ( dom g) by XBOOLE_0:def 4;

 (gb . x) in { +infty } by A7, FUNCT_2: 38;

 then (g . x) in { +infty } by A8, FUNCT_1: 47;

 then

 A11: x in (gg " { +infty }) by A10, FUNCT_2: 38;

 x in B by A9, XBOOLE_0:def 4;

 hence x in ((g " { +infty }) /\ B) by A11, XBOOLE_0:def 4;

 end;

 then ((g | B) " { +infty }) c= ((g " { +infty }) /\ B);

 then

 A12: ((g | B) " { +infty }) = ((g " { +infty }) /\ B) by A6;

 now

 let x be object;

 assume

 A13: x in ((g " { -infty }) /\ B);

 then

 A14: x in B by XBOOLE_0:def 4;



 A15: x in (g " { -infty }) by A13, XBOOLE_0:def 4;

 then x in ( dom gg) by FUNCT_2: 38;

 then x in (( dom gg) /\ B) by A14, XBOOLE_0:def 4;

 then

 A16: x in ( dom (gg | B)) by RELAT_1: 61;

 (gg . x) in { -infty } by A15, FUNCT_2: 38;

 then (gb . x) in { -infty } by A16, FUNCT_1: 47;

 hence x in ((g | B) " { -infty }) by A16, FUNCT_2: 38;

 end;

 then

 A17: ((g " { -infty }) /\ B) c= ((g | B) " { -infty });

 now

 let x be object;

 assume

 A18: x in ((g | B) " { -infty });

 then

 A19: x in ( dom gb) by FUNCT_2: 38;

 then

 A20: x in (( dom g) /\ B) by RELAT_1: 61;

 then

 A21: x in ( dom g) by XBOOLE_0:def 4;

 (gb . x) in { -infty } by A18, FUNCT_2: 38;

 then (g . x) in { -infty } by A19, FUNCT_1: 47;

 then

 A22: x in (gg " { -infty }) by A21, FUNCT_2: 38;

 x in B by A20, XBOOLE_0:def 4;

 hence x in ((g " { -infty }) /\ B) by A22, XBOOLE_0:def 4;

 end;

 then ((g | B) " { -infty }) c= ((g " { -infty }) /\ B);

 then

 A23: ((g | B) " { -infty }) = ((g " { -infty }) /\ B) by A17;

 for x be object st x in ( dom f) holds (f . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider ff = f as Function of ( dom f), ExtREAL by FUNCT_2: 3;

 for x be object st x in ( dom (f | B)) holds ((f | B) . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider fb = (f | B) as Function of ( dom (f | B)), ExtREAL by FUNCT_2: 3;

 now

 let x be object;

 assume

 A24: x in ((f " { +infty }) /\ B);

 then

 A25: x in B by XBOOLE_0:def 4;



 A26: x in (f " { +infty }) by A24, XBOOLE_0:def 4;

 then x in ( dom ff) by FUNCT_2: 38;

 then x in (( dom ff) /\ B) by A25, XBOOLE_0:def 4;

 then

 A27: x in ( dom (ff | B)) by RELAT_1: 61;

 (ff . x) in { +infty } by A26, FUNCT_2: 38;

 then (fb . x) in { +infty } by A27, FUNCT_1: 47;

 hence x in ((f | B) " { +infty }) by A27, FUNCT_2: 38;

 end;

 then

 A28: ((f " { +infty }) /\ B) c= ((f | B) " { +infty });

 now

 let x be object;

 assume

 A29: x in ((f " { -infty }) /\ B);

 then

 A30: x in B by XBOOLE_0:def 4;



 A31: x in (f " { -infty }) by A29, XBOOLE_0:def 4;

 then x in ( dom ff) by FUNCT_2: 38;

 then x in (( dom ff) /\ B) by A30, XBOOLE_0:def 4;

 then

 A32: x in ( dom (ff | B)) by RELAT_1: 61;

 (ff . x) in { -infty } by A31, FUNCT_2: 38;

 then (fb . x) in { -infty } by A32, FUNCT_1: 47;

 hence x in ((f | B) " { -infty }) by A32, FUNCT_2: 38;

 end;

 then

 A33: ((f " { -infty }) /\ B) c= ((f | B) " { -infty });

 now

 let x be object;

 assume

 A34: x in ((f | B) " { -infty });

 then

 A35: x in ( dom fb) by FUNCT_2: 38;

 then

 A36: x in (( dom f) /\ B) by RELAT_1: 61;

 then

 A37: x in ( dom f) by XBOOLE_0:def 4;

 (fb . x) in { -infty } by A34, FUNCT_2: 38;

 then (f . x) in { -infty } by A35, FUNCT_1: 47;

 then

 A38: x in (ff " { -infty }) by A37, FUNCT_2: 38;

 x in B by A36, XBOOLE_0:def 4;

 hence x in ((f " { -infty }) /\ B) by A38, XBOOLE_0:def 4;

 end;

 then ((f | B) " { -infty }) c= ((f " { -infty }) /\ B);

 then ((f | B) " { -infty }) = ((f " { -infty }) /\ B) by A33;



 then

 A39: (((f | B) " { -infty }) /\ ((g | B) " { +infty })) = ((((f " { -infty }) /\ B) /\ (g " { +infty })) /\ B) by A12, XBOOLE_1: 16

 .= ((((f " { -infty }) /\ (g " { +infty })) /\ B) /\ B) by XBOOLE_1: 16

 .= (((f " { -infty }) /\ (g " { +infty })) /\ (B /\ B)) by XBOOLE_1: 16;

 now

 let x be object;

 assume

 A40: x in ((f | B) " { +infty });

 then

 A41: x in ( dom fb) by FUNCT_2: 38;

 then

 A42: x in (( dom f) /\ B) by RELAT_1: 61;

 then

 A43: x in ( dom f) by XBOOLE_0:def 4;

 (fb . x) in { +infty } by A40, FUNCT_2: 38;

 then (f . x) in { +infty } by A41, FUNCT_1: 47;

 then

 A44: x in (ff " { +infty }) by A43, FUNCT_2: 38;

 x in B by A42, XBOOLE_0:def 4;

 hence x in ((f " { +infty }) /\ B) by A44, XBOOLE_0:def 4;

 end;

 then ((f | B) " { +infty }) c= ((f " { +infty }) /\ B);

 then ((f | B) " { +infty }) = ((f " { +infty }) /\ B) by A28;



 then (((f | B) " { +infty }) /\ ((g | B) " { -infty })) = ((((f " { +infty }) /\ B) /\ (g " { -infty })) /\ B) by A23, XBOOLE_1: 16

 .= ((((f " { +infty }) /\ (g " { -infty })) /\ B) /\ B) by XBOOLE_1: 16

 .= (((f " { +infty }) /\ (g " { -infty })) /\ (B /\ B)) by XBOOLE_1: 16;

 then

 A45: ((((f | B) " { -infty }) /\ ((g | B) " { +infty })) \/ (((f | B) " { +infty }) /\ ((g | B) " { -infty }))) = ((((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty }))) /\ B) by A39, XBOOLE_1: 23;

 (( dom (f | B)) /\ ( dom (g | B))) = ((( dom f) /\ B) /\ ( dom (g | B))) by RELAT_1: 61

 .= ((( dom f) /\ B) /\ (( dom g) /\ B)) by RELAT_1: 61

 .= (((( dom f) /\ B) /\ ( dom g)) /\ B) by XBOOLE_1: 16

 .= (((( dom f) /\ ( dom g)) /\ B) /\ B) by XBOOLE_1: 16

 .= ((( dom f) /\ ( dom g)) /\ (B /\ B)) by XBOOLE_1: 16;



 then

 A46: ( dom ((f | B) + (g | B))) = (((( dom f) /\ ( dom g)) /\ B) \ ((((f | B) " { -infty }) /\ ((g | B) " { +infty })) \/ (((f | B) " { +infty }) /\ ((g | B) " { -infty })))) by MESFUNC1:def 3

 .= (((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) /\ B) by A45, XBOOLE_1: 50

 .= (( dom (f + g)) /\ B) by MESFUNC1:def 3

 .= B by A1, XBOOLE_1: 28;

 ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then ( dom (f + g)) c= (( dom f) /\ ( dom g)) by XBOOLE_1: 36;

 then

 A47: B c= (( dom f) /\ ( dom g)) by A1;

 ( dom (g | B)) = (( dom g) /\ B) by RELAT_1: 61;

 then

 A48: ( dom (g | B)) = B by A47, XBOOLE_1: 18, XBOOLE_1: 28;



 A49: ( dom ((f + g) | B)) = (( dom (f + g)) /\ B) by RELAT_1: 61;

 then

 A50: ( dom ((f + g) | B)) = B by A1, XBOOLE_1: 28;

 ( dom (f | B)) = (( dom f) /\ B) by RELAT_1: 61;

 then

 A51: ( dom (f | B)) = B by A47, XBOOLE_1: 18, XBOOLE_1: 28;

 now

 let x be object;

 assume

 A52: x in ( dom ((f + g) | B));



 hence (((f + g) | B) . x) = ((f + g) . x) by FUNCT_1: 47

 .= ((f . x) + (g . x)) by A1, A50, A52, MESFUNC1:def 3

 .= (((f | B) . x) + (g . x)) by A50, A51, A52, FUNCT_1: 47

 .= (((f | B) . x) + ((g | B) . x)) by A50, A48, A52, FUNCT_1: 47

 .= (((f | B) + (g | B)) . x) by A50, A46, A52, MESFUNC1:def 3;

 end;

 hence thesis by A1, A49, A46, FUNCT_1: 2, XBOOLE_1: 28;

 end;

 theorem :: MESFUNC5:30



 Th30: for X be non empty set, f be PartFunc of X, ExtREAL , a be R_eal holds ( eq_dom (f,a)) = (f " {a})

 proof

 let X be non empty set;

 let f be PartFunc of X, ExtREAL ;

 let a be R_eal;

 now

 let x be object;

 assume

 A1: x in (f " {a});

 then (f . x) in {a} by FUNCT_1:def 7;

 then

 A2: (f . x) = a by TARSKI:def 1;

 x in ( dom f) by A1, FUNCT_1:def 7;

 hence x in ( eq_dom (f,a)) by A2, MESFUNC1:def 15;

 end;

 then

 A3: (f " {a}) c= ( eq_dom (f,a));

 now

 let x be object;

 assume

 A4: x in ( eq_dom (f,a));

 then (f . x) = a by MESFUNC1:def 15;

 then

 A5: (f . x) in {a} by TARSKI:def 1;

 x in ( dom f) by A4, MESFUNC1:def 15;

 hence x in (f " {a}) by A5, FUNCT_1:def 7;

 end;

 then ( eq_dom (f,a)) c= (f " {a});

 hence thesis by A3;

 end;

 begin

 theorem :: MESFUNC5:31



 Th31: for X be non empty set, S be SigmaField of X, f,g be PartFunc of X, ExtREAL , A be Element of S st f is without-infty & g is without-infty & f is A -measurable & g is A -measurable holds (f + g) is A -measurable

 proof

 let X be non empty set, S be SigmaField of X, f,g be PartFunc of X, ExtREAL , A be Element of S;

 assume that

 A1: f is without-infty and

 A2: g is without-infty and

 A3: f is A -measurable and

 A4: g is A -measurable;

 for r be Real holds (A /\ ( less_dom ((f + g),r))) in S

 proof

 let r be Real;

 consider F be Function of RAT , S such that

 A5: for p be Rational holds (F . p) = ((A /\ ( less_dom (f,p))) /\ (A /\ ( less_dom (g,(r - p qua Complex))))) by A3, A4, MESFUNC2: 6;

 ex G be sequence of S st ( rng F) = ( rng G) by MESFUNC1: 5, MESFUNC2: 5;

 then

 A6: ( rng F) is N_Sub_set_fam of X by MEASURE1: 23;

 (A /\ ( less_dom ((f + g),r))) = ( union ( rng F)) by A1, A2, A5, Th18;

 hence thesis by A6, MEASURE1:def 5;

 end;

 hence thesis by MESFUNC1:def 16;

 end;

 theorem :: MESFUNC5:32

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & ( dom f) = {} holds ex F be Finite_Sep_Sequence of S, a,x be FinSequence of ExtREAL st (F,a) are_Re-presentation_of f & (a . 1) = 0 & (for n be Nat st 2 <= n & n in ( dom a) holds 0 < (a . n) & (a . n) < +infty ) & ( dom x) = ( dom F) & (for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n))) & ( Sum x) = 0

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: ( dom f) = {} ;

 for x be object st x in ( dom f) holds 0 <= (f . x) by A2;

 then f is nonnegative by SUPINF_2: 52;

 then

 consider F be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL such that

 A3: (F,a) are_Re-presentation_of f and

 A4: (a . 1) = 0 and

 A5: for n be Nat st 2 <= n & n in ( dom a) holds 0 < (a . n) & (a . n) < +infty by A1, MESFUNC3: 14;

 deffunc F( Nat) = ((a . $1) * ((M * F) . $1));

 consider x be FinSequence of ExtREAL such that

 A6: ( len x) = ( len F) and

 A7: for n be Nat st n in ( dom x) holds (x . n) = F(n) from FINSEQ_2:sch 1;



 A8: ( dom x) = ( Seg ( len F)) by A6, FINSEQ_1:def 3;

 then

 A9: ( dom x) = ( dom F) by FINSEQ_1:def 3;

 take F, a, x;

 consider sumx be sequence of ExtREAL such that

 A10: ( Sum x) = (sumx . ( len x)) and

 A11: (sumx . 0 ) = 0 and

 A12: for i be Nat st i < ( len x) holds (sumx . (i + 1)) = ((sumx . i) + (x . (i + 1))) by EXTREAL1:def 2;

 defpred P[ Nat] means $1 <= ( len x) implies (sumx . $1) = 0 ;



 A13: ( union ( rng F)) = {} by A2, A3, MESFUNC3:def 1;



 A14: for n be Nat st n in ( dom F) holds (F . n) = {}

 proof

 let n be Nat;

 assume n in ( dom F);

 then

 A15: (F . n) in ( rng F) by FUNCT_1: 3;

 assume (F . n) <> {} ;

 then ex v be object st v in (F . n) by XBOOLE_0:def 1;

 hence contradiction by A13, A15, TARSKI:def 4;

 end;



 A16: for i be Nat st P[i] holds P[(i + 1)]

 proof

 let i be Nat;

 assume

 A17: P[i];

 assume

 A18: (i + 1) <= ( len x);

 reconsider i as Element of NAT by ORDINAL1:def 12;

 i < ( len x) by A18, NAT_1: 13;

 then

 A19: (sumx . (i + 1)) = ((sumx . i) + (x . (i + 1))) by A12;

 1 <= (i + 1) by NAT_1: 11;

 then

 A20: (i + 1) in ( dom x) by A18, FINSEQ_3: 25;

 then (F . (i + 1)) = {} by A9, A14;

 then (M . (F . (i + 1))) = 0 by VALUED_0:def 19;

 then

 A21: ((M * F) . (i + 1)) = 0 by A9, A20, FUNCT_1: 13;

 (x . (i + 1)) = ((a . (i + 1)) * ((M * F) . (i + 1))) by A7, A20

 .= 0 by A21;

 hence thesis by A17, A18, A19, NAT_1: 13;

 end;



 A22: P[ 0 ] by A11;

 for i be Nat holds P[i] from NAT_1:sch 2( A22, A16);

 hence thesis by A3, A4, A5, A7, A8, A10, FINSEQ_1:def 3;

 end;

 theorem :: MESFUNC5:33



 Th33: for X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL , A be Element of S, r,s be Real st f is A -measurable & A c= ( dom f) holds ((A /\ ( great_eq_dom (f,r))) /\ ( less_dom (f,s))) in S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let f be PartFunc of X, ExtREAL ;

 let A be Element of S;

 let r,s be Real;

 assume that

 A1: f is A -measurable and

 A2: A c= ( dom f);



 A3: (A /\ ( less_dom (f,s))) in S by A1, MESFUNC1:def 16;



 A4: ((A /\ ( great_eq_dom (f,r))) /\ (A /\ ( less_dom (f,s)))) = (((A /\ ( great_eq_dom (f,r))) /\ A) /\ ( less_dom (f,s))) by XBOOLE_1: 16

 .= ((( great_eq_dom (f,r)) /\ (A /\ A)) /\ ( less_dom (f,s))) by XBOOLE_1: 16;

 (A /\ ( great_eq_dom (f,r))) in S by A1, A2, MESFUNC1: 27;

 hence thesis by A3, A4, FINSUB_1:def 2;

 end;

 theorem :: MESFUNC5:34



 Th34: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st f is_simple_func_in S holds (f | A) is_simple_func_in S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let A be Element of S;

 assume

 A1: f is_simple_func_in S;

 then

 consider F be Finite_Sep_Sequence of S such that

 A2: ( dom f) = ( union ( rng F)) and

 A3: for n be Nat, x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y) by MESFUNC2:def 4;

 deffunc FA( Nat) = ((F . $1) /\ A);

 consider G be FinSequence such that

 A4: ( len G) = ( len F) & for n be Nat st n in ( dom G) holds (G . n) = FA(n) from FINSEQ_1:sch 2;



 A5: ( rng G) c= S

 proof

 let P be object;

 assume P in ( rng G);

 then

 consider k be Nat such that

 A6: k in ( dom G) and

 A7: P = (G . k) by FINSEQ_2: 10;

 k in ( Seg ( len F)) by A4, A6, FINSEQ_1:def 3;

 then k in ( dom F) by FINSEQ_1:def 3;

 then

 A8: (F . k) in ( rng F) by FUNCT_1: 3;

 (G . k) = ((F . k) /\ A) by A4, A6;

 hence thesis by A7, A8, FINSUB_1:def 2;

 end;



 A9: ( dom G) = ( Seg ( len F)) by A4, FINSEQ_1:def 3;

 reconsider G as FinSequence of S by A5, FINSEQ_1:def 4;

 for i,j be Nat st i in ( dom G) & j in ( dom G) & i <> j holds (G . i) misses (G . j)

 proof

 let i,j be Nat;

 assume that

 A10: i in ( dom G) and

 A11: j in ( dom G) and

 A12: i <> j;

 j in ( Seg ( len F)) by A4, A11, FINSEQ_1:def 3;

 then

 A13: j in ( dom F) by FINSEQ_1:def 3;

 i in ( Seg ( len F)) by A4, A10, FINSEQ_1:def 3;

 then i in ( dom F) by FINSEQ_1:def 3;

 then

 A14: (F . i) misses (F . j) by A12, A13, MESFUNC3: 4;



 A15: (G . j) = ((F . j) /\ A) by A4, A11;

 (G . i) = ((F . i) /\ A) by A4, A10;



 then ((G . i) /\ (G . j)) = ((((F . i) /\ A) /\ (F . j)) /\ A) by A15, XBOOLE_1: 16

 .= ((((F . i) /\ (F . j)) /\ A) /\ A) by XBOOLE_1: 16

 .= (( {} /\ A) /\ A) by A14;

 hence thesis;

 end;

 then

 reconsider G as Finite_Sep_Sequence of S by MESFUNC3: 4;

 for v be object st v in ( union ( rng G)) holds v in ( dom (f | A))

 proof

 let v be object;

 assume v in ( union ( rng G));

 then

 consider W be set such that

 A16: v in W and

 A17: W in ( rng G) by TARSKI:def 4;

 consider k be Nat such that

 A18: k in ( dom G) and

 A19: W = (G . k) by A17, FINSEQ_2: 10;

 k in ( Seg ( len F)) by A4, A18, FINSEQ_1:def 3;

 then k in ( dom F) by FINSEQ_1:def 3;

 then

 A20: (F . k) in ( rng F) by FUNCT_1: 3;



 A21: (G . k) = ((F . k) /\ A) by A4, A18;

 then v in (F . k) by A16, A19, XBOOLE_0:def 4;

 then

 A22: v in ( union ( rng F)) by A20, TARSKI:def 4;

 v in A by A16, A19, A21, XBOOLE_0:def 4;

 then v in (( dom f) /\ A) by A2, A22, XBOOLE_0:def 4;

 hence thesis by RELAT_1: 61;

 end;

 then

 A23: ( union ( rng G)) c= ( dom (f | A));

 for v be object st v in ( dom (f | A)) holds v in ( union ( rng G))

 proof

 let v be object;

 assume v in ( dom (f | A));

 then

 A24: v in (( dom f) /\ A) by RELAT_1: 61;

 then

 A25: v in A by XBOOLE_0:def 4;

 v in ( dom f) by A24, XBOOLE_0:def 4;

 then

 consider W be set such that

 A26: v in W and

 A27: W in ( rng F) by A2, TARSKI:def 4;

 consider k be Nat such that

 A28: k in ( dom F) and

 A29: W = (F . k) by A27, FINSEQ_2: 10;



 A30: k in ( Seg ( len F)) by A28, FINSEQ_1:def 3;

 then k in ( dom G) by A4, FINSEQ_1:def 3;

 then

 A31: (G . k) in ( rng G) by FUNCT_1: 3;

 (G . k) = ((F . k) /\ A) by A4, A9, A30;

 then v in (G . k) by A25, A26, A29, XBOOLE_0:def 4;

 hence thesis by A31, TARSKI:def 4;

 end;

 then ( dom (f | A)) c= ( union ( rng G));

 then

 A32: ( dom (f | A)) = ( union ( rng G)) by A23;



 A33: for n be Nat, x,y be Element of X st n in ( dom G) & x in (G . n) & y in (G . n) holds ((f | A) . x) = ((f | A) . y)

 proof

 let n be Nat;

 let x,y be Element of X;

 assume that

 A34: n in ( dom G) and

 A35: x in (G . n) and

 A36: y in (G . n);



 A37: (G . n) in ( rng G) by A34, FUNCT_1: 3;

 then

 A38: x in ( dom (f | A)) by A32, A35, TARSKI:def 4;



 A39: (G . n) = ((F . n) /\ A) by A4, A34;

 then

 A40: y in (F . n) by A36, XBOOLE_0:def 4;

 n in ( Seg ( len F)) by A4, A34, FINSEQ_1:def 3;

 then

 A41: n in ( dom F) by FINSEQ_1:def 3;

 x in (F . n) by A35, A39, XBOOLE_0:def 4;

 then (f . x) = (f . y) by A3, A40, A41;

 then

 A42: ((f | A) . x) = (f . y) by A38, FUNCT_1: 47;

 y in ( dom (f | A)) by A32, A36, A37, TARSKI:def 4;

 hence thesis by A42, FUNCT_1: 47;

 end;

 f is real-valued by A1, MESFUNC2:def 4;

 hence thesis by A32, A33, MESFUNC2:def 4;

 end;

 theorem :: MESFUNC5:35



 Th35: for X be non empty set, S be SigmaField of X, A be Element of S, F be Finite_Sep_Sequence of S, G be FinSequence st ( dom F) = ( dom G) & (for n be Nat st n in ( dom F) holds (G . n) = ((F . n) /\ A)) holds G is Finite_Sep_Sequence of S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let A be Element of S;

 let F be Finite_Sep_Sequence of S;

 let G be FinSequence;

 assume that

 A1: ( dom F) = ( dom G) and

 A2: for n be Nat st n in ( dom F) holds (G . n) = ((F . n) /\ A);

 ( rng G) c= S

 proof

 let v be object;

 assume v in ( rng G);

 then

 consider k be object such that

 A3: k in ( dom G) and

 A4: v = (G . k) by FUNCT_1:def 3;



 A5: (F . k) in ( rng F) by A1, A3, FUNCT_1: 3;

 (G . k) = ((F . k) /\ A) by A1, A2, A3;

 hence thesis by A4, A5, FINSUB_1:def 2;

 end;

 then

 reconsider G as FinSequence of S by FINSEQ_1:def 4;

 now

 let i,j be Nat;

 assume that

 A6: i in ( dom G) and

 A7: j in ( dom G) and

 A8: i <> j;



 A9: (F . i) misses (F . j) by A8, PROB_2:def 2;



 A10: (G . j) = ((F . j) /\ A) by A1, A2, A7;

 (G . i) = ((F . i) /\ A) by A1, A2, A6;

 hence (G . i) misses (G . j) by A10, A9, XBOOLE_1: 76;

 end;

 hence thesis by MESFUNC3: 4;

 end;

 theorem :: MESFUNC5:36



 Th36: for X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL , A be Element of S, F,G be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL st ( dom F) = ( dom G) & (for n be Nat st n in ( dom F) holds (G . n) = ((F . n) /\ A)) & (F,a) are_Re-presentation_of f holds (G,a) are_Re-presentation_of (f | A)

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let f be PartFunc of X, ExtREAL ;

 let A be Element of S;

 let F,G be Finite_Sep_Sequence of S;

 let a be FinSequence of ExtREAL ;

 assume that

 A1: ( dom F) = ( dom G) and

 A2: for n be Nat st n in ( dom F) holds (G . n) = ((F . n) /\ A) and

 A3: (F,a) are_Re-presentation_of f;



 A4: ( dom G) = ( dom a) by A1, A3, MESFUNC3:def 1;

 now

 let v be object;

 assume v in ( union ( rng G));

 then

 consider C be set such that

 A5: v in C and

 A6: C in ( rng G) by TARSKI:def 4;

 consider k be object such that

 A7: k in ( dom G) and

 A8: C = (G . k) by A6, FUNCT_1:def 3;



 A9: (F . k) in ( rng F) by A1, A7, FUNCT_1: 3;



 A10: (G . k) = ((F . k) /\ A) by A1, A2, A7;

 then v in (F . k) by A5, A8, XBOOLE_0:def 4;

 then v in ( union ( rng F)) by A9, TARSKI:def 4;

 then

 A11: v in ( dom f) by A3, MESFUNC3:def 1;

 v in A by A5, A8, A10, XBOOLE_0:def 4;

 then v in (( dom f) /\ A) by A11, XBOOLE_0:def 4;

 hence v in ( dom (f | A)) by RELAT_1: 61;

 end;

 then

 A12: ( union ( rng G)) c= ( dom (f | A));



 A13: for k be Nat st k in ( dom G) holds for x be object st x in (G . k) holds ((f | A) . x) = (a . k)

 proof



 A14: for k be Nat st k in ( dom G) holds for x be set st x in (G . k) holds (f . x) = (a . k)

 proof

 let k be Nat;

 assume

 A15: k in ( dom G);

 let x be set;

 assume x in (G . k);

 then x in ((F . k) /\ A) by A1, A2, A15;

 then x in (F . k) by XBOOLE_0:def 4;

 hence thesis by A1, A3, A15, MESFUNC3:def 1;

 end;

 let k be Nat;

 assume

 A16: k in ( dom G);

 let x be object;

 assume

 A17: x in (G . k);

 (G . k) in ( rng G) by A16, FUNCT_1: 3;

 then x in ( union ( rng G)) by A17, TARSKI:def 4;

 then ((f | A) . x) = (f . x) by A12, FUNCT_1: 47;

 hence thesis by A16, A17, A14;

 end;

 now

 let v be object;

 assume v in ( dom (f | A));

 then

 A18: v in (( dom f) /\ A) by RELAT_1: 61;

 then v in ( dom f) by XBOOLE_0:def 4;

 then v in ( union ( rng F)) by A3, MESFUNC3:def 1;

 then

 consider C be set such that

 A19: v in C and

 A20: C in ( rng F) by TARSKI:def 4;

 consider k be Nat such that

 A21: k in ( dom F) and

 A22: C = (F . k) by A20, FINSEQ_2: 10;



 A23: (G . k) = ((F . k) /\ A) by A2, A21;



 A24: (G . k) in ( rng G) by A1, A21, FUNCT_1: 3;

 v in A by A18, XBOOLE_0:def 4;

 then v in ((F . k) /\ A) by A19, A22, XBOOLE_0:def 4;

 hence v in ( union ( rng G)) by A23, A24, TARSKI:def 4;

 end;

 then ( dom (f | A)) c= ( union ( rng G));

 then ( dom (f | A)) = ( union ( rng G)) by A12;

 hence thesis by A4, A13, MESFUNC3:def 1;

 end;

 theorem :: MESFUNC5:37



 Th37: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S holds ( dom f) is Element of S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 assume f is_simple_func_in S;

 then ex F be Finite_Sep_Sequence of S st ( dom f) = ( union ( rng F)) & for n be Nat, x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y) by MESFUNC2:def 4;

 hence thesis by MESFUNC2: 31;

 end;



 Lm2: for Y be set, p be FinSequence holds (for i be Nat st i in ( dom p) holds (p . i) in Y) implies p is FinSequence of Y

 proof

 let Y be set;

 let p be FinSequence;

 assume

 A1: for i be Nat st i in ( dom p) holds (p . i) in Y;

 let b be object;

 assume b in ( rng p);

 then ex i be Nat st i in ( dom p) & (p . i) = b by FINSEQ_2: 10;

 hence thesis by A1;

 end;



 Lm3: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & ( dom f) <> {} & g is_simple_func_in S & ( dom g) = ( dom f) holds (f + g) is_simple_func_in S & ( dom (f + g)) <> {}

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: f is_simple_func_in S and

 A2: ( dom f) <> {} and

 A3: g is_simple_func_in S and

 A4: ( dom g) = ( dom f);

 consider F be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL such that

 A5: (F,a) are_Re-presentation_of f by A1, MESFUNC3: 12;

 set la = ( len F);



 A6: ( dom f) = ( union ( rng F)) by A5, MESFUNC3:def 1;

 consider G be Finite_Sep_Sequence of S, b be FinSequence of ExtREAL such that

 A7: (G,b) are_Re-presentation_of g by A3, MESFUNC3: 12;

 set lb = ( len G);

 deffunc FG1( Nat) = ((F . ((($1 -' 1) div lb) + 1)) /\ (G . ((($1 -' 1) mod lb) + 1)));

 consider FG be FinSequence such that

 A8: ( len FG) = (la * lb) and

 A9: for k be Nat st k in ( dom FG) holds (FG . k) = FG1(k) from FINSEQ_1:sch 2;



 A10: ( dom FG) = ( Seg (la * lb)) by A8, FINSEQ_1:def 3;

 now

 reconsider lb9 = lb as Nat;

 let k be Nat;

 set i = (((k -' 1) div lb) + 1);

 set j = (((k -' 1) mod lb) + 1);



 A11: lb9 divides (la * lb) by NAT_D:def 3;

 assume

 A12: k in ( dom FG);

 then

 A13: k in ( Seg (la * lb)) by A8, FINSEQ_1:def 3;

 then

 A14: k <= (la * lb) by FINSEQ_1: 1;

 then (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 then

 A15: ((k -' 1) div lb) <= (((la * lb) -' 1) div lb) by NAT_2: 24;

 1 <= k by A13, FINSEQ_1: 1;

 then

 A16: 1 <= (la * lb) by A14, XXREAL_0: 2;



 A17: lb <> 0 by A13;

 then ((k -' 1) mod lb) < lb by NAT_D: 1;

 then

 A18: j <= lb by NAT_1: 13;

 lb >= ( 0 + 1) by A17, NAT_1: 13;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A11, A16, NAT_2: 15;

 then (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by A15, XREAL_1: 19;

 then

 A19: i <= la by A17, NAT_D: 18;

 i >= ( 0 + 1) by XREAL_1: 6;

 then i in ( Seg la) by A19;

 then i in ( dom F) by FINSEQ_1:def 3;

 then

 A20: (F . i) in ( rng F) by FUNCT_1: 3;

 j >= ( 0 + 1) by XREAL_1: 6;

 then j in ( dom G) by A18, FINSEQ_3: 25;

 then

 A21: (G . j) in ( rng G) by FUNCT_1: 3;

 (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A9, A12;

 hence (FG . k) in S by A20, A21, MEASURE1: 34;

 end;

 then

 reconsider FG as FinSequence of S by Lm2;



 A22: for k,l be Nat st k in ( dom FG) & l in ( dom FG) & k <> l holds (FG . k) misses (FG . l)

 proof



 A23: lb divides (la * lb) by NAT_D:def 3;

 let k,l be Nat;

 assume that

 A24: k in ( dom FG) and

 A25: l in ( dom FG) and

 A26: k <> l;



 A27: k in ( Seg (la * lb)) by A8, A24, FINSEQ_1:def 3;

 then

 A28: 1 <= k by FINSEQ_1: 1;

 set m = (((l -' 1) mod lb) + 1);

 set n = (((l -' 1) div lb) + 1);

 set j = (((k -' 1) mod lb) + 1);

 set i = (((k -' 1) div lb) + 1);



 A29: (FG . k) = ((F . i) /\ (G . j)) by A9, A24;



 A30: k <= (la * lb) by A27, FINSEQ_1: 1;

 then

 A31: 1 <= (la * lb) by A28, XXREAL_0: 2;



 A32: lb <> 0 by A27;

 then lb >= ( 0 + 1) by NAT_1: 13;

 then

 A33: (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A23, A31, NAT_2: 15;



 A34: l in ( Seg (la * lb)) by A8, A25, FINSEQ_1:def 3;

 then

 A35: 1 <= l by FINSEQ_1: 1;

 A36:

 now

 ((l -' 1) + 1) = ((((n - 1) * lb) + (m - 1)) + 1) by A32, NAT_D: 2;

 then

 A37: ((l - 1) + 1) = (((n - 1) * lb) + m) by A35, XREAL_1: 233;

 assume that

 A38: i = n and

 A39: j = m;

 ((k -' 1) + 1) = ((((i - 1) * lb) + (j - 1)) + 1) by A32, NAT_D: 2;

 then ((k - 1) + 1) = (((i - 1) * lb) + j) by A28, XREAL_1: 233;

 hence contradiction by A26, A38, A39, A37;

 end;

 (k -' 1) <= ((la * lb) -' 1) by A30, NAT_D: 42;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A33, NAT_2: 24;

 then (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 then

 A40: i <= la by A32, NAT_D: 18;

 i >= ( 0 + 1) by XREAL_1: 6;

 then i in ( Seg la) by A40;

 then

 A41: i in ( dom F) by FINSEQ_1:def 3;



 A42: j >= ( 0 + 1) by XREAL_1: 6;

 ((k -' 1) mod lb) < lb by A32, NAT_D: 1;

 then j <= lb by NAT_1: 13;

 then

 A43: j in ( dom G) by A42, FINSEQ_3: 25;



 A44: m >= ( 0 + 1) by XREAL_1: 6;

 ((l -' 1) mod lb) < lb by A32, NAT_D: 1;

 then m <= lb by NAT_1: 13;

 then

 A45: m in ( dom G) by A44, FINSEQ_3: 25;



 A46: n >= ( 0 + 1) by XREAL_1: 6;

 l <= (la * lb) by A34, FINSEQ_1: 1;

 then (l -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 then ((l -' 1) div lb) <= (((la * lb) div lb) - 1) by A33, NAT_2: 24;

 then (((l -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 then n <= la by A32, NAT_D: 18;

 then n in ( Seg la) by A46;

 then

 A47: n in ( dom F) by FINSEQ_1:def 3;

 per cases by A36;

 suppose

 A48: i <> n;

 ((FG . k) /\ (FG . l)) = (((F . i) /\ (G . j)) /\ ((F . n) /\ (G . m))) by A9, A25, A29;

 then ((FG . k) /\ (FG . l)) = ((F . i) /\ ((G . j) /\ ((F . n) /\ (G . m)))) by XBOOLE_1: 16;

 then ((FG . k) /\ (FG . l)) = ((F . i) /\ ((F . n) /\ ((G . j) /\ (G . m)))) by XBOOLE_1: 16;

 then

 A49: ((FG . k) /\ (FG . l)) = (((F . i) /\ (F . n)) /\ ((G . j) /\ (G . m))) by XBOOLE_1: 16;

 (F . i) misses (F . n) by A41, A47, A48, MESFUNC3: 4;

 then ((FG . k) /\ (FG . l)) = ( {} /\ ((G . j) /\ (G . m))) by A49;

 hence thesis;

 end;

 suppose

 A50: j <> m;

 ((FG . k) /\ (FG . l)) = (((F . i) /\ (G . j)) /\ ((F . n) /\ (G . m))) by A9, A25, A29;

 then ((FG . k) /\ (FG . l)) = ((F . i) /\ ((G . j) /\ ((F . n) /\ (G . m)))) by XBOOLE_1: 16;

 then ((FG . k) /\ (FG . l)) = ((F . i) /\ ((F . n) /\ ((G . j) /\ (G . m)))) by XBOOLE_1: 16;

 then

 A51: ((FG . k) /\ (FG . l)) = (((F . i) /\ (F . n)) /\ ((G . j) /\ (G . m))) by XBOOLE_1: 16;

 (G . j) misses (G . m) by A43, A45, A50, MESFUNC3: 4;

 then ((FG . k) /\ (FG . l)) = (((F . i) /\ (F . n)) /\ {} ) by A51;

 hence thesis;

 end;

 end;



 A52: g is real-valued by A3, MESFUNC2:def 4;

 then

 A53: ( dom (f + g)) = (( dom f) /\ ( dom g)) by MESFUNC2: 2;

 reconsider FG as Finite_Sep_Sequence of S by A22, MESFUNC3: 4;



 A54: ( dom g) = ( union ( rng G)) by A7, MESFUNC3:def 1;



 A55: ( dom f) = ( union ( rng FG))

 proof

 now

 let z be object;

 assume

 A56: z in ( dom f);

 then

 consider Y be set such that

 A57: z in Y and

 A58: Y in ( rng F) by A6, TARSKI:def 4;

 consider i be Nat such that

 A59: i in ( dom F) and

 A60: Y = (F . i) by A58, FINSEQ_2: 10;



 A61: i in ( Seg ( len F)) by A59, FINSEQ_1:def 3;

 then 1 <= i by FINSEQ_1: 1;

 then

 consider i9 be Nat such that

 A62: i = (1 qua Complex + i9) by NAT_1: 10;

 consider Z be set such that

 A63: z in Z and

 A64: Z in ( rng G) by A4, A54, A56, TARSKI:def 4;

 consider j be Nat such that

 A65: j in ( dom G) and

 A66: Z = (G . j) by A64, FINSEQ_2: 10;



 A67: j in ( Seg ( len G)) by A65, FINSEQ_1:def 3;

 then

 A68: 1 <= j by FINSEQ_1: 1;

 then

 consider j9 be Nat such that

 A69: j = (1 qua Complex + j9) by NAT_1: 10;

 ((i9 * lb) + j) in NAT by ORDINAL1:def 12;

 then

 reconsider k = (((i - 1) * lb) + j) as Element of NAT by A62;

 i <= la by A61, FINSEQ_1: 1;

 then (i - 1) <= (la - 1) by XREAL_1: 9;

 then ((i - 1) * lb) <= ((la - 1) * lb) by XREAL_1: 64;

 then

 A70: k <= (((la - 1) * lb) + j) by XREAL_1: 6;



 A71: j <= lb by A67, FINSEQ_1: 1;

 then

 A72: j9 < lb by A69, NAT_1: 13;



 A73: k >= ( 0 + j) by A62, XREAL_1: 6;



 then

 A74: (k -' 1) = (k - 1) by A68, XREAL_1: 233, XXREAL_0: 2

 .= ((i9 * lb) + j9) by A62, A69;

 then

 A75: i = (((k -' 1) div lb) + 1) by A62, A72, NAT_D:def 1;

 (((la - 1) * lb) + j) <= (((la - 1) * lb) + lb) by A71, XREAL_1: 6;

 then

 A76: k <= (la * lb) by A70, XXREAL_0: 2;

 k >= 1 by A68, A73, XXREAL_0: 2;

 then

 A77: k in ( Seg (la * lb)) by A76;

 then k in ( dom FG) by A8, FINSEQ_1:def 3;

 then

 A78: (FG . k) in ( rng FG) by FUNCT_1:def 3;



 A79: j = (((k -' 1) mod lb) + 1) by A69, A74, A72, NAT_D:def 2;

 z in ((F . i) /\ (G . j)) by A57, A60, A63, A66, XBOOLE_0:def 4;

 then z in (FG . k) by A9, A10, A75, A79, A77;

 hence z in ( union ( rng FG)) by A78, TARSKI:def 4;

 end;

 hence ( dom f) c= ( union ( rng FG));

 reconsider lb9 = lb as Nat;

 let z be object;



 A80: lb9 divides (la * lb) by NAT_D:def 3;

 assume z in ( union ( rng FG));

 then

 consider Y be set such that

 A81: z in Y and

 A82: Y in ( rng FG) by TARSKI:def 4;

 consider k be Nat such that

 A83: k in ( dom FG) and

 A84: Y = (FG . k) by A82, FINSEQ_2: 10;



 A85: k in ( Seg ( len FG)) by A83, FINSEQ_1:def 3;

 then

 A86: k <= (la * lb) by A8, FINSEQ_1: 1;

 then

 A87: (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 set j = (((k -' 1) mod lb) + 1);

 set i = (((k -' 1) div lb) + 1);



 A88: i >= ( 0 + 1) by NAT_1: 13;

 1 <= k by A85, FINSEQ_1: 1;

 then

 A89: 1 <= (la * lb) by A86, XXREAL_0: 2;



 A90: lb <> 0 by A8, A85;

 then lb >= ( 0 + 1) by NAT_1: 13;

 then (((la * lb) -' 1) div lb9) = (((la * lb) div lb) - 1) by A80, A89, NAT_2: 15;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A87, NAT_2: 24;

 then

 A91: i <= ((la * lb) div lb) by XREAL_1: 19;

 ((la * lb) div lb) = la by A90, NAT_D: 18;

 then i in ( Seg la) by A91, A88;

 then i in ( dom F) by FINSEQ_1:def 3;

 then

 A92: (F . i) in ( rng F) by FUNCT_1:def 3;

 (FG . k) = ((F . i) /\ (G . j)) by A9, A83;

 then z in (F . i) by A81, A84, XBOOLE_0:def 4;

 hence thesis by A6, A92, TARSKI:def 4;

 end;



 A93: for k be Nat, x,y be Element of X st k in ( dom FG) & x in (FG . k) & y in (FG . k) holds ((f + g) . x) = ((f + g) . y)

 proof



 A94: lb divides (la * lb) by NAT_D:def 3;

 let k be Nat;

 let x,y be Element of X;

 assume that

 A95: k in ( dom FG) and

 A96: x in (FG . k) and

 A97: y in (FG . k);

 set j = (((k -' 1) mod lb) + 1);



 A98: (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A9, A95;

 then

 A99: y in (G . j) by A97, XBOOLE_0:def 4;

 set i = (((k -' 1) div lb) + 1);



 A100: i >= ( 0 + 1) by XREAL_1: 6;



 A101: k in ( Seg ( len FG)) by A95, FINSEQ_1:def 3;

 then

 A102: 1 <= k by FINSEQ_1: 1;



 A103: lb > 0 by A8, A101;

 then

 A104: lb >= ( 0 + 1) by NAT_1: 13;



 A105: k <= (la * lb) by A8, A101, FINSEQ_1: 1;

 then

 A106: (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 1 <= (la * lb) by A102, A105, XXREAL_0: 2;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A104, A94, NAT_2: 15;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A106, NAT_2: 24;

 then

 A107: (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 lb <> 0 by A8, A101;

 then i <= la by A107, NAT_D: 18;

 then i in ( Seg la) by A100;

 then

 A108: i in ( dom F) by FINSEQ_1:def 3;

 x in (F . i) by A96, A98, XBOOLE_0:def 4;

 then

 A109: (f . x) = (a . i) by A5, A108, MESFUNC3:def 1;



 A110: j >= ( 0 + 1) by XREAL_1: 6;

 ((k -' 1) mod lb) < lb by A103, NAT_D: 1;

 then j <= lb by NAT_1: 13;

 then j in ( Seg lb) by A110;

 then

 A111: j in ( dom G) by FINSEQ_1:def 3;

 y in (F . i) by A97, A98, XBOOLE_0:def 4;

 then

 A112: (f . y) = (a . i) by A5, A108, MESFUNC3:def 1;



 A113: (FG . k) in ( rng FG) by A95, FUNCT_1:def 3;

 then x in ( dom (f + g)) by A4, A55, A53, A96, TARSKI:def 4;

 then

 A114: ((f + g) . x) = ((f . x) + (g . x)) by MESFUNC1:def 3;

 x in (G . j) by A96, A98, XBOOLE_0:def 4;

 then ((f + g) . x) = ((a . i) + (b . j)) by A7, A109, A111, A114, MESFUNC3:def 1;

 then

 A115: ((f + g) . x) = ((f . y) + (g . y)) by A7, A99, A111, A112, MESFUNC3:def 1;

 y in ( dom (f + g)) by A4, A55, A53, A97, A113, TARSKI:def 4;

 hence thesis by A115, MESFUNC1:def 3;

 end;

 now

 let x be Element of X;

 assume

 A116: x in ( dom (f + g));

 then

 A117: |.(g . x).| < +infty by A4, A52, A53, MESFUNC2:def 1;

 |.((f + g) . x).| = |.((f . x) + (g . x)).| by A116, MESFUNC1:def 3;

 then

 A118: |.((f + g) . x).| <= ( |.(f . x).| + |.(g . x).|) by EXTREAL1: 24;

 f is real-valued by A1, MESFUNC2:def 4;

 then |.(f . x).| < +infty by A4, A53, A116, MESFUNC2:def 1;

 then ( |.(f . x).| + |.(g . x).|) <> +infty by A117, XXREAL_3: 16;

 hence |.((f + g) . x).| < +infty by A118, XXREAL_0: 2, XXREAL_0: 4;

 end;

 then (f + g) is real-valued by MESFUNC2:def 1;

 hence (f + g) is_simple_func_in S by A4, A55, A53, A93, MESFUNC2:def 4;

 thus thesis by A2, A4, A53;

 end;

 theorem :: MESFUNC5:38



 Th38: for X be non empty set, S be SigmaField of X, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & g is_simple_func_in S holds (f + g) is_simple_func_in S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: g is_simple_func_in S;

 per cases ;

 suppose

 A3: ( dom (f + g)) = {} ;

 reconsider EMPTY = {} as Element of S by PROB_1: 4;

 set F = <*EMPTY*>;



 A4: ( dom F) = ( Seg 1) by FINSEQ_1: 38;

 A5:

 now

 let i,j be Nat;

 assume that

 A6: i in ( dom F) and

 A7: j in ( dom F) and

 A8: i <> j;

 i = 1 by A4, A6, FINSEQ_1: 2, TARSKI:def 1;

 hence (F . i) misses (F . j) by A4, A7, A8, FINSEQ_1: 2, TARSKI:def 1;

 end;



 A9: for n be Nat st n in ( dom F) holds (F . n) = EMPTY

 proof

 let n be Nat;

 assume n in ( dom F);

 then n = 1 by A4, FINSEQ_1: 2, TARSKI:def 1;

 hence thesis by FINSEQ_1: 40;

 end;

 reconsider F as Finite_Sep_Sequence of S by A5, MESFUNC3: 4;

 ( union ( rng F)) = ( union ( bool {} )) by FINSEQ_1: 39, ZFMISC_1: 1;

 then

 A10: ( dom (f + g)) = ( union ( rng F)) by A3, ZFMISC_1: 81;

 for x be Element of X st x in ( dom (f + g)) holds |.((f + g) . x).| < +infty by A3;

 then

 A11: (f + g) is real-valued by MESFUNC2:def 1;

 for n be Nat holds for x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds ((f + g) . x) = ((f + g) . y) by A9;

 hence thesis by A11, A10, MESFUNC2:def 4;

 end;

 suppose

 A12: ( dom (f + g)) <> {} ;



 A13: ((f | ( dom (f + g))) " { +infty }) = (( dom (f + g)) /\ (f " { +infty })) by FUNCT_1: 70;

 g is without+infty by A2, Th14;

 then not +infty in ( rng g);

 then

 A14: (g " { +infty }) = {} by FUNCT_1: 72;



 A15: ((g | ( dom (f + g))) " { +infty }) = (( dom (f + g)) /\ (g " { +infty })) by FUNCT_1: 70;

 f is without+infty by A1, Th14;

 then not +infty in ( rng f);

 then

 A16: (f " { +infty }) = {} by FUNCT_1: 72;

 then

 A17: ((( dom f) /\ ( dom g)) \ (((f " { +infty }) /\ (g " { -infty })) \/ ((f " { -infty }) /\ (g " { +infty })))) = (( dom f) /\ ( dom g)) by A14;

 then

 A18: ( dom (f + g)) = (( dom f) /\ ( dom g)) by MESFUNC1:def 3;

 ( dom (f | ( dom (f + g)))) = (( dom f) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A19: ( dom (f | ( dom (f + g)))) = ((( dom f) /\ ( dom f)) /\ ( dom g)) by A18, XBOOLE_1: 16;

 then

 A20: ( dom (f | ( dom (f + g)))) = ( dom (f + g)) by A17, MESFUNC1:def 3;



 A21: ( dom g) is Element of S by A2, Th37;

 ( dom f) is Element of S by A1, Th37;

 then

 A22: ( dom (f + g)) in S by A18, A21, FINSUB_1:def 2;

 then

 A23: (g | ( dom (f + g))) is_simple_func_in S by A2, Th34;

 ( dom (g | ( dom (f + g)))) = (( dom g) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A24: ( dom (g | ( dom (f + g)))) = ((( dom g) /\ ( dom g)) /\ ( dom f)) by A18, XBOOLE_1: 16;

 then

 A25: ( dom (g | ( dom (f + g)))) = ( dom (f + g)) by A17, MESFUNC1:def 3;



 A26: ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g))))) = ((( dom (f | ( dom (f + g)))) /\ ( dom (g | ( dom (f + g))))) \ ((((f | ( dom (f + g))) " { +infty }) /\ ((g | ( dom (f + g))) " { -infty })) \/ (((f | ( dom (f + g))) " { -infty }) /\ ((g | ( dom (f + g))) " { +infty })))) by MESFUNC1:def 3

 .= ( dom (f + g)) by A16, A14, A17, A19, A24, A13, A15, MESFUNC1:def 3;



 A27: for x be Element of X st x in ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g))))) holds (((f | ( dom (f + g))) + (g | ( dom (f + g)))) . x) = ((f + g) . x)

 proof

 let x be Element of X;

 assume

 A28: x in ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g)))));



 then (((f | ( dom (f + g))) + (g | ( dom (f + g)))) . x) = (((f | ( dom (f + g))) . x) + ((g | ( dom (f + g))) . x)) by MESFUNC1:def 3

 .= ((f . x) + ((g | ( dom (f + g))) . x)) by A26, A28, FUNCT_1: 49

 .= ((f . x) + (g . x)) by A26, A28, FUNCT_1: 49;

 hence thesis by A26, A28, MESFUNC1:def 3;

 end;

 (f | ( dom (f + g))) is_simple_func_in S by A1, A22, Th34;

 then ((f | ( dom (f + g))) + (g | ( dom (f + g)))) is_simple_func_in S by A12, A23, A20, A25, Lm3;

 hence thesis by A26, A27, PARTFUN1: 5;

 end;

 end;

 theorem :: MESFUNC5:39



 Th39: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real st f is_simple_func_in S holds (c (#) f) is_simple_func_in S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let c be Real;

 set g = (c (#) f);

 assume

 A1: f is_simple_func_in S;

 then

 consider G be Finite_Sep_Sequence of S such that

 A2: ( dom f) = ( union ( rng G)) and

 A3: for n be Nat, x,y be Element of X st n in ( dom G) & x in (G . n) & y in (G . n) holds (f . x) = (f . y) by MESFUNC2:def 4;



 A4: f is real-valued by A1, MESFUNC2:def 4;

 now

 let x be Element of X;

 assume

 A5: x in ( dom g);

 (c * (f . x)) <> -infty by A4;

 then (g . x) <> -infty by A5, MESFUNC1:def 6;

 then -infty < (g . x) by XXREAL_0: 6;

 then

 A6: ( - +infty ) < (g . x) by XXREAL_3:def 3;

 (c * (f . x)) <> +infty by A4;

 then (g . x) <> +infty by A5, MESFUNC1:def 6;

 then (g . x) < +infty by XXREAL_0: 4;

 hence |.(g . x).| < +infty by A6, EXTREAL1: 22;

 end;

 then

 A7: g is real-valued by MESFUNC2:def 1;



 A8: ( dom g) = ( dom f) by MESFUNC1:def 6;

 now

 let n be Nat;

 let x,y be Element of X;

 assume that

 A9: n in ( dom G) and

 A10: x in (G . n) and

 A11: y in (G . n);



 A12: (G . n) in ( rng G) by A9, FUNCT_1: 3;

 then y in ( dom g) by A8, A2, A11, TARSKI:def 4;

 then

 A13: (g . y) = (c * (f . y)) by MESFUNC1:def 6;

 x in ( dom g) by A8, A2, A10, A12, TARSKI:def 4;

 then (g . x) = (c * (f . x)) by MESFUNC1:def 6;

 hence (g . x) = (g . y) by A3, A9, A10, A11, A13;

 end;

 hence thesis by A8, A7, A2, MESFUNC2:def 4;

 end;

 theorem :: MESFUNC5:40



 Th40: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & g is_simple_func_in S & (for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x)) holds (f - g) is nonnegative

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: f is_simple_func_in S and

 A2: g is_simple_func_in S and

 A3: for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x);

 g is without-infty by A2, Th14;

 then not -infty in ( rng g);

 then

 A4: (g " { -infty }) = {} by FUNCT_1: 72;

 f is without+infty by A1, Th14;

 then not +infty in ( rng f);

 then

 A5: (f " { +infty }) = {} by FUNCT_1: 72;

 then ((( dom f) /\ ( dom g)) \ (((f " { +infty }) /\ (g " { +infty })) \/ ((f " { -infty }) /\ (g " { -infty })))) = (( dom f) /\ ( dom g)) by A4;

 then

 A6: ( dom (f - g)) = (( dom f) /\ ( dom g)) by MESFUNC1:def 4;

 for x be set st x in (( dom f) /\ ( dom g)) holds (g . x) <= (f . x) & -infty < (g . x) & (f . x) < +infty

 proof

 let x be set;

 assume

 A7: x in (( dom f) /\ ( dom g));

 hence (g . x) <= (f . x) by A3, A6;

 x in ( dom g) by A7, XBOOLE_0:def 4;

 then not (g . x) in { -infty } by A4, FUNCT_1:def 7;

 then not (g . x) = -infty by TARSKI:def 1;

 hence -infty < (g . x) by XXREAL_0: 6;

 x in ( dom f) by A7, XBOOLE_0:def 4;

 then not (f . x) in { +infty } by A5, FUNCT_1:def 7;

 then not (f . x) = +infty by TARSKI:def 1;

 hence thesis by XXREAL_0: 4;

 end;

 hence thesis by Th21;

 end;

 theorem :: MESFUNC5:41



 Th41: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, c be R_eal st c <> +infty & c <> -infty holds ex f be PartFunc of X, ExtREAL st f is_simple_func_in S & ( dom f) = A & for x be object st x in A holds (f . x) = c

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let A be Element of S;

 let c be R_eal;

 assume that

 A1: c <> +infty and

 A2: c <> -infty ;

 -infty < c by A2, XXREAL_0: 6;

 then

 A3: ( - +infty ) < c by XXREAL_3:def 3;

 deffunc F( object) = c;

 defpred P[ object] means $1 in A;



 A4: for x be object st P[x] holds F(x) in ExtREAL ;

 consider f be PartFunc of X, ExtREAL such that

 A5: (for x be object holds x in ( dom f) iff x in X & P[x]) & for x be object st x in ( dom f) holds (f . x) = F(x) from PARTFUN1:sch 3( A4);

 c < +infty by A1, XXREAL_0: 4;

 then |.c.| < +infty by A3, EXTREAL1: 22;

 then for x be Element of X st x in ( dom f) holds |.(f . x).| < +infty by A5;

 then

 A6: f is real-valued by MESFUNC2:def 1;

 take f;



 A7: A c= ( dom f) by A5;

 set F = <*( dom f)*>;



 A8: ( dom f) c= A by A5;



 A9: ( rng F) = {( dom f)} by FINSEQ_1: 38;

 then

 A10: ( rng F) = {A} by A8, A7, XBOOLE_0:def 10;

 ( rng F) c= S

 proof

 let z be object;

 assume z in ( rng F);

 then z = A by A10, TARSKI:def 1;

 hence thesis;

 end;

 then

 reconsider F as FinSequence of S by FINSEQ_1:def 4;

 now

 let i,j be Nat;

 assume that

 A11: i in ( dom F) and

 A12: j in ( dom F) and

 A13: i <> j;



 A14: ( dom F) = ( Seg 1) by FINSEQ_1: 38;

 then i = 1 by A11, FINSEQ_1: 2, TARSKI:def 1;

 hence (F . i) misses (F . j) by A12, A13, A14, FINSEQ_1: 2, TARSKI:def 1;

 end;

 then

 reconsider F as Finite_Sep_Sequence of S by MESFUNC3: 4;

 A15:

 now

 let n be Nat;

 let x,y be Element of X;

 assume that

 A16: n in ( dom F) and

 A17: x in (F . n) and

 A18: y in (F . n);

 ( dom F) = ( Seg 1) by FINSEQ_1: 38;

 then

 A19: n = 1 by A16, FINSEQ_1: 2, TARSKI:def 1;

 then x in ( dom f) by A17, FINSEQ_1: 40;

 then

 A20: (f . x) = c by A5;

 y in ( dom f) by A18, A19, FINSEQ_1: 40;

 hence (f . x) = (f . y) by A5, A20;

 end;

 ( dom f) = ( union ( rng F)) by A9, ZFMISC_1: 25;

 hence f is_simple_func_in S by A6, A15, MESFUNC2:def 4;

 thus ( dom f) = A by A8, A7;

 thus thesis by A5;

 end;

 theorem :: MESFUNC5:42



 Th42: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , B,BF be Element of S st f is B -measurable & BF = (( dom f) /\ B) holds (f | B) is BF -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , B,BF be Element of S such that

 A1: f is B -measurable and

 A2: BF = (( dom f) /\ B);

 now

 let r be Real;

 A3:

 now

 let x be object;

 reconsider xx = x as set by TARSKI: 1;

 (x in ( dom (f | B)) & ex y be R_eal st y = ((f | B) . x) & y < r) iff x in (( dom f) /\ B) & ex y be R_eal st y = ((f | B) . x) & y < r by RELAT_1: 61;

 then

 A4: x in BF & x in ( less_dom ((f | B),r)) iff x in B & x in ( dom f) & ((f | B) . xx) < r by A2, MESFUNC1:def 11, XBOOLE_0:def 4;

 x in B & x in ( dom f) implies ((f . x) < r iff ((f | B) . x) < r) by FUNCT_1: 49;

 then x in (BF /\ ( less_dom ((f | B),r))) iff x in B & x in ( less_dom (f,r)) by A4, MESFUNC1:def 11, XBOOLE_0:def 4;

 hence x in (BF /\ ( less_dom ((f | B),r))) iff x in (B /\ ( less_dom (f,r))) by XBOOLE_0:def 4;

 end;

 then

 A5: (B /\ ( less_dom (f,r))) c= (BF /\ ( less_dom ((f | B),r)));

 (BF /\ ( less_dom ((f | B),r))) c= (B /\ ( less_dom (f,r))) by A3;

 then (BF /\ ( less_dom ((f | B),r))) = (B /\ ( less_dom (f,r))) by A5;

 hence (BF /\ ( less_dom ((f | B),r))) in S by A1, MESFUNC1:def 16;

 end;

 hence thesis by MESFUNC1:def 16;

 end;

 theorem :: MESFUNC5:43



 Th43: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f,g be PartFunc of X, ExtREAL st A c= ( dom f) & f is A -measurable & g is A -measurable & f is without-infty & g is without-infty holds (( max+ (f + g)) + ( max- f)) is A -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: A c= ( dom f) and

 A2: f is A -measurable and

 A3: g is A -measurable and

 A4: f is without-infty and

 A5: g is without-infty;

 (f + g) is A -measurable by A2, A3, A4, A5, Th31;

 then

 A6: ( max+ (f + g)) is A -measurable by MESFUNC2: 25;



 A7: ( max- f) is nonnegative by Lm1;



 A8: ( max+ (f + g)) is nonnegative by Lm1;

 ( max- f) is A -measurable by A1, A2, MESFUNC2: 26;

 hence thesis by A6, A8, A7, Th31;

 end;

 theorem :: MESFUNC5:44



 Th44: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f,g be PartFunc of X, ExtREAL st A c= (( dom f) /\ ( dom g)) & f is A -measurable & g is A -measurable & f is without-infty & g is without-infty holds (( max- (f + g)) + ( max+ f)) is A -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: A c= (( dom f) /\ ( dom g)) and

 A2: f is A -measurable and

 A3: g is A -measurable and

 A4: f is without-infty and

 A5: g is without-infty;



 A6: ( dom (f + g)) = (( dom f) /\ ( dom g)) by A4, A5, Th16;

 (f + g) is A -measurable by A2, A3, A4, A5, Th31;

 then

 A7: ( max- (f + g)) is A -measurable by A1, A6, MESFUNC2: 26;



 A8: ( max- (f + g)) is nonnegative by Lm1;



 A9: ( max+ f) is nonnegative by Lm1;

 ( max+ f) is A -measurable by A2, MESFUNC2: 25;

 hence thesis by A7, A8, A9, Th31;

 end;

 theorem :: MESFUNC5:45



 Th45: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be set st A in S holds 0 <= (M . A)

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let A be set;

 reconsider E = {} as Element of S by PROB_1: 4;

 assume A in S;

 then

 reconsider A as Element of S;

 (M . E) <= (M . A) by MEASURE1: 31, XBOOLE_1: 2;

 hence thesis by VALUED_0:def 19;

 end;



 Lm4: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , r be Real st ( dom f) in S & (for x be object st x in ( dom f) holds (f . x) = r) holds f is_simple_func_in S

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let r be Real;

 assume that

 A1: ( dom f) in S and

 A2: for x be object st x in ( dom f) holds (f . x) = r;

 reconsider Df = ( dom f) as Element of S by A1;

 set F = <*Df*>;



 A3: ( dom F) = ( Seg 1) by FINSEQ_1: 38;

 A4:

 now

 let i,j be Nat;

 assume that

 A5: i in ( dom F) and

 A6: j in ( dom F) and

 A7: i <> j;

 i = 1 by A3, A5, FINSEQ_1: 2, TARSKI:def 1;

 hence (F . i) misses (F . j) by A3, A6, A7, FINSEQ_1: 2, TARSKI:def 1;

 end;



 A8: for n be Nat st n in ( dom F) holds (F . n) = Df

 proof

 let n be Nat;

 assume n in ( dom F);

 then n = 1 by A3, FINSEQ_1: 2, TARSKI:def 1;

 hence thesis by FINSEQ_1: 40;

 end;

 reconsider F as Finite_Sep_Sequence of S by A4, MESFUNC3: 4;

 A9:

 now

 let n be Nat;

 let x,y be Element of X;

 assume that

 A10: n in ( dom F) and

 A11: x in (F . n) and

 A12: y in (F . n);



 A13: (F . n) = Df by A8, A10;

 then (f . x) = r by A2, A11;

 hence (f . x) = (f . y) by A2, A12, A13;

 end;

 F = <*(F . 1)*> by FINSEQ_1: 40;

 then

 A14: ( rng F) = {(F . 1)} by FINSEQ_1: 38;



 A15: r in REAL by XREAL_0:def 1;

 now

 let x be Element of X;

 assume x in ( dom f);

 then

 A16: (f . x) = r by A2;

 then -infty < (f . x) by XXREAL_0: 12, A15;

 then

 A17: ( - +infty ) < (f . x) by XXREAL_3:def 3;

 (f . x) < +infty by A16, XXREAL_0: 9, A15;

 hence |.(f . x).| < +infty by A17, EXTREAL1: 22;

 end;

 then

 A18: f is real-valued by MESFUNC2:def 1;

 1 in ( Seg 1);

 then (F . 1) = Df by A3, A8;

 then ( dom f) = ( union ( rng F)) by A14, ZFMISC_1: 25;

 hence thesis by A18, A9, MESFUNC2:def 4;

 end;

 theorem :: MESFUNC5:46



 Th46: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable) & (ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable) & (f " { +infty }) in S & (f " { -infty }) in S & (g " { +infty }) in S & (g " { -infty }) in S holds ( dom (f + g)) in S

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable and

 A2: ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable and

 A3: (f " { +infty }) in S and

 A4: (f " { -infty }) in S and

 A5: (g " { +infty }) in S and

 A6: (g " { -infty }) in S;



 A7: ((f " { +infty }) /\ (g " { -infty })) in S by A3, A6, MEASURE1: 34;

 ((f " { -infty }) /\ (g " { +infty })) in S by A4, A5, MEASURE1: 34;

 then (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty }))) in S by A7, MEASURE1: 34;

 then

 A8: (X \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) in S by MEASURE1: 34;

 consider E2 be Element of S such that

 A9: E2 = ( dom g) and g is E2 -measurable by A2;

 consider E1 be Element of S such that

 A10: E1 = ( dom f) and f is E1 -measurable by A1;



 A11: ((E1 /\ E2) /\ (X \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty }))))) = (((E1 /\ E2) /\ X) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by XBOOLE_1: 49

 .= ((E1 /\ E2) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by XBOOLE_1: 28;

 ( dom (f + g)) = ((E1 /\ E2) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by A10, A9, MESFUNC1:def 3;

 hence thesis by A8, A11, MEASURE1: 34;

 end;



 Lm5: for X be non empty set, S be SigmaField of X, A be Element of S, f be PartFunc of X, ExtREAL , r be Real holds (A /\ ( less_dom (f,r))) = ( less_dom ((f | A),r))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let A be Element of S;

 let f be PartFunc of X, ExtREAL ;

 let r be Real;

 now

 let v be object;

 assume

 A1: v in (A /\ ( less_dom (f,r)));

 then

 A2: v in ( less_dom (f,r)) by XBOOLE_0:def 4;



 A3: v in A by A1, XBOOLE_0:def 4;

 then

 A4: (f . v) = ((f | A) . v) by FUNCT_1: 49;

 v in ( dom f) by A2, MESFUNC1:def 11;

 then v in (A /\ ( dom f)) by A3, XBOOLE_0:def 4;

 then

 A5: v in ( dom (f | A)) by RELAT_1: 61;

 (f . v) < r by A2, MESFUNC1:def 11;

 hence v in ( less_dom ((f | A),r)) by A5, A4, MESFUNC1:def 11;

 end;

 hence (A /\ ( less_dom (f,r))) c= ( less_dom ((f | A),r));

 let v be object;

 reconsider vv = v as set by TARSKI: 1;

 assume

 A6: v in ( less_dom ((f | A),r));

 then

 A7: v in ( dom (f | A)) by MESFUNC1:def 11;

 then

 A8: v in (( dom f) /\ A) by RELAT_1: 61;

 then

 A9: v in ( dom f) by XBOOLE_0:def 4;

 ((f | A) . vv) < r by A6, MESFUNC1:def 11;

 then ex w be R_eal st w = (f . vv) & w < r by A7, FUNCT_1: 47;

 then

 A10: v in ( less_dom (f,r)) by A9, MESFUNC1:def 11;

 v in A by A8, XBOOLE_0:def 4;

 hence thesis by A10, XBOOLE_0:def 4;

 end;



 Lm6: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f be PartFunc of X, ExtREAL holds (f | A) is A -measurable iff f is A -measurable

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let A be Element of S;

 let f be PartFunc of X, ExtREAL ;

 now

 assume

 A1: (f | A) is A -measurable;

 now

 let r be Real;

 (A /\ ( less_dom ((f | A),r))) in S by A1, MESFUNC1:def 16;

 then (A /\ (A /\ ( less_dom (f,r)))) in S by Lm5;

 then ((A /\ A) /\ ( less_dom (f,r))) in S by XBOOLE_1: 16;

 hence (A /\ ( less_dom (f,r))) in S;

 end;

 hence f is A -measurable by MESFUNC1:def 16;

 end;

 hence (f | A) is A -measurable implies f is A -measurable;

 assume

 A2: f is A -measurable;

 now

 let r be Real;

 ((A /\ A) /\ ( less_dom (f,r))) in S by A2, MESFUNC1:def 16;

 then (A /\ (A /\ ( less_dom (f,r)))) in S by XBOOLE_1: 16;

 hence (A /\ ( less_dom ((f | A),r))) in S by Lm5;

 end;

 hence thesis by MESFUNC1:def 16;

 end;



 Lm7: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable) & (ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable) & ( dom f) = ( dom g) holds ex DFPG be Element of S st DFPG = ( dom (f + g)) & (f + g) is DFPG -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable and

 A2: ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable and

 A3: ( dom f) = ( dom g);

 consider Gf be Element of S such that

 A4: Gf = ( dom g) and

 A5: g is Gf -measurable by A2;

 now

 let v be object;

 assume

 A6: v in (g " { -infty });

 then

 A7: v in ( dom g) by FUNCT_1:def 7;

 (g . v) in { -infty } by A6, FUNCT_1:def 7;

 then (g . v) = -infty by TARSKI:def 1;

 then v in ( eq_dom (g, -infty )) by A7, MESFUNC1:def 15;

 hence v in (Gf /\ ( eq_dom (g, -infty ))) by A4, A7, XBOOLE_0:def 4;

 end;

 then

 A8: (g " { -infty }) c= (Gf /\ ( eq_dom (g, -infty )));

 now

 let v be object;

 assume v in (Gf /\ ( eq_dom (g, -infty )));

 then

 A9: v in ( eq_dom (g, -infty )) by XBOOLE_0:def 4;

 then (g . v) = -infty by MESFUNC1:def 15;

 then

 A10: (g . v) in { -infty } by TARSKI:def 1;

 v in ( dom g) by A9, MESFUNC1:def 15;

 hence v in (g " { -infty }) by A10, FUNCT_1:def 7;

 end;

 then

 A11: (Gf /\ ( eq_dom (g, -infty ))) c= (g " { -infty });

 (Gf /\ ( eq_dom (g, -infty ))) in S by A5, MESFUNC1: 34;

 then

 A12: (g " { -infty }) in S by A8, A11, XBOOLE_0:def 10;

 now

 let v be object;

 assume

 A13: v in (g " { +infty });

 then

 A14: v in ( dom g) by FUNCT_1:def 7;

 (g . v) in { +infty } by A13, FUNCT_1:def 7;

 then (g . v) = +infty by TARSKI:def 1;

 then v in ( eq_dom (g, +infty )) by A14, MESFUNC1:def 15;

 hence v in (Gf /\ ( eq_dom (g, +infty ))) by A4, A14, XBOOLE_0:def 4;

 end;

 then

 A15: (g " { +infty }) c= (Gf /\ ( eq_dom (g, +infty )));

 now

 let v be object;

 assume v in (Gf /\ ( eq_dom (g, +infty )));

 then

 A16: v in ( eq_dom (g, +infty )) by XBOOLE_0:def 4;

 then (g . v) = +infty by MESFUNC1:def 15;

 then

 A17: (g . v) in { +infty } by TARSKI:def 1;

 v in ( dom g) by A16, MESFUNC1:def 15;

 hence v in (g " { +infty }) by A17, FUNCT_1:def 7;

 end;

 then

 A18: (Gf /\ ( eq_dom (g, +infty ))) c= (g " { +infty });



 A19: ((f " { +infty }) /\ (g " { -infty })) c= (g " { -infty }) by XBOOLE_1: 17;



 A20: ((f " { -infty }) /\ (g " { +infty })) c= (f " { -infty }) by XBOOLE_1: 17;



 A21: ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 (Gf /\ ( eq_dom (g, +infty ))) in S by A4, A5, MESFUNC1: 33;

 then

 A22: (g " { +infty }) in S by A15, A18, XBOOLE_0:def 10;

 then

 reconsider NG = ((g " { +infty }) \/ (g " { -infty })) as Element of S by A12, PROB_1: 3;

 consider E0 be Element of S such that

 A23: E0 = ( dom f) and

 A24: f is E0 -measurable by A1;



 A25: (E0 /\ ( eq_dom (f, +infty ))) in S by A23, A24, MESFUNC1: 33;

 now

 let v be object;

 assume v in (E0 /\ ( eq_dom (f, +infty )));

 then

 A26: v in ( eq_dom (f, +infty )) by XBOOLE_0:def 4;

 then (f . v) = +infty by MESFUNC1:def 15;

 then

 A27: (f . v) in { +infty } by TARSKI:def 1;

 v in ( dom f) by A26, MESFUNC1:def 15;

 hence v in (f " { +infty }) by A27, FUNCT_1:def 7;

 end;

 then

 A28: (E0 /\ ( eq_dom (f, +infty ))) c= (f " { +infty });

 now

 let v be object;

 assume

 A29: v in (f " { +infty });

 then

 A30: v in ( dom f) by FUNCT_1:def 7;

 (f . v) in { +infty } by A29, FUNCT_1:def 7;

 then (f . v) = +infty by TARSKI:def 1;

 then v in ( eq_dom (f, +infty )) by A30, MESFUNC1:def 15;

 hence v in (E0 /\ ( eq_dom (f, +infty ))) by A23, A30, XBOOLE_0:def 4;

 end;

 then (f " { +infty }) c= (E0 /\ ( eq_dom (f, +infty )));

 then

 A31: (f " { +infty }) = (E0 /\ ( eq_dom (f, +infty ))) by A28;

 now

 let v be object;

 assume v in (E0 /\ ( eq_dom (f, -infty )));

 then

 A32: v in ( eq_dom (f, -infty )) by XBOOLE_0:def 4;

 then (f . v) = -infty by MESFUNC1:def 15;

 then

 A33: (f . v) in { -infty } by TARSKI:def 1;

 v in ( dom f) by A32, MESFUNC1:def 15;

 hence v in (f " { -infty }) by A33, FUNCT_1:def 7;

 end;

 then

 A34: (E0 /\ ( eq_dom (f, -infty ))) c= (f " { -infty });

 now

 let v be object;

 assume

 A35: v in (f " { -infty });

 then

 A36: v in ( dom f) by FUNCT_1:def 7;

 (f . v) in { -infty } by A35, FUNCT_1:def 7;

 then (f . v) = -infty by TARSKI:def 1;

 then v in ( eq_dom (f, -infty )) by A36, MESFUNC1:def 15;

 hence v in (E0 /\ ( eq_dom (f, -infty ))) by A23, A36, XBOOLE_0:def 4;

 end;

 then

 A37: (f " { -infty }) c= (E0 /\ ( eq_dom (f, -infty )));

 then

 A38: (f " { -infty }) = (E0 /\ ( eq_dom (f, -infty ))) by A34;



 A39: (E0 /\ ( eq_dom (f, -infty ))) in S by A24, MESFUNC1: 34;

 then

 A40: (f " { -infty }) in S by A37, A34, XBOOLE_0:def 10;

 then

 reconsider NF = ((f " { +infty }) \/ (f " { -infty })) as Element of S by A25, A31, PROB_1: 3;

 reconsider NFG = (NF \/ NG) as Element of S;

 reconsider E = (E0 \ NFG) as Element of S;

 reconsider DFPG = ( dom (f + g)) as Element of S by A1, A2, A25, A31, A40, A22, A12, Th46;

 set g1 = (g | E);

 set f1 = (f | E);



 A41: (( dom f) /\ E) = E by A23, XBOOLE_1: 28, XBOOLE_1: 36;

 (g " { -infty }) c= NG by XBOOLE_1: 7;

 then

 A42: ((f " { +infty }) /\ (g " { -infty })) c= NG by A19;

 (f " { -infty }) c= NF by XBOOLE_1: 7;

 then ((f " { -infty }) /\ (g " { +infty })) c= NF by A20;

 then (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty }))) c= (NF \/ NG) by A42, XBOOLE_1: 13;

 then

 A43: E c= ( dom (f + g)) by A3, A23, A21, XBOOLE_1: 34;

 then

 A44: ((f + g) | E) = (f1 + g1) by Th29;



 A45: ( dom (f1 + g1)) = E by A43, Th29;



 A46: E = ( dom (f1 + g1)) by A43, Th29;



 A47: for r be Real holds (DFPG /\ ( less_dom ((f + g),r))) = (((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) \/ ((g " { -infty }) /\ (DFPG \ (f " { +infty }))))

 proof

 let r be Real;

 set SL = (DFPG /\ ( less_dom ((f + g),r)));

 set SR = (((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) \/ ((g " { -infty }) /\ (DFPG \ (f " { +infty }))));

 A48:

 now

 let x be object;

 reconsider xx = x as set by TARSKI: 1;

 assume x in SR;

 then

 A49: x in ((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) or x in ((g " { -infty }) /\ (DFPG \ (f " { +infty }))) by XBOOLE_0:def 3;

 per cases by A49, XBOOLE_0:def 3;

 suppose

 A50: x in (E /\ ( less_dom ((f1 + g1),r)));

 then

 A51: x in E by XBOOLE_0:def 4;

 x in ( less_dom ((f1 + g1),r)) by A50, XBOOLE_0:def 4;

 then ((f1 + g1) . xx) < r by MESFUNC1:def 11;

 then ((f + g) . xx) < r by A44, A45, A51, FUNCT_1: 47;

 then x in ( less_dom ((f + g),r)) by A43, A51, MESFUNC1:def 11;

 hence x in SL by A43, A51, XBOOLE_0:def 4;

 end;

 suppose

 A52: x in ((f " { -infty }) /\ (DFPG \ (g " { +infty }))) or x in ((g " { -infty }) /\ (DFPG \ (f " { +infty })));

 per cases by A52;

 suppose

 A53: x in ((f " { -infty }) /\ (DFPG \ (g " { +infty })));

 r in REAL by XREAL_0:def 1;

 then

 A54: -infty < r by XXREAL_0: 12;



 A55: x in (DFPG \ (g " { +infty })) by A53, XBOOLE_0:def 4;

 then

 A56: x in DFPG by XBOOLE_0:def 5;

 then x in ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then

 A57: x in (( dom f) /\ ( dom g)) by XBOOLE_0:def 5;

 then x in ( dom g) by XBOOLE_0:def 4;

 then x in (g " { +infty }) iff (g . x) in { +infty } by FUNCT_1:def 7;

 then

 A58: x in (g " { +infty }) iff (g . x) = +infty by TARSKI:def 1;

 x in ( dom f) by A57, XBOOLE_0:def 4;

 then x in (f " { -infty }) iff (f . x) in { -infty } by FUNCT_1:def 7;

 then x in (f " { -infty }) iff (f . x) = -infty by TARSKI:def 1;

 then ((f . xx) + (g . xx)) = -infty by A53, A55, A58, XBOOLE_0:def 4, XBOOLE_0:def 5, XXREAL_3:def 2;

 then ((f + g) . xx) < r by A56, A54, MESFUNC1:def 3;

 then x in ( less_dom ((f + g),r)) by A56, MESFUNC1:def 11;

 hence x in (DFPG /\ ( less_dom ((f + g),r))) by A56, XBOOLE_0:def 4;

 end;

 suppose

 A59: x in ((g " { -infty }) /\ (DFPG \ (f " { +infty })));

 r in REAL by XREAL_0:def 1;

 then

 A60: -infty < r by XXREAL_0: 12;



 A61: x in (DFPG \ (f " { +infty })) by A59, XBOOLE_0:def 4;

 then

 A62: x in DFPG by XBOOLE_0:def 5;



 A63: x in DFPG by A61, XBOOLE_0:def 5;

 then x in ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then

 A64: x in (( dom f) /\ ( dom g)) by XBOOLE_0:def 5;

 then x in ( dom g) by XBOOLE_0:def 4;

 then x in (g " { -infty }) iff (g . x) in { -infty } by FUNCT_1:def 7;

 then

 A65: x in (g " { -infty }) iff (g . x) = -infty by TARSKI:def 1;

 x in ( dom f) by A64, XBOOLE_0:def 4;

 then x in (f " { +infty }) iff (f . x) in { +infty } by FUNCT_1:def 7;

 then x in (f " { +infty }) iff (f . x) = +infty by TARSKI:def 1;

 then ((f . xx) + (g . xx)) = -infty by A59, A61, A65, XBOOLE_0:def 4, XBOOLE_0:def 5, XXREAL_3:def 2;

 then ((f + g) . xx) < r by A62, A60, MESFUNC1:def 3;

 then x in ( less_dom ((f + g),r)) by A62, MESFUNC1:def 11;

 hence x in (DFPG /\ ( less_dom ((f + g),r))) by A63, XBOOLE_0:def 4;

 end;

 end;

 end;

 now

 let x be object;

 reconsider xx = x as set by TARSKI: 1;

 assume

 A66: x in SL;

 then

 A67: x in DFPG by XBOOLE_0:def 4;

 then

 A68: x in ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then

 A69: not x in (((f " { +infty }) /\ (g " { -infty })) \/ ((f " { -infty }) /\ (g " { +infty }))) by XBOOLE_0:def 5;

 then

 A70: not x in ((f " { +infty }) /\ (g " { -infty })) by XBOOLE_0:def 3;

 x in ( less_dom ((f + g),r)) by A66, XBOOLE_0:def 4;

 then

 A71: ((f + g) . xx) < r by MESFUNC1:def 11;

 then

 A72: ((f . xx) + (g . xx)) < r by A67, MESFUNC1:def 3;



 A73: not x in ((f " { -infty }) /\ (g " { +infty })) by A69, XBOOLE_0:def 3;



 A74: x in (( dom f) /\ ( dom g)) by A68, XBOOLE_0:def 5;

 then

 A75: x in ( dom f) by XBOOLE_0:def 4;

 then

 A76: x in (f " { -infty }) iff (f . x) in { -infty } by FUNCT_1:def 7;



 A77: x in (f " { +infty }) iff (f . x) in { +infty } by A75, FUNCT_1:def 7;

 then

 A78: x in (f " { +infty }) iff (f . x) = +infty by TARSKI:def 1;



 A79: x in ( dom g) by A74, XBOOLE_0:def 4;

 then

 A80: x in (g " { -infty }) iff (g . x) in { -infty } by FUNCT_1:def 7;



 A81: x in (g " { +infty }) iff (g . x) in { +infty } by A79, FUNCT_1:def 7;

 then

 A82: x in (g " { +infty }) iff (g . x) = +infty by TARSKI:def 1;

 per cases ;

 suppose

 A83: (f . x) = -infty ;

 then x in (DFPG \ (g " { +infty })) by A67, A76, A81, A73, TARSKI:def 1, XBOOLE_0:def 4, XBOOLE_0:def 5;

 then x in ((f " { -infty }) /\ (DFPG \ (g " { +infty }))) by A76, A83, TARSKI:def 1, XBOOLE_0:def 4;

 then x in ((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) by XBOOLE_0:def 3;

 hence x in SR by XBOOLE_0:def 3;

 end;

 suppose

 A84: (f . x) <> -infty ;

 per cases ;

 suppose

 A85: (g . x) = -infty ;

 then x in (DFPG \ (f " { +infty })) by A67, A77, A80, A70, TARSKI:def 1, XBOOLE_0:def 4, XBOOLE_0:def 5;

 then x in ((g " { -infty }) /\ (DFPG \ (f " { +infty }))) by A80, A85, TARSKI:def 1, XBOOLE_0:def 4;

 hence x in SR by XBOOLE_0:def 3;

 end;

 suppose

 A86: (g . x) <> -infty ;

 then not x in ((f " { -infty }) \/ (f " { +infty })) by A76, A78, A72, A84, TARSKI:def 1, XBOOLE_0:def 3, XXREAL_3: 52;

 then not x in (((f " { -infty }) \/ (f " { +infty })) \/ (g " { -infty })) by A80, A86, TARSKI:def 1, XBOOLE_0:def 3;

 then not x in ((((f " { -infty }) \/ (f " { +infty })) \/ (g " { -infty })) \/ (g " { +infty })) by A82, A72, A84, XBOOLE_0:def 3, XXREAL_3: 52;

 then not x in NFG by XBOOLE_1: 4;

 then

 A87: x in E by A23, A75, XBOOLE_0:def 5;

 then ((f1 + g1) . x) = ((f + g) . x) by A44, A45, FUNCT_1: 47;

 then x in ( less_dom ((f1 + g1),r)) by A46, A71, A87, MESFUNC1:def 11;

 then x in (E /\ ( less_dom ((f1 + g1),r))) by A87, XBOOLE_0:def 4;

 then x in ((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) by XBOOLE_0:def 3;

 hence x in SR by XBOOLE_0:def 3;

 end;

 end;

 end;

 hence thesis by A48, TARSKI: 2;

 end;

 A88:

 now

 let x be set;

 for x be object st x in ( dom f) holds (f . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider ff = f as Function of ( dom f), ExtREAL by FUNCT_2: 3;

 assume

 A89: x in ( dom f1);

 then

 A90: x in (( dom f) /\ E) by RELAT_1: 61;

 then

 A91: x in ( dom f) by XBOOLE_0:def 4;

 x in E by A90, XBOOLE_0:def 4;

 then

 A92: not x in NFG by XBOOLE_0:def 5;

 A93:

 now

 assume (f1 . x) = -infty ;

 then (f . x) = -infty by A89, FUNCT_1: 47;

 then (ff . x) in { -infty } by TARSKI:def 1;

 then

 A94: x in (ff " { -infty }) by A91, FUNCT_2: 38;

 (f " { -infty }) c= NF by XBOOLE_1: 7;

 hence contradiction by A92, A94, XBOOLE_0:def 3;

 end;

 now

 assume (f1 . x) = +infty ;

 then (f . x) = +infty by A89, FUNCT_1: 47;

 then (f . x) in { +infty } by TARSKI:def 1;

 then

 A95: x in (ff " { +infty }) by A91, FUNCT_2: 38;

 (f " { +infty }) c= NF by XBOOLE_1: 7;

 hence contradiction by A92, A95, XBOOLE_0:def 3;

 end;

 hence -infty < (f1 . x) & (f1 . x) < +infty by A93, XXREAL_0: 4, XXREAL_0: 6;

 end;

 now

 let x be Element of X;



 A96: ( - +infty ) = -infty by XXREAL_3:def 3;

 assume

 A97: x in ( dom f1);

 then

 A98: (f1 . x) < +infty by A88;

 -infty < (f1 . x) by A88, A97;

 hence |.(f1 . x).| < +infty by A98, A96, EXTREAL1: 22;

 end;

 then

 A99: f1 is real-valued by MESFUNC2:def 1;

 A100:

 now

 let x be set;

 for x be object st x in ( dom g) holds (g . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider gg = g as Function of ( dom g), ExtREAL by FUNCT_2: 3;

 assume

 A101: x in ( dom g1);

 then

 A102: x in (( dom g) /\ E) by RELAT_1: 61;

 then

 A103: x in ( dom g) by XBOOLE_0:def 4;

 x in E by A102, XBOOLE_0:def 4;

 then

 A104: not x in NFG by XBOOLE_0:def 5;

 A105:

 now

 assume (g1 . x) = -infty ;

 then (g . x) = -infty by A101, FUNCT_1: 47;

 then (gg . x) in { -infty } by TARSKI:def 1;

 then

 A106: x in (gg " { -infty }) by A103, FUNCT_2: 38;

 (g " { -infty }) c= NG by XBOOLE_1: 7;

 hence contradiction by A104, A106, XBOOLE_0:def 3;

 end;

 now

 assume (g1 . x) = +infty ;

 then (g . x) = +infty by A101, FUNCT_1: 47;

 then (gg . x) in { +infty } by TARSKI:def 1;

 then

 A107: x in (gg " { +infty }) by A103, FUNCT_2: 38;

 (g " { +infty }) c= NG by XBOOLE_1: 7;

 hence contradiction by A104, A107, XBOOLE_0:def 3;

 end;

 hence -infty < (g1 . x) & (g1 . x) < +infty by A105, XXREAL_0: 4, XXREAL_0: 6;

 end;

 now

 let x be Element of X;



 A108: ( - +infty ) = -infty by XXREAL_3:def 3;

 assume

 A109: x in ( dom g1);

 then

 A110: (g1 . x) < +infty by A100;

 -infty < (g1 . x) by A100, A109;

 hence |.(g1 . x).| < +infty by A110, A108, EXTREAL1: 22;

 end;

 then

 A111: g1 is real-valued by MESFUNC2:def 1;

 f is E -measurable by A1, A23, MESFUNC1: 30, XBOOLE_1: 36;

 then

 A112: f1 is E -measurable by A41, Th42;



 A113: (( dom g) /\ E) = E by A3, A23, XBOOLE_1: 28, XBOOLE_1: 36;

 g is E -measurable by A2, A3, A23, MESFUNC1: 30, XBOOLE_1: 36;

 then g1 is E -measurable by A113, Th42;

 then

 A114: (f1 + g1) is E -measurable by A112, A99, A111, MESFUNC2: 7;

 now

 let r be Real;



 A115: (E /\ ( less_dom ((f1 + g1),r))) in S by A114, MESFUNC1:def 16;

 (DFPG \ (f " { +infty })) in S by A25, A31, PROB_1: 6;

 then

 A116: ((g " { -infty }) /\ (DFPG \ (f " { +infty }))) in S by A12, FINSUB_1:def 2;

 (DFPG \ (g " { +infty })) in S by A22, PROB_1: 6;

 then ((f " { -infty }) /\ (DFPG \ (g " { +infty }))) in S by A39, A38, FINSUB_1:def 2;

 then (((f " { -infty }) /\ (DFPG \ (g " { +infty }))) \/ ((g " { -infty }) /\ (DFPG \ (f " { +infty })))) in S by A116, PROB_1: 3;

 then

 A117: ((E /\ ( less_dom ((f1 + g1),r))) \/ (((f " { -infty }) /\ (DFPG \ (g " { +infty }))) \/ ((g " { -infty }) /\ (DFPG \ (f " { +infty }))))) in S by A115, PROB_1: 3;

 (DFPG /\ ( less_dom ((f + g),r))) = (((E /\ ( less_dom ((f1 + g1),r))) \/ ((f " { -infty }) /\ (DFPG \ (g " { +infty })))) \/ ((g " { -infty }) /\ (DFPG \ (f " { +infty })))) by A47;

 hence (DFPG /\ ( less_dom ((f + g),r))) in S by A117, XBOOLE_1: 4;

 end;

 then (f + g) is DFPG -measurable by MESFUNC1:def 16;

 hence thesis;

 end;

 theorem :: MESFUNC5:47



 Th47: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable) & (ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable) holds ex E be Element of S st E = ( dom (f + g)) & (f + g) is E -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable and

 A2: ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable;

 consider E1 be Element of S such that

 A3: E1 = ( dom f) and

 A4: f is E1 -measurable by A1;

 consider E2 be Element of S such that

 A5: E2 = ( dom g) and

 A6: g is E2 -measurable by A2;

 set E3 = (E1 /\ E2);

 set g1 = (g | E3);



 A7: (g1 " { -infty }) = (E3 /\ (g " { -infty })) by FUNCT_1: 70;

 set f1 = (f | E3);

 ( dom f1) = (( dom f) /\ E3) by RELAT_1: 61;

 then

 A8: ( dom f1) = E3 by A3, XBOOLE_1: 17, XBOOLE_1: 28;

 g is E3 -measurable by A6, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A9: g1 is E3 -measurable by Lm6;



 A10: (g1 " { +infty }) = (E3 /\ (g " { +infty })) by FUNCT_1: 70;

 ( dom g1) = (( dom g) /\ E3) by RELAT_1: 61;

 then

 A11: ( dom g1) = E3 by A5, XBOOLE_1: 17, XBOOLE_1: 28;

 (f1 " { +infty }) = (E3 /\ (f " { +infty })) by FUNCT_1: 70;



 then

 A12: ((f1 " { +infty }) /\ (g1 " { -infty })) = ((f " { +infty }) /\ (E3 /\ (E3 /\ (g " { -infty })))) by A7, XBOOLE_1: 16

 .= ((f " { +infty }) /\ ((E3 /\ E3) /\ (g " { -infty }))) by XBOOLE_1: 16

 .= (((f " { +infty }) /\ (g " { -infty })) /\ E3) by XBOOLE_1: 16;



 A13: ( dom (f1 + g1)) = ((( dom f1) /\ ( dom g1)) \ (((f1 " { -infty }) /\ (g1 " { +infty })) \/ ((f1 " { +infty }) /\ (g1 " { -infty })))) by MESFUNC1:def 3;

 f is E3 -measurable by A4, MESFUNC1: 30, XBOOLE_1: 17;

 then f1 is E3 -measurable by Lm6;

 then

 consider E be Element of S such that

 A14: E = ( dom (f1 + g1)) and

 A15: (f1 + g1) is E -measurable by A9, A8, A11, Lm7;

 take E;



 A16: ( dom ((f + g) | E)) = (( dom (f + g)) /\ E) by RELAT_1: 61;

 (f1 " { -infty }) = (E3 /\ (f " { -infty })) by FUNCT_1: 70;



 then ((f1 " { -infty }) /\ (g1 " { +infty })) = ((f " { -infty }) /\ (E3 /\ (E3 /\ (g " { +infty })))) by A10, XBOOLE_1: 16

 .= ((f " { -infty }) /\ ((E3 /\ E3) /\ (g " { +infty }))) by XBOOLE_1: 16

 .= (((f " { -infty }) /\ (g " { +infty })) /\ E3) by XBOOLE_1: 16;

 then

 A17: (((f1 " { -infty }) /\ (g1 " { +infty })) \/ ((f1 " { +infty }) /\ (g1 " { -infty }))) = (E3 /\ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by A12, XBOOLE_1: 23;



 A18: ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then

 A19: ( dom (f + g)) = E by A3, A5, A8, A11, A14, A13, A17, XBOOLE_1: 47;

 now

 let v be Element of X;

 assume

 A20: v in ( dom ((f + g) | E));

 then

 A21: v in (( dom (f + g)) /\ E) by RELAT_1: 61;

 then

 A22: v in ( dom (f + g)) by XBOOLE_0:def 4;



 A23: (((f + g) | E) . v) = ((f + g) . v) by A20, FUNCT_1: 47

 .= ((f . v) + (g . v)) by A22, MESFUNC1:def 3;



 A24: v in E by A21, XBOOLE_0:def 4;



 A25: E c= E3 by A8, A11, A14, A13, XBOOLE_1: 36;

 ((f1 + g1) . v) = ((f1 . v) + (g1 . v)) by A14, A19, A16, A20, MESFUNC1:def 3

 .= ((f . v) + (g1 . v)) by A8, A24, A25, FUNCT_1: 47;

 hence (((f + g) | E) . v) = ((f1 + g1) . v) by A11, A24, A25, A23, FUNCT_1: 47;

 end;

 then ((f + g) | E) = (f1 + g1) by A14, A19, A16, PARTFUN1: 5;

 hence thesis by A3, A5, A8, A11, A14, A15, A13, A17, A18, Lm6, XBOOLE_1: 47;

 end;

 theorem :: MESFUNC5:48



 Th48: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st ( dom f) = A holds f is B -measurable iff f is (A /\ B) -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S such that

 A1: ( dom f) = A;

 A2:

 now

 let r be Real;

 A3:

 now

 let x be object;

 x in (A /\ ( less_dom (f,r))) iff x in A & x in ( less_dom (f,r)) by XBOOLE_0:def 4;

 hence x in (A /\ ( less_dom (f,r))) iff x in ( less_dom (f,r)) by A1, MESFUNC1:def 11;

 end;

 then

 A4: ( less_dom (f,r)) c= (A /\ ( less_dom (f,r)));

 (A /\ ( less_dom (f,r))) c= ( less_dom (f,r)) by A3;

 hence (A /\ ( less_dom (f,r))) = ( less_dom (f,r)) by A4;

 end;

 hereby

 assume

 A5: f is B -measurable;

 now

 let r be Real;

 ((A /\ B) /\ ( less_dom (f,r))) = (B /\ (A /\ ( less_dom (f,r)))) by XBOOLE_1: 16

 .= (B /\ ( less_dom (f,r))) by A2;

 hence ((A /\ B) /\ ( less_dom (f,r))) in S by A5, MESFUNC1:def 16;

 end;

 hence f is (A /\ B) -measurable by MESFUNC1:def 16;

 end;

 assume

 A6: f is (A /\ B) -measurable;

 now

 let r be Real;

 ((A /\ B) /\ ( less_dom (f,r))) = (B /\ (A /\ ( less_dom (f,r)))) by XBOOLE_1: 16

 .= (B /\ ( less_dom (f,r))) by A2;

 hence (B /\ ( less_dom (f,r))) in S by A6, MESFUNC1:def 16;

 end;

 hence thesis by MESFUNC1:def 16;

 end;

 theorem :: MESFUNC5:49



 Th49: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st ( dom f) = A) holds for c be Real, B be Element of S st f is B -measurable holds (c (#) f) is B -measurable

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;

 assume ex A be Element of S st A = ( dom f);

 then

 consider A be Element of S such that

 A1: A = ( dom f);

 let c be Real, B be Element of S;

 assume f is B -measurable;

 then f is (A /\ B) -measurable by A1, Th48;

 then

 A2: (c (#) f) is (A /\ B) -measurable by A1, MESFUNC1: 37, XBOOLE_1: 17;

 ( dom (c (#) f)) = A by A1, MESFUNC1:def 6;

 hence thesis by A2, Th48;

 end;

 begin

 definition

 mode ExtREAL_sequence is sequence of ExtREAL ;

 end

 definition

 let seq be ExtREAL_sequence;

 :: MESFUNC5:def8

 attr seq is convergent_to_finite_number means ex g be Real st for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - g) qua ExtReal.| < p;

 end

 definition

 let seq be ExtREAL_sequence;

 :: MESFUNC5:def9

 attr seq is convergent_to_+infty means for g be Real st 0 < g holds ex n be Nat st for m be Nat st n <= m holds g <= (seq . m);

 end

 definition

 let seq be ExtREAL_sequence;

 :: MESFUNC5:def10

 attr seq is convergent_to_-infty means for g be Real st g < 0 holds ex n be Nat st for m be Nat st n <= m holds (seq . m) <= g;

 end

 theorem :: MESFUNC5:50



 Th50: for seq be ExtREAL_sequence st seq is convergent_to_+infty holds not seq is convergent_to_-infty & not seq is convergent_to_finite_number

 proof

 let seq be ExtREAL_sequence;

 assume

 A1: seq is convergent_to_+infty;

 hereby

 assume seq is convergent_to_-infty;

 then

 consider n1 be Nat such that

 A2: for m be Nat st n1 <= m holds (seq . m) <= ( - 1);

 consider n2 be Nat such that

 A3: for m be Nat st n2 <= m holds 1 <= (seq . m) by A1;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 (seq . m) <= ( - 1) by A2, XXREAL_0: 25;

 hence contradiction by A3, XXREAL_0: 25;

 end;

 assume seq is convergent_to_finite_number;

 then

 consider g be Real such that

 A4: for p be Real st 0 0 ;

 then

 consider n1 be Nat such that

 A6: for m be Nat st n1 <= m holds |.((seq . m) - g).| < g by A4;

 A7:

 now

 let m be Nat;

 assume n1 <= m;

 then |.((seq . m) - g) qua ExtReal.| < g by A6;

 then ((seq . m) - g1) < g by EXTREAL1: 21;

 then (seq . m) < (g + g) by XXREAL_3: 54;

 hence (seq . m) < (2 * g);

 end;

 consider n2 be Nat such that

 A8: for m be Nat st n2 <= m holds (2 * g) <= (seq . m) by A1, A5;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 (seq . m) < (2 * g) by A7, XXREAL_0: 25;

 hence contradiction by A8, XXREAL_0: 25;

 end;

 suppose

 A9: g = 0 ;

 consider n1 be Nat such that

 A10: for m be Nat st n1 <= m holds |.((seq . m) - g).| < 1 by A4;

 consider n2 be Nat such that

 A11: for m be Nat st n2 <= m holds 1 <= (seq . m) by A1;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 reconsider jj = 1 as R_eal by XXREAL_0:def 1;

 set m = ( max (n1,n2));

 |.((seq . m) - g1).| < jj by A10, XXREAL_0: 25;

 then ((seq . m) - g1) < jj by EXTREAL1: 21;

 then (seq . m) < (1 + g) by XXREAL_3: 54;

 then (seq . m) < 1 by A9;

 hence contradiction by A11, XXREAL_0: 25;

 end;

 suppose

 A12: g < 0 ;

 consider n1 be Nat such that

 A13: for m be Nat st n1 <= m holds |.((seq . m) - g).| < ( - g1) by A4, A12;

 A14:

 now

 let m be Element of NAT ;

 assume n1 <= m;

 then |.((seq . m) - g1).| < ( - g1) by A13;

 then ((seq . m) - g1) < ( - g1) by EXTREAL1: 21;

 then (seq . m) < (g - g1) by XXREAL_3: 54;

 hence (seq . m) < 0 by XXREAL_3: 7;

 end;

 consider n2 be Nat such that

 A15: for m be Nat st n2 <= m holds 1 <= (seq . m) by A1;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 (seq . m) < 0 by A14, XXREAL_0: 25;

 hence contradiction by A15, XXREAL_0: 25;

 end;

 end;

 theorem :: MESFUNC5:51



 Th51: for seq be ExtREAL_sequence st seq is convergent_to_-infty holds not seq is convergent_to_+infty & not seq is convergent_to_finite_number

 proof

 let seq be ExtREAL_sequence;

 assume

 A1: seq is convergent_to_-infty;

 hereby

 assume seq is convergent_to_+infty;

 then

 consider n1 be Nat such that

 A2: for m be Nat st n1 <= m holds 1 <= (seq . m);

 consider n2 be Nat such that

 A3: for m be Nat st n2 <= m holds (seq . m) <= ( - 1) by A1;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 (seq . m) <= ( - 1) by A3, XXREAL_0: 25;

 hence contradiction by A2, XXREAL_0: 25;

 end;

 assume seq is convergent_to_finite_number;

 then

 consider g be Real such that

 A4: for p be Real st 0 0 ;

 then

 consider n1 be Nat such that

 A6: for m be Nat st n1 <= m holds |.((seq . m) - g).| < g by A4;

 A7:

 now

 let m be Element of NAT ;

 assume n1 <= m;

 then |.((seq . m) - g1).| < g by A6;

 then ( - g1) < ((seq . m) - g) by EXTREAL1: 21;

 then (( - g) + g) < (seq . m) by XXREAL_3: 53;

 hence 0 < (seq . m);

 end;

 consider n2 be Nat such that

 A8: for m be Nat st n2 <= m holds (seq . m) <= ( - g1) by A1, A5;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 0 < (seq . m) by A7, XXREAL_0: 25;

 hence contradiction by A5, A8, XXREAL_0: 25;

 end;

 suppose

 A9: g = 0 ;

 consider n1 be Nat such that

 A10: for m be Nat st n1 <= m holds |.((seq . m) - g).| < 1 by A4;

 consider n2 be Nat such that

 A11: for m be Nat st n2 <= m holds (seq . m) <= ( - 1) by A1;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 reconsider jj = 1 as R_eal by XXREAL_0:def 1;

 set m = ( max (n1,n2));

 |.((seq . m) - g1).| < 1 by A10, XXREAL_0: 25;

 then ( - jj) < ((seq . m) - g1) by EXTREAL1: 21;

 then (( - 1) + g) < (seq . m) by XXREAL_3: 53;

 then ( - 1) < (seq . m) by A9;

 then ( - 1) < (seq . m);

 hence contradiction by A11, XXREAL_0: 25;

 end;

 suppose

 A12: g < 0 ;

 then

 consider n1 be Nat such that

 A13: for m be Nat st n1 <= m holds |.((seq . m) - g).| < ( - g1) by A4;

 A14:

 now

 let m be Element of NAT ;

 assume n1 <= m;

 then |.((seq . m) - g1).| < ( - g1) by A13;

 then ( - ( - g1)) < ((seq . m) - g) by EXTREAL1: 21;

 then (g1 + g) < (seq . m) by XXREAL_3: 53;

 then (g + g) < (seq . m);

 hence (2 * g) < (seq . m);

 end;

 consider n2 be Nat such that

 A15: for m be Nat st n2 <= m holds (seq . m) <= (2 * g) by A1, A12;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));

 (seq . m) <= (2 * g) by A15, XXREAL_0: 25;

 hence contradiction by A14, XXREAL_0: 25;

 end;

 end;

 definition

 let seq be ExtREAL_sequence;

 :: MESFUNC5:def11

 attr seq is convergent means seq is convergent_to_finite_number or seq is convergent_to_+infty or seq is convergent_to_-infty;

 end

 definition

 let seq be ExtREAL_sequence;

 assume

 A1: seq is convergent;

 :: MESFUNC5:def12

 func lim seq -> R_eal means

 : Def12: (ex g be Real st it = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - it ).| < p) & seq is convergent_to_finite_number) or it = +infty & seq is convergent_to_+infty or it = -infty & seq is convergent_to_-infty;

 existence

 proof

 per cases by A1;

 suppose

 A2: seq is convergent_to_finite_number;

 then

 consider g be Real such that

 A3: for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - g).| < p;

 reconsider g as R_eal by XXREAL_0:def 1;

 take g;

 thus thesis by A2, A3;

 end;

 suppose seq is convergent_to_+infty or seq is convergent_to_-infty;

 hence thesis;

 end;

 end;

 uniqueness

 proof

 defpred P[ R_eal] means (ex g be Real st $1 = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - $1).| < p) & seq is convergent_to_finite_number) or ($1 = +infty & seq is convergent_to_+infty) or ($1 = -infty & seq is convergent_to_-infty);

 given g1,g2 be R_eal such that

 A4: P[g1] and

 A5: P[g2] and

 A6: g1 <> g2;

 per cases by A1;

 suppose

 A7: seq is convergent_to_finite_number;

 then

 consider g be Real such that

 A8: g1 = g and

 A9: for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - g1).| < p and seq is convergent_to_finite_number by A4, Th50, Th51;

 consider h be Real such that

 A10: g2 = h and

 A11: for p be Real st 0 0 by A6, A8, A10;

 then

 A12: |.(g - h).| > 0 ;

 then

 consider n1 be Nat such that

 A13: for m be Nat st n1 <= m holds |.((seq . m) - g1).| < ( |.(g - h).| / 2) by A9;

 consider n2 be Nat such that

 A14: for m be Nat st n2 <= m holds |.((seq . m) - g2).| < ( |.(g - h).| / 2) by A11, A12;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set m = ( max (n1,n2));



 A15: |.((seq . m) - g1).| < ( |.(g - h).| / 2) by A13, XXREAL_0: 25;



 A16: |.((seq . m) - g2).| < ( |.(g - h).| / 2) by A14, XXREAL_0: 25;

 reconsider g, h as Complex;



 A17: ((seq . m) - g2) < ( |.(g - h).| / 2) by A16, EXTREAL1: 21;



 A18: ( - ( |.(g - h).| / 2) qua ExtReal) < ((seq . m) - g2) by A16, EXTREAL1: 21;

 then

 reconsider w = ((seq . m) - g2) as Element of REAL by A17, XXREAL_0: 48;



 A19: ((seq . m) - g2) in REAL by A18, A17, XXREAL_0: 48;

 then

 A20: (seq . m) <> +infty by A10;



 A21: (( - (seq . m)) + g1) = ( - ((seq . m) - g1)) by XXREAL_3: 26;

 then

 A22: |.(( - (seq . m)) + g1).| < ( |.(g - h).| / 2) by A15, EXTREAL1: 29;

 then

 A23: (( - (seq . m)) + g1) < ( |.(g - h).| / 2) by EXTREAL1: 21;

 ( - ( |.(g - h).| / 2) qua ExtReal) < (( - (seq . m)) + g1) by A22, EXTREAL1: 21;

 then

 A24: (( - (seq . m)) + g1) in REAL by A23, XXREAL_0: 48;



 A25: (seq . m) <> -infty by A10, A19;

 |.(g1 - g2).| = |.((g1 + 0. ) - g2).| by XXREAL_3: 4

 .= |.((g1 + ((seq . m) + ( - (seq . m)))) - g2).| by XXREAL_3: 7

 .= |.(((( - (seq . m)) + g1) + (seq . m)) - g2).| by A8, A20, A25, XXREAL_3: 29

 .= |.((( - (seq . m)) + g1) + ((seq . m) - g2)).| by A10, A24, XXREAL_3: 30;

 then |.(g1 - g2).| <= ( |.(( - (seq . m)) + g1).| + |.((seq . m) - g2).|) by EXTREAL1: 24;

 then

 A26: |.(g1 - g2).| <= ( |.((seq . m) - g1).| + |.((seq . m) - g2).|) by A21, EXTREAL1: 29;

 |.w.| in REAL by XREAL_0:def 1;

 then |.((seq . m) - g2).| in REAL ;

 then

 A27: ( |.((seq . m) - g1).| + |.((seq . m) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + |.((seq . m) - g2).|) by A15, XXREAL_3: 43;

 ( |.(g - h).| / 2) in REAL by XREAL_0:def 1;

 then ( |.(g - h).| / 2) in REAL ;

 then (( |.(g - h).| / 2) qua ExtReal + |.((seq . m) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + ( |.(g - h).| / 2)) by A16, XXREAL_3: 43;

 then

 A28: ( |.((seq . m) - g1).| + |.((seq . m) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + ( |.(g - h).| / 2)) by A27, XXREAL_0: 2;

 (g - h) = (g1 - g2) by A8, A10, SUPINF_2: 3;

 then |.(g - h).| = |.(g1 - g2).| by EXTREAL1: 12;

 then |.(g - h).| < (( |.(g - h).| / 2) + ( |.(g - h).| / 2)) by A28, A26;

 hence contradiction;

 end;

 suppose seq is convergent_to_+infty or seq is convergent_to_-infty;

 hence contradiction by A4, A5, A6, Th50, Th51;

 end;

 end;

 end

 theorem :: MESFUNC5:52



 Th52: for seq be ExtREAL_sequence, r be Real st (for n be Nat holds (seq . n) = r) holds seq is convergent_to_finite_number & ( lim seq) = r

 proof

 let seq be ExtREAL_sequence;

 let r be Real;

 assume

 A1: for n be Nat holds (seq . n) = r;

 A2:

 now

 reconsider n = 1 as Nat;

 let p be Real;

 assume

 A3: 0 < p;

 take n;

 let m be Nat such that n <= m;

 (seq . m) = r by A1;

 then ((seq . m) - r) = 0 by XXREAL_3: 7;

 then |.((seq . m) - r).| = 0 by EXTREAL1: 16;

 hence |.((seq . m) - r).| < p by A3;

 end;

 hence

 A4: seq is convergent_to_finite_number;

 then

 A5: seq is convergent;

 reconsider r as R_eal by XXREAL_0:def 1;

 (ex g be Real st r = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - r).| < p) & seq is convergent_to_finite_number) by A2, A4;

 hence thesis by Def12, A5;

 end;

 theorem :: MESFUNC5:53



 Th53: for F be FinSequence of ExtREAL st (for n be Nat st n in ( dom F) holds 0 <= (F . n)) holds 0 <= ( Sum F)

 proof

 let F be FinSequence of ExtREAL ;

 consider sumf be sequence of ExtREAL such that

 A1: ( Sum F) = (sumf . ( len F)) and

 A2: (sumf . 0 ) = 0 and

 A3: for n be Nat st n < ( len F) holds (sumf . (n + 1)) = ((sumf . n) + (F . (n + 1))) by EXTREAL1:def 2;

 defpred P[ Nat] means $1 <= ( len F) implies 0 <= (sumf . $1);

 assume

 A4: for n be Nat st n in ( dom F) holds 0 <= (F . n);



 A5: for n be Nat st P[n] holds P[(n + 1)]

 proof

 let n be Nat;

 assume

 A6: P[n];

 assume

 A7: (n + 1) <= ( len F);

 reconsider n as Element of NAT by ORDINAL1:def 12;

 1 <= (n + 1) by NAT_1: 11;

 then (n + 1) in ( Seg ( len F)) by A7;

 then (n + 1) in ( dom F) by FINSEQ_1:def 3;

 then

 A8: 0 <= (F . (n + 1)) by A4;

 n < ( len F) by A7, NAT_1: 13;

 then (sumf . (n + 1)) = ((sumf . n) + (F . (n + 1))) by A3;

 hence thesis by A6, A7, A8, NAT_1: 13;

 end;



 A9: P[ 0 ] by A2;

 for n be Nat holds P[n] from NAT_1:sch 2( A9, A5);

 hence thesis by A1;

 end;

 theorem :: MESFUNC5:54



 Th54: for L be ExtREAL_sequence st (for n,m be Nat st n <= m holds (L . n) <= (L . m)) holds L is convergent & ( lim L) = ( sup ( rng L))

 proof

 let L be ExtREAL_sequence;

 assume

 A1: for n,m be Nat st n <= m holds (L . n) <= (L . m);

 A2:

 now

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 ( dom L) = NAT by FUNCT_2:def 1;

 then

 A3: (L . n1) in ( rng L) by FUNCT_1:def 3;

 ( sup ( rng L)) is UpperBound of ( rng L) by XXREAL_2:def 3;

 hence (L . n) <= ( sup ( rng L)) by A3, XXREAL_2:def 1;

 end;

 per cases ;

 suppose

 A4: not ex k0 be Nat st -infty < (L . k0);

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex n be object st n in ( dom L) & x = (L . n) by FUNCT_1:def 3;

 hence x <= -infty by A4;

 end;

 then

 A5: -infty is UpperBound of ( rng L) by XXREAL_2:def 1;

 for y be UpperBound of ( rng L) holds -infty <= y by XXREAL_0: 5;

 then

 A6: -infty = ( sup ( rng L)) by A5, XXREAL_2:def 3;

 now

 reconsider N0 = 0 as Nat;

 let K be Real such that K < 0 ;

 take N0;

 let n be Nat such that N0 <= n;

 (L . n) = -infty by A4, XXREAL_0: 6;

 hence (L . n) <= K by XXREAL_0: 5;

 end;

 then

 A7: L is convergent_to_-infty;

 then L is convergent;

 hence thesis by A7, A6, Def12;

 end;

 suppose ex k0 be Nat st -infty < (L . k0);

 then

 consider k0 be Nat such that

 A8: -infty < (L . k0);

 reconsider k0 as Element of NAT by ORDINAL1:def 12;

 per cases ;

 suppose ex K be Real st for n be Nat holds (L . n) < K;

 then

 consider K be Real such that

 A9: for n be Nat holds (L . n) < K;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex z be object st z in ( dom L) & x = (L . z) by FUNCT_1:def 3;

 hence x <= K by A9;

 end;

 then K is UpperBound of ( rng L) by XXREAL_2:def 1;

 then

 A10: ( sup ( rng L)) <= K by XXREAL_2:def 3;

 K in REAL by XREAL_0:def 1;

 then

 A11: ( sup ( rng L)) <> +infty by A10, XXREAL_0: 9;



 A12: ( sup ( rng L)) <> -infty by A2, A8;

 then

 reconsider h = ( sup ( rng L)) as Element of REAL by A11, XXREAL_0: 14;



 A13: for p be Real st 0 < p holds ex N0 be Nat st for m be Nat st N0 <= m holds |.((L . m) - ( sup ( rng L))).| < p

 proof

 let p be Real;

 assume

 A14: 0 < p;

 reconsider e = p as R_eal by XXREAL_0:def 1;

 ( sup ( rng L)) in REAL by A12, A11, XXREAL_0: 14;

 then

 consider y be ExtReal such that

 A15: y in ( rng L) and

 A16: (( sup ( rng L)) - e) < y by A14, MEASURE6: 6;

 consider x be object such that

 A17: x in ( dom L) and

 A18: y = (L . x) by A15, FUNCT_1:def 3;

 reconsider N1 = x as Element of NAT by A17;

 set N0 = ( max (N1,k0));

 take N0;

 hereby

 let n be Nat;



 A19: N1 <= N0 by XXREAL_0: 25;

 assume N0 <= n;

 then N1 <= n by A19, XXREAL_0: 2;

 then (L . N1) <= (L . n) by A1;

 then (( sup ( rng L)) - e) < (L . n) by A16, A18, XXREAL_0: 2;

 then ( sup ( rng L)) < ((L . n) + e) by XXREAL_3: 54;

 then (( sup ( rng L)) - (L . n)) < e by XXREAL_3: 55;

 then ( - e) < ( - (( sup ( rng L)) - (L . n))) by XXREAL_3: 38;

 then

 A20: ( - e) < ((L . n) - ( sup ( rng L))) by XXREAL_3: 26;



 A21: (L . n) <= ( sup ( rng L)) by A2;

 A22:

 now

 assume

 A23: ( sup ( rng L)) = (( sup ( rng L)) + e);

 ((e + ( sup ( rng L))) + ( - ( sup ( rng L)))) = (e + (( sup ( rng L)) + ( - ( sup ( rng L))))) by A12, A11, XXREAL_3: 29

 .= (e + 0 ) by XXREAL_3: 7

 .= e by XXREAL_3: 4;

 hence contradiction by A14, A23, XXREAL_3: 7;

 end;

 (( sup ( rng L)) + 0 qua ExtReal) <= (( sup ( rng L)) + e) by A14, XXREAL_3: 36;

 then ( sup ( rng L)) <= (( sup ( rng L)) + e) by XXREAL_3: 4;

 then ( sup ( rng L)) < (( sup ( rng L)) + e) by A22, XXREAL_0: 1;

 then (L . n) < (( sup ( rng L)) + e) by A21, XXREAL_0: 2;

 then ((L . n) - ( sup ( rng L))) < e by XXREAL_3: 55;

 hence |.((L . n) - ( sup ( rng L))).| < p by A20, EXTREAL1: 22;

 end;

 end;



 A24: h = ( sup ( rng L));

 then

 A25: L is convergent_to_finite_number by A13;

 hence L is convergent;

 hence thesis by A13, A24, A25, Def12;

 end;

 suppose

 A26: not (ex K be Real st 0 < K & for n be Nat holds (L . n) < K);

 now

 let K be Real;

 assume 0 < K;

 then

 consider N0 be Nat such that

 A27: K <= (L . N0) by A26;

 now

 let n be Nat;

 assume N0 <= n;

 then (L . N0) <= (L . n) by A1;

 hence K <= (L . n) by A27, XXREAL_0: 2;

 end;

 hence ex N0 be Nat st for n be Nat st N0 <= n holds K <= (L . n);

 end;

 then

 A28: L is convergent_to_+infty;

 hence

 A29: L is convergent;

 now

 assume

 A30: ( sup ( rng L)) <> +infty ;

 (L . k0) <= ( sup ( rng L)) by A2;

 then

 reconsider h = ( sup ( rng L)) as Element of REAL by A8, A30, XXREAL_0: 14;

 set K = ( max ( 0 ,h));

 0 <= K by XXREAL_0: 25;

 then

 consider N0 be Nat such that

 A31: (K + 1) <= (L . N0) by A26;

 (h + 0 ) < (K + 1) by XREAL_1: 8, XXREAL_0: 25;

 then ( sup ( rng L)) < (L . N0) by A31, XXREAL_0: 2;

 hence contradiction by A2;

 end;

 hence thesis by A28, A29, Def12;

 end;

 end;

 end;

 theorem :: MESFUNC5:55



 Th55: for L,G be ExtREAL_sequence st (for n be Nat holds (L . n) <= (G . n)) holds ( sup ( rng L)) <= ( sup ( rng G))

 proof

 let L,G be ExtREAL_sequence;

 assume

 A1: for n be Nat holds (L . n) <= (G . n);

 A2:

 now

 let n be Element of NAT ;

 ( dom G) = NAT by FUNCT_2:def 1;

 then

 A3: (G . n) in ( rng G) by FUNCT_1:def 3;



 A4: (L . n) <= (G . n) by A1;

 ( sup ( rng G)) is UpperBound of ( rng G) by XXREAL_2:def 3;

 then (G . n) <= ( sup ( rng G)) by A3, XXREAL_2:def 1;

 hence (L . n) <= ( sup ( rng G)) by A4, XXREAL_0: 2;

 end;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex z be object st z in ( dom L) & x = (L . z) by FUNCT_1:def 3;

 hence x <= ( sup ( rng G)) by A2;

 end;

 then ( sup ( rng G)) is UpperBound of ( rng L) by XXREAL_2:def 1;

 hence thesis by XXREAL_2:def 3;

 end;

 theorem :: MESFUNC5:56



 Th56: for L be ExtREAL_sequence holds for n be Nat holds (L . n) <= ( sup ( rng L))

 proof

 let L be ExtREAL_sequence;

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 ( dom L) = NAT by FUNCT_2:def 1;

 then

 A1: (L . n) in ( rng L) by FUNCT_1:def 3;

 ( sup ( rng L)) is UpperBound of ( rng L) by XXREAL_2:def 3;

 hence thesis by A1, XXREAL_2:def 1;

 end;

 theorem :: MESFUNC5:57



 Th57: for L be ExtREAL_sequence, K be R_eal st (for n be Nat holds (L . n) <= K) holds ( sup ( rng L)) <= K

 proof

 let L be ExtREAL_sequence, K be R_eal;

 assume

 A1: for n be Nat holds (L . n) <= K;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex z be object st z in ( dom L) & x = (L . z) by FUNCT_1:def 3;

 hence x <= K by A1;

 end;

 then K is UpperBound of ( rng L) by XXREAL_2:def 1;

 hence thesis by XXREAL_2:def 3;

 end;

 theorem :: MESFUNC5:58

 for L be ExtREAL_sequence, K be R_eal st K <> +infty & (for n be Nat holds (L . n) <= K) holds ( sup ( rng L)) < +infty

 proof

 let L be ExtREAL_sequence, K be R_eal;

 assume that

 A1: K <> +infty and

 A2: for n be Nat holds (L . n) <= K;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex z be object st z in ( dom L) & x = (L . z) by FUNCT_1:def 3;

 hence x <= K by A2;

 end;

 then K is UpperBound of ( rng L) by XXREAL_2:def 1;

 then ( sup ( rng L)) <= K by XXREAL_2:def 3;

 hence thesis by A1, XXREAL_0: 2, XXREAL_0: 4;

 end;

 theorem :: MESFUNC5:59



 Th59: for L be ExtREAL_sequence st L is without-infty holds ( sup ( rng L)) <> +infty iff ex K be Real st 0 < K & for n be Nat holds (L . n) <= K

 proof

 let L be ExtREAL_sequence;

 assume L is without-infty;

 then

 A1: -infty < (L . 1);



 A2: ( dom L) = NAT by FUNCT_2:def 1;

 then

 A3: (L . 1) <= ( sup ( rng L)) by FUNCT_1: 3, XXREAL_2: 4;

 A4:

 now

 assume ( sup ( rng L)) <> +infty ;

 then not ( sup ( rng L)) in { -infty , +infty } by A1, A3, TARSKI:def 2;

 then ( sup ( rng L)) in REAL by XBOOLE_0:def 3, XXREAL_0:def 4;

 then

 reconsider S = ( sup ( rng L)) as Real;

 take K = ( max (S,1));

 thus 0 < K by XXREAL_0: 25;

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 then

 A5: (L . n) <= ( sup ( rng L)) by A2, FUNCT_1: 3, XXREAL_2: 4;

 S <= K by XXREAL_0: 25;

 hence (L . n) <= K by A5, XXREAL_0: 2;

 end;

 now

 given K be Real such that 0 < K and

 A6: for n be Nat holds (L . n) <= K;

 now

 let w be ExtReal;

 assume w in ( rng L);

 then ex v be object st v in ( dom L) & w = (L . v) by FUNCT_1:def 3;

 hence w <= K by A6;

 end;

 then K is UpperBound of ( rng L) by XXREAL_2:def 1;

 then

 A7: ( sup ( rng L)) <= K by XXREAL_2:def 3;

 K in REAL by XREAL_0:def 1;

 hence ( sup ( rng L)) <> +infty by A7, XXREAL_0: 9;

 end;

 hence thesis by A4;

 end;

 theorem :: MESFUNC5:60



 Th60: for L be ExtREAL_sequence, c be ExtReal st (for n be Nat holds (L . n) = c) holds L is convergent & ( lim L) = c & ( lim L) = ( sup ( rng L))

 proof

 let L be ExtREAL_sequence;

 let c be ExtReal;

 reconsider cc = c as R_eal by XXREAL_0:def 1;



 A1: ( dom L) = NAT by FUNCT_2:def 1;

 c in ExtREAL by XXREAL_0:def 1;

 then

 A2: c in REAL or c in { -infty , +infty } by XBOOLE_0:def 3, XXREAL_0:def 4;

 assume

 A3: for n be Nat holds (L . n) = c;

 then

 A4: (L . 1) = c;

 now

 let v be ExtReal;

 assume v in ( rng L);

 then ex n be object st n in ( dom L) & v = (L . n) by FUNCT_1:def 3;

 hence v <= c by A3;

 end;

 then

 A5: c is UpperBound of ( rng L) by XXREAL_2:def 1;

 per cases by A2, TARSKI:def 2;

 suppose c in REAL ;

 then

 reconsider rc = c as Real;

 A6:

 now

 reconsider n = 0 as Nat;

 let p be Real;

 assume

 A7: 0 < p;

 take n;

 let m be Nat such that n <= m;

 ((L . m) - rc) = ((L . m) - (L . m)) by A3;

 then ((L . m) - rc) = 0 by XXREAL_3: 7;

 hence |.((L . m) - rc).| < p by A7, EXTREAL1: 16;

 end;

 then

 A8: L is convergent_to_finite_number;

 hence L is convergent;

 then ( lim L) = cc by A6, A8, Def12;

 hence ( lim L) = c;

 hence thesis by A5, A1, A4, FUNCT_1: 3, XXREAL_2: 55;

 end;

 suppose

 A9: c = -infty ;

 for p be Real st p < 0 holds ex n be Nat st for m be Nat st n <= m holds (L . m) <= p

 proof

 let p be Real such that p < 0 ;

 take 0 ;



 A10: p in REAL by XREAL_0:def 1;

 now

 let m be Nat such that 0 <= m;

 (L . m) = -infty by A3, A9;

 hence (L . m) < p by A10, XXREAL_0: 12;

 end;

 hence thesis;

 end;

 then

 A11: L is convergent_to_-infty;

 hence L is convergent;

 hence ( lim L) = c by A9, A11, Def12;

 hence thesis by A5, A1, A4, FUNCT_1: 3, XXREAL_2: 55;

 end;

 suppose

 A12: c = +infty ;

 for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds p <= (L . m)

 proof

 let p be Real such that 0 < p;

 take 0 ;



 A13: p in REAL by XREAL_0:def 1;

 now

 let m be Nat such that 0 <= m;

 (L . m) = +infty by A3, A12;

 hence p < (L . m) by A13, XXREAL_0: 9;

 end;

 hence thesis;

 end;

 then

 A14: L is convergent_to_+infty;

 hence L is convergent;

 hence ( lim L) = c by A12, A14, Def12;

 hence thesis by A5, A1, A4, FUNCT_1: 3, XXREAL_2: 55;

 end;

 end;



 Lm8: for J be ExtREAL_sequence st J is without-infty holds ( sup ( rng J)) in REAL or ( sup ( rng J)) = +infty

 proof

 let J be ExtREAL_sequence;

 assume J is without-infty;

 then

 A1: -infty < (J . 1);

 ( dom J) = NAT by FUNCT_2:def 1;

 then not -infty is UpperBound of ( rng J) by A1, FUNCT_1: 3, XXREAL_2:def 1;

 then ( sup ( rng J)) <> -infty by XXREAL_2:def 3;

 hence thesis by XXREAL_0: 14;

 end;

 theorem :: MESFUNC5:61



 Th61: for J,K,L be ExtREAL_sequence st (for n,m be Nat st n <= m holds (J . n) <= (J . m)) & (for n,m be Nat st n <= m holds (K . n) <= (K . m)) & J is without-infty & K is without-infty & (for n be Nat holds ((J . n) + (K . n)) = (L . n)) holds L is convergent & ( lim L) = ( sup ( rng L)) & ( lim L) = (( lim J) + ( lim K)) & ( sup ( rng L)) = (( sup ( rng K)) + ( sup ( rng J)))

 proof

 let J,K,L be ExtREAL_sequence;

 assume that

 A1: for n,m be Nat st n <= m holds (J . n) <= (J . m) and

 A2: for n,m be Nat st n <= m holds (K . n) <= (K . m) and

 A3: J is without-infty and

 A4: K is without-infty and

 A5: for n be Nat holds ((J . n) + (K . n)) = (L . n);



 A6: ( dom K) = NAT by FUNCT_2:def 1;



 A7: ( dom J) = NAT by FUNCT_2:def 1;

 A8:

 now

 per cases by A3, Lm8;

 suppose

 A9: ( sup ( rng J)) in REAL ;

 then

 reconsider SJ = ( sup ( rng J)) as Real;

 per cases by A4, Lm8;

 suppose

 A10: ( sup ( rng K)) in REAL ;

 then

 reconsider SK = ( sup ( rng K)) as Real;

 A11:

 now

 let p be Real;

 assume

 A12: 0 < p;

 then

 consider SJ9 be ExtReal such that

 A13: SJ9 in ( rng J) and

 A14: (( sup ( rng J)) - (p / 2)) < SJ9 by A9, MEASURE6: 6;

 consider nj be object such that

 A15: nj in ( dom J) and

 A16: SJ9 = (J . nj) by A13, FUNCT_1:def 3;

 reconsider nj as Element of NAT by A15;

 consider SK9 be ExtReal such that

 A17: SK9 in ( rng K) and

 A18: (( sup ( rng K)) - (p / 2)) < SK9 by A10, A12, MEASURE6: 6;

 consider nk be object such that

 A19: nk in ( dom K) and

 A20: SK9 = (K . nk) by A17, FUNCT_1:def 3;

 reconsider nk as Element of NAT by A19;

 reconsider n = ( max (nj,nk)) as Nat;

 take n;

 hereby

 reconsider SJ9, SK9 as R_eal by XXREAL_0:def 1;

 let m be Nat;

 assume

 A21: n <= m;

 nk <= n by XXREAL_0: 25;

 then nk <= m by A21, XXREAL_0: 2;

 then SK9 <= (K . m) by A2, A20;

 then

 A22: (SK - (K . m)) <= (SK - SK9) by XXREAL_3: 37;

 nj <= n by XXREAL_0: 25;

 then nj <= m by A21, XXREAL_0: 2;

 then SJ9 <= (J . m) by A1, A16;

 then (SJ - (J . m)) <= (SJ - SJ9) by XXREAL_3: 37;

 then

 A23: ((SJ - (J . m)) + (SK - (K . m))) <= ((SJ - SJ9) + (SK - SK9)) by A22, XXREAL_3: 36;

 SJ in REAL by XREAL_0:def 1;

 then

 A24: SJ < +infty by XXREAL_0: 9;

 reconsider s1 = SK as Element of REAL by XREAL_0:def 1;

 reconsider m1 = m as Element of NAT by ORDINAL1:def 12;



 A25: ( - ((L . m) - (SJ + SK))) = ((SJ + SK) - (L . m)) by XXREAL_3: 26;



 A26: (p / 2) in REAL by XREAL_0:def 1;

 SK < ((p / 2) + SK9) by A18, XXREAL_3: 54;

 then (SK - SK9) < (p / 2) by XXREAL_3: 55;

 then

 A27: ((p / 2) + (SK - SK9)) < ((p / 2) + (p / 2)) by XXREAL_3: 43, A26;

 SJ < ((p / 2) + SJ9) by A14, XXREAL_3: 54;

 then

 A28: (SJ - SJ9) < (p / 2) by XXREAL_3: 55;

 nk <= n by XXREAL_0: 25;

 then nk <= m by A21, XXREAL_0: 2;

 then

 A29: (K . nk) <= (K . m) by A2;



 A30: SK in REAL by XREAL_0:def 1;

 then

 A31: SK < +infty by XXREAL_0: 9;

 (K . m1) in ( rng K) by A6, FUNCT_1: 3;

 then

 A32: (K . m) <= SK by XXREAL_2: 4;

 then

 A33: (K . m) < +infty by A30, XXREAL_0: 2, XXREAL_0: 9;

 -infty < SK9 by A4, A20;

 then

 reconsider s0 = SK9 as Element of REAL by A20, A33, A29, XXREAL_0: 14;



 A34: (L . m) = ((J . m) + (K . m)) by A5;

 (J . m1) in ( rng J) by A7, FUNCT_1: 3;

 then

 A35: (J . m) <= SJ by XXREAL_2: 4;

 then ((J . m) + (K . m)) <= (SJ + SK) by A32, XXREAL_3: 36;

 then ((L . m) - (SJ + SK)) <= 0 by A34, A25, XXREAL_3: 40;

 then

 A36: |.((L . m) - (SJ + SK)).| = ((SJ + SK) - (L . m)) by A25, EXTREAL1: 18;

 (SK - SK9) = (s1 - s0) by SUPINF_2: 3;

 then ((SJ - SJ9) + (SK - SK9)) < ((p / 2) + (SK - SK9)) by A28, XXREAL_3: 43;

 then

 A37: ((SJ - SJ9) + (SK - SK9)) < ((p / 2) + (p / 2)) by A27, XXREAL_0: 2;

 -infty < (K . m) by A4;



 then ((SJ - (J . m)) + (SK - (K . m))) = (((SJ - (J . m)) + SK) - (K . m)) by A33, XXREAL_3: 30

 .= (((SK + SJ) - (J . m)) - (K . m)) by XXREAL_3: 30

 .= ((SK + SJ) - ((J . m) + (K . m))) by A24, A31, A35, A32, XXREAL_3: 31

 .= ((SK + SJ) - (L . m)) by A5;

 hence |.((L . m) - (SJ + SK)).| < p by A36, A37, A23, XXREAL_0: 2;

 end;

 end;

 then

 A38: L is convergent_to_finite_number;

 hence L is convergent;

 hence ( lim L) = (( sup ( rng J)) + ( sup ( rng K))) by A11, A38, Def12;

 end;

 suppose

 A39: ( sup ( rng K)) = +infty ;

 for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds p <= (L . m)

 proof

 reconsider supj = ( sup ( rng J)) as Element of REAL by A9;

 let p be Real;

 reconsider p92 = (p / 2), p9 = p as Element of REAL by XREAL_0:def 1;

 assume 0 < p;

 then

 consider j be ExtReal such that

 A40: j in ( rng J) and

 A41: (( sup ( rng J)) - (p / 2)) < j by A9, MEASURE6: 6;

 consider n1 be object such that

 A42: n1 in ( dom J) and

 A43: j = (J . n1) by A40, FUNCT_1:def 3;



 A44: (supj - p92) = (( sup ( rng J)) - (p / 2)) by SUPINF_2: 3;

 then

 A45: (p9 - (supj - p92)) = (p - (( sup ( rng J)) - (p / 2))) by SUPINF_2: 3;

 then (p - (( sup ( rng J)) - (p / 2))) < ( sup ( rng K)) by A39, XXREAL_0: 9;

 then

 consider k be Element of ExtREAL such that

 A46: k in ( rng K) and

 A47: (p - (( sup ( rng J)) - (p / 2))) < k by XXREAL_2: 94;

 p9 = ((p9 - (supj - p92)) + (supj - p92));

 then

 A48: ((p - (( sup ( rng J)) - (p / 2))) + (( sup ( rng J)) - (p / 2))) = p9 by A44, A45, SUPINF_2: 1;

 reconsider n1 as Element of NAT by A42;

 consider n2 be object such that

 A49: n2 in ( dom K) and

 A50: k = (K . n2) by A46, FUNCT_1:def 3;

 reconsider n2 as Element of NAT by A49;

 set n = ( max (n1,n2));

 (J . n1) <= (J . n) by A1, XXREAL_0: 25;

 then

 A51: (( sup ( rng J)) - (p / 2)) < (J . n) by A41, A43, XXREAL_0: 2;

 (K . n2) <= (K . n) by A2, XXREAL_0: 25;

 then

 A52: (p - (( sup ( rng J)) - (p / 2))) < (K . n) by A47, A50, XXREAL_0: 2;



 A53: p < ((J . n) + (K . n)) by A51, A52, A48, XXREAL_3: 64;

 now

 let m be Nat;

 assume

 A54: n <= m;

 then

 A55: (K . n) <= (K . m) by A2;

 (J . n) <= (J . m) by A1, A54;

 then ((J . n) + (K . n)) <= ((J . m) + (K . m)) by A55, XXREAL_3: 36;

 then ((J . n) + (K . n)) <= (L . m) by A5;

 hence p <= (L . m) by A53, XXREAL_0: 2;

 end;

 hence thesis;

 end;

 then

 A56: L is convergent_to_+infty;

 hence L is convergent;

 then ( lim L) = +infty by A56, Def12;

 hence ( lim L) = (( sup ( rng J)) + ( sup ( rng K))) by A9, A39, XXREAL_3:def 2;

 end;

 end;

 suppose

 A57: ( sup ( rng J)) = +infty ;

 now

 let p be Real;

 assume

 A58: 0 < p;

 per cases by A4, Lm8;

 suppose

 A59: ( sup ( rng K)) in REAL ;

 then

 reconsider supk = ( sup ( rng K)) as Element of REAL ;

 reconsider p92 = (p / 2), p9 = p as Element of REAL by XREAL_0:def 1;



 A60: (supk - p92) = (( sup ( rng K)) - (p / 2)) by SUPINF_2: 3;

 then

 A61: (p9 - (supk - p92)) = (p - (( sup ( rng K)) - (p / 2))) by SUPINF_2: 3;

 then (p - (( sup ( rng K)) - (p / 2))) < ( sup ( rng J)) by A57, XXREAL_0: 9;

 then

 consider j be Element of ExtREAL such that

 A62: j in ( rng J) and

 A63: (p - (( sup ( rng K)) - (p / 2))) < j by XXREAL_2: 94;

 p9 = ((p9 - (supk - p92)) + (supk - p92));

 then

 A64: ((p - (( sup ( rng K)) - (p / 2))) + (( sup ( rng K)) - (p / 2))) = p9 by A60, A61, SUPINF_2: 1;

 consider k be ExtReal such that

 A65: k in ( rng K) and

 A66: (( sup ( rng K)) - (p / 2)) < k by A58, A59, MEASURE6: 6;

 consider n1 be object such that

 A67: n1 in ( dom K) and

 A68: k = (K . n1) by A65, FUNCT_1:def 3;

 consider n2 be object such that

 A69: n2 in ( dom J) and

 A70: j = (J . n2) by A62, FUNCT_1:def 3;

 reconsider n1 as Element of NAT by A67;

 reconsider n2 as Element of NAT by A69;

 set n = ( max (n1,n2));

 (J . n2) <= (J . n) by A1, XXREAL_0: 25;

 then

 A71: (p - (( sup ( rng K)) - (p / 2))) < (J . n) by A63, A70, XXREAL_0: 2;

 (K . n1) <= (K . n) by A2, XXREAL_0: 25;

 then

 A72: (( sup ( rng K)) - (p / 2)) < (K . n) by A66, A68, XXREAL_0: 2;



 A73: p < ((J . n) + (K . n)) by A72, A71, A64, XXREAL_3: 64;

 now

 let m be Nat;

 assume

 A74: n <= m;

 then

 A75: (K . n) <= (K . m) by A2;

 (J . n) <= (J . m) by A1, A74;

 then ((J . n) + (K . n)) <= ((J . m) + (K . m)) by A75, XXREAL_3: 36;

 then ((J . n) + (K . n)) <= (L . m) by A5;

 hence p <= (L . m) by A73, XXREAL_0: 2;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds p <= (L . m);

 end;

 suppose ( sup ( rng K)) = +infty ;

 then

 consider n1 be Nat such that

 A76: (p / 2) < (K . n1) by A4, A58, Th59;

 consider n2 be Nat such that

 A77: (p / 2) < (J . n2) by A3, A57, A58, Th59;

 reconsider n1, n2 as Element of NAT by ORDINAL1:def 12;

 set n = ( max (n1,n2));

 (K . n1) <= (K . n) by A2, XXREAL_0: 25;

 then

 A78: (p / 2) < (K . n) by A76, XXREAL_0: 2;

 (J . n2) <= (J . n) by A1, XXREAL_0: 25;

 then

 A79: (p / 2) < (J . n) by A77, XXREAL_0: 2;

 ((p / 2) + (p / 2)) < ((J . n) + (K . n)) by A79, A78, XXREAL_3: 64;

 then p < ((J . n) + (K . n));

 then

 A80: p < (L . n) by A5;

 now

 let m be Nat;

 assume

 A81: n <= m;

 then

 A82: (K . n) <= (K . m) by A2;

 (J . n) <= (J . m) by A1, A81;

 then ((J . n) + (K . n)) <= ((J . m) + (K . m)) by A82, XXREAL_3: 36;

 then ((J . n) + (K . n)) <= (L . m) by A5;

 then (L . n) <= (L . m) by A5;

 hence p <= (L . m) by A80, XXREAL_0: 2;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds p <= (L . m);

 end;

 end;

 then

 A83: L is convergent_to_+infty;

 hence L is convergent;

 then

 A84: ( lim L) = +infty by A83, Def12;



 A85: (K . 0 ) <= ( sup ( rng K)) by A6, FUNCT_1: 3, XXREAL_2: 4;

 -infty < (K . 0 ) by A4;

 hence ( lim L) = (( sup ( rng J)) + ( sup ( rng K))) by A57, A84, A85, XXREAL_3:def 2;

 end;

 end;

 hence L is convergent;

 A86:

 now

 let n,m be Nat;

 assume

 A87: n <= m;

 then

 A88: (K . n) <= (K . m) by A2;

 (J . n) <= (J . m) by A1, A87;

 then ((J . n) + (K . n)) <= ((J . m) + (K . m)) by A88, XXREAL_3: 36;

 then (L . n) <= ((J . m) + (K . m)) by A5;

 hence (L . n) <= (L . m) by A5;

 end;

 hence ( lim L) = ( sup ( rng L)) by Th54;

 ( lim J) = ( sup ( rng J)) by A1, Th54;

 hence thesis by A2, A8, A86, Th54;

 end;

 theorem :: MESFUNC5:62



 Th62: for L,K be ExtREAL_sequence, c be Real st 0 <= c & L is without-infty & (for n be Nat holds (K . n) = (c * (L . n))) holds ( sup ( rng K)) = (c * ( sup ( rng L))) & K is without-infty

 proof

 let L,K be ExtREAL_sequence;

 let c be Real;

 assume that

 A1: 0 <= c and

 A2: L is without-infty and

 A3: for n be Nat holds (K . n) = (c * (L . n));

 now

 per cases by A2, Lm8;

 suppose

 A4: ( sup ( rng L)) in REAL ;



 A5: for y be UpperBound of ( rng K) holds (c * ( sup ( rng L))) <= y

 proof

 let y be UpperBound of ( rng K);

 reconsider y as R_eal by XXREAL_0:def 1;



 A6: ( dom L) = NAT by FUNCT_2:def 1;



 A7: ( dom K) = NAT by FUNCT_2:def 1;

 per cases ;

 suppose

 A8: c = 0 ;



 A9: (K . 1) <= y by A7, FUNCT_1: 3, XXREAL_2:def 1;

 (K . 1) = (c * (L . 1)) by A3;

 hence thesis by A8, A9;

 end;

 suppose

 A10: c <> 0 ;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then

 consider n be object such that

 A11: n in ( dom L) and

 A12: x = (L . n) by FUNCT_1:def 3;

 reconsider n as Element of NAT by A11;



 A13: (K . n) in ( rng K) by A7, FUNCT_1: 3;

 (K . n) = (c * (L . n)) by A3;

 then ((c * (L . n)) / c) <= (y / c) by A1, A10, A13, XXREAL_2:def 1, XXREAL_3: 79;

 hence x <= (y / c) by A10, A12, XXREAL_3: 88;

 end;

 then (y / c) is UpperBound of ( rng L) by XXREAL_2:def 1;

 then

 A14: ( sup ( rng L)) <= (y / c) by XXREAL_2:def 3;

 A15:

 now

 assume

 A16: y = -infty ;

 (K . 1) in ( rng K) by A7, FUNCT_1: 3;

 then (K . 1) = -infty by A16, XXREAL_0: 6, XXREAL_2:def 1;

 then

 A17: (c * (L . 1)) = -infty by A3;

 (L . 1) <= ( sup ( rng L)) by A6, FUNCT_1: 3, XXREAL_2: 4;

 then

 A18: (L . 1) < +infty by A4, XXREAL_0: 2, XXREAL_0: 9;

 -infty < (L . 1) by A2;

 hence contradiction by A17, A18, XXREAL_3: 70;

 end;

 per cases by A15, XXREAL_0: 14;

 suppose y = +infty ;

 hence thesis by XXREAL_0: 4;

 end;

 suppose y in REAL ;

 then

 reconsider ry = y as Real;

 reconsider sl = ( sup ( rng L)) as Real by A4;

 (y / c) = (ry / c);

 then (sl * c) <= ry by A1, A10, A14, XREAL_1: 83;

 hence thesis;

 end;

 end;

 end;

 now

 let x be ExtReal;



 A19: ( sup ( rng L)) is UpperBound of ( rng L) by XXREAL_2:def 3;

 assume x in ( rng K);

 then

 consider m be object such that

 A20: m in ( dom K) and

 A21: x = (K . m) by FUNCT_1:def 3;

 reconsider m as Element of NAT by A20;

 ( dom L) = NAT by FUNCT_2:def 1;

 then

 A22: (L . m) <= ( sup ( rng L)) by A19, FUNCT_1: 3, XXREAL_2:def 1;

 x = (c * (L . m)) by A3, A21;

 hence x <= (c * ( sup ( rng L))) by A1, A22, XXREAL_3: 71;

 end;

 then (c * ( sup ( rng L))) is UpperBound of ( rng K) by XXREAL_2:def 1;

 hence ( sup ( rng K)) = (c * ( sup ( rng L))) by A5, XXREAL_2:def 3;

 end;

 suppose

 A23: ( sup ( rng L)) = +infty ;

 per cases ;

 suppose

 A24: c = 0 ;

 A25:

 now

 let n be Nat;

 (K . n) = (c * (L . n)) by A3;

 hence (K . n) = 0 by A24;

 end;

 then ( lim K) = ( sup ( rng K)) by Th60;

 hence ( sup ( rng K)) = (c * ( sup ( rng L))) by A24, A25, Th60;

 end;

 suppose

 A26: c <> 0 ;

 now

 let n be object;

 -infty < (L . n) by A2;

 then

 A27: ( -infty * c) < (c * (L . n)) by A1, A26, XXREAL_3: 72;

 per cases ;

 suppose n in ( dom K);

 then

 reconsider n1 = n as Element of NAT ;

 ( -infty * c) = -infty by A1, A26, XXREAL_3:def 5;

 then -infty < (K . n1) by A3, A27;

 hence -infty < (K . n);

 end;

 suppose not n in ( dom K);

 hence -infty < (K . n) by FUNCT_1:def 2;

 end;

 end;

 then

 A28: K is without-infty;

 A29:

 now

 let k be Real;

 reconsider k1 = k as Real;



 A30: ((k / c) * c) = ((k1 / c) * c)

 .= (k1 / (c / c)) by XCMPLX_1: 82

 .= k by A26, XCMPLX_1: 51;

 assume 0 < k;

 then

 consider n be Nat such that

 A31: (k / c) < (L . n) by A1, A2, A23, A26, Th59;

 ((k / c) * c) < (c * (L . n)) by A1, A26, A31, XXREAL_3: 72;

 then k < (K . n) by A3, A30;

 hence ex n be Nat st not (K . n) <= k;

 end;

 (c * ( sup ( rng L))) = +infty by A1, A23, A26, XXREAL_3:def 5;

 hence ( sup ( rng K)) = (c * ( sup ( rng L))) by A28, A29, Th59;

 end;

 end;

 end;

 hence ( sup ( rng K)) = (c * ( sup ( rng L)));

 now

 let n be object;



 A32: (L . n) = +infty implies (c * (L . n)) <> -infty by A1;

 -infty < (L . n) by A2;

 then

 A33: -infty <> (c * (L . n)) by A32, XXREAL_3: 70;

 per cases ;

 suppose n in ( dom K);

 then

 reconsider n1 = n as Element of NAT ;

 (K . n1) <> -infty by A3, A33;

 hence -infty < (K . n) by XXREAL_0: 6;

 end;

 suppose not n in ( dom K);

 hence -infty < (K . n) by FUNCT_1:def 2;

 end;

 end;

 hence thesis;

 end;

 theorem :: MESFUNC5:63



 Th63: for L,K be ExtREAL_sequence, c be Real st 0 <= c & (for n,m be Nat st n <= m holds (L . n) <= (L . m)) & (for n be Nat holds (K . n) = (c * (L . n))) & L is without-infty holds (for n,m be Nat st n <= m holds (K . n) <= (K . m)) & K is without-infty & K is convergent & ( lim K) = ( sup ( rng K)) & ( lim K) = (c * ( lim L))

 proof

 let L,K be ExtREAL_sequence, c be Real;

 assume that

 A1: 0 <= c and

 A2: for n,m be Nat st n <= m holds (L . n) <= (L . m) and

 A3: for n be Nat holds (K . n) = (c * (L . n)) and

 A4: L is without-infty;



 A5: ( sup ( rng L)) = ( lim L) by A2, Th54;

 now

 let n,m be Nat;

 assume n <= m;

 then (c * (L . n)) <= (c * (L . m)) by A1, A2, XXREAL_3: 71;

 then (K . n) <= (c * (L . m)) by A3;

 hence (K . n) <= (K . m) by A3;

 end;

 thus K is without-infty by A1, A3, A4, Th62;

 thus K is convergent & ( lim K) = ( sup ( rng K)) by A6, Th54;

 ( sup ( rng K)) = ( lim K) by A6, Th54;

 hence thesis by A1, A3, A4, A5, Th62;

 end;

 begin

 definition

 let X be non empty set, H be Functional_Sequence of X, ExtREAL , x be Element of X;

 :: MESFUNC5:def13

 func H # x -> ExtREAL_sequence means

 : Def13: for n be Nat holds (it . n) = ((H . n) . x);

 existence

 proof

 deffunc U( Nat) = ((H . $1) . x);

 consider f be sequence of ExtREAL such that

 A1: for n be Element of NAT holds (f . n) = U(n) from FUNCT_2:sch 4;

 take f;

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence thesis by A1;

 end;

 uniqueness

 proof

 let S1,S2 be ExtREAL_sequence such that

 A2: for n be Nat holds (S1 . n) = ((H . n) . x) and

 A3: for n be Nat holds (S2 . n) = ((H . n) . x);

 now

 let n be Element of NAT ;

 (S1 . n) = ((H . n) . x) by A2;

 hence (S1 . n) = (S2 . n) by A3;

 end;

 hence thesis by FUNCT_2: 63;

 end;

 end

 definition

 let D1,D2 be set, F be sequence of ( PFuncs (D1,D2)), n be Nat;

 :: original: .

 redefine

 func F . n -> PartFunc of D1, D2 ;

 coherence

 proof

 n in NAT by ORDINAL1:def 12;

 then n in ( dom F) by FUNCT_2:def 1;

 then (F . n) in ( rng F) by FUNCT_1:def 3;

 hence thesis;

 end;

 end

 theorem :: MESFUNC5:64



 Th64: for X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & f qua ext-real-valued Function is nonnegative holds ex F be Functional_Sequence of X, ExtREAL st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x)) & for x be Element of X st x in ( dom f) holds (F # x) is convergent & ( lim (F # x)) = (f . x)

 proof

 let X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL such that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: f is nonnegative;

 defpred PF[ Element of NAT , PartFunc of X, ExtREAL ] means ( dom $2) = ( dom f) & (for x be Element of X st x in ( dom f) holds (for k be Nat st 1 <= k & k <= ((2 |^ $1) * $1) & ((k - 1) / (2 |^ $1)) <= (f . x) & (f . x) < (k / (2 |^ $1)) holds ($2 . x) = ((k - 1) / (2 |^ $1))) & ($1 <= (f . x) implies ($2 . x) = $1));



 A3: for n be Element of NAT holds ex y be Element of ( PFuncs (X, ExtREAL )) st PF[n, y]

 proof

 let n be Element of NAT ;

 reconsider nn = n as Nat;

 defpred PP[ object, object] means (for k be Nat st 1 <= k & k <= ((2 |^ n) * n) & ((k - 1) / (2 |^ n)) <= (f . $1) & (f . $1) < (k / (2 |^ n)) holds $2 = ((k - 1) / (2 |^ n))) & (n <= (f . $1) implies $2 = n);



 A4: for x be object st x in ( dom f) holds ex y be object st PP[x, y]

 proof

 let x be object;

 assume x in ( dom f);

 per cases ;

 suppose

 A5: (f . x) < n;

 0 <= (f . x) by A2, SUPINF_2: 51;

 then

 consider k be Nat such that 1 <= k and k <= ((2 |^ nn) * nn) and

 A6: ((k - 1) / (2 |^ nn)) <= (f . x) qua ExtReal and

 A7: (f . x) < (k / (2 |^ nn)) by A5, Th4;

 take y = ((k - 1) / (2 |^ n));

 now

 let k1 be Nat;

 assume that 1 <= k1 and k1 <= ((2 |^ n) * n) and

 A8: ((k1 - 1) / (2 |^ n)) <= (f . x) and

 A9: (f . x) < (k1 / (2 |^ n));

 A10:

 now

 assume k1 < k;

 then (k1 + 1) <= k by NAT_1: 13;

 then k1 <= (k - 1) by XREAL_1: 19;

 then (k1 / (2 |^ n)) <= ((k - 1) / (2 |^ n)) by XREAL_1: 72;

 hence contradiction by A6, A9, XXREAL_0: 2;

 end;

 now

 assume k < k1;

 then (k + 1) <= k1 by NAT_1: 13;

 then k <= (k1 - 1) by XREAL_1: 19;

 then (k / (2 |^ n)) <= ((k1 - 1) / (2 |^ n)) by XREAL_1: 72;

 hence contradiction by A7, A8, XXREAL_0: 2;

 end;

 hence y = ((k1 - 1) / (2 |^ n)) by A10, XXREAL_0: 1;

 end;

 hence thesis by A5;

 end;

 suppose

 A11: n <= (f . x);

 reconsider y = nn as Real;

 take y;

 thus for k be Nat st 1 <= k & k <= ((2 |^ n) * n) & ((k - 1) / (2 |^ n)) <= (f . x) & (f . x) < (k / (2 |^ n)) holds y = ((k - 1) / (2 |^ n))

 proof

 let k be Nat such that 1 <= k and

 A12: k <= ((2 |^ n) * n) and ((k - 1) / (2 |^ n)) <= (f . x) and

 A13: (f . x) < (k / (2 |^ n));

 reconsider p = (f . x) as ExtReal;

 k <= ((2 |^ nn) * nn) by A12;

 then (k / (2 |^ nn)) <= p by A11, Th5;

 then (k / (2 |^ n)) <= p;

 hence y = ((k - 1) / (2 |^ n)) by A13;

 end;

 thus thesis;

 end;

 end;

 consider fn be Function such that

 A14: ( dom fn) = ( dom f) & for x be object st x in ( dom f) holds PP[x, (fn . x)] from CLASSES1:sch 1( A4);

 now

 let w be object;

 assume w in ( rng fn);

 then

 consider v be object such that

 A15: v in ( dom fn) and

 A16: w = (fn . v) by FUNCT_1:def 3;

 per cases ;

 suppose n <= (f . v);

 then (fn . v) = n by A14, A15;

 hence w in ExtREAL by A16, XXREAL_0:def 1;

 end;

 suppose

 A17: (f . v) < n;

 0 <= (f . v) by A2, SUPINF_2: 51;

 then

 consider k be Nat such that

 A18: 1 <= k and

 A19: k <= ((2 |^ nn) * nn) and

 A20: ((k - 1) / (2 |^ nn)) <= (f . v) and

 A21: (f . v) < (k / (2 |^ nn)) by A17, Th4;

 (fn . v) = ((k - 1) / (2 |^ n)) by A14, A15, A18, A19, A20, A21;

 hence w in ExtREAL by A16;

 end;

 end;

 then ( rng fn) c= ExtREAL ;

 then

 reconsider fn as PartFunc of ( dom f), ExtREAL by A14, RELSET_1: 4;

 reconsider fn as PartFunc of X, ExtREAL by A14, RELSET_1: 5;

 reconsider y = fn as Element of ( PFuncs (X, ExtREAL )) by PARTFUN1: 45;

 take y;

 thus thesis by A14;

 end;

 consider F be sequence of ( PFuncs (X, ExtREAL )) such that

 A22: for n be Element of NAT holds PF[n, (F . n)] from FUNCT_2:sch 3( A3);



 A23: for n be Element of NAT holds ( dom (F . n)) = ( dom f) by A22;



 A24: for n be Element of NAT , x be Element of X st x in ( dom f) holds (for k be Nat st 1 <= k & k <= ((2 |^ n) * n) & ((k - 1) / (2 |^ n)) <= (f . x) & (f . x) < (k / (2 |^ n)) holds ((F . n) . x) = ((k - 1) / (2 |^ n))) & (n <= (f . x) implies ((F . n) . x) = n) by A22;

 A25:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence ( dom (F . n)) = ( dom f) by A23;

 end;

 reconsider F as Functional_Sequence of X, ExtREAL ;

 consider A be Element of S such that

 A26: A = ( dom f) and

 A27: f is A -measurable by A1;



 A28: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let n,m be Nat such that

 A29: n <= m;

 reconsider nn = n, mm = m as Element of NAT by ORDINAL1:def 12;

 let x be Element of X such that

 A30: x in ( dom f);

 per cases ;

 suppose

 A31: m <= (f . x);

 then

 A32: nn <= (f . x) by A29, XXREAL_0: 2;

 ((F . mm) . x) = m by A24, A30, A31;

 hence thesis by A24, A29, A30, A32;

 end;

 suppose

 A33: (f . x) < m;



 A34: 0 <= (f . x) by A2, SUPINF_2: 51;

 then

 consider M be Nat such that

 A35: 1 <= M and

 A36: M <= ((2 |^ m) * m) and

 A37: ((M - 1) / (2 |^ m)) <= (f . x) and

 A38: (f . x) < (M / (2 |^ m)) by A33, Th4;

 reconsider M as Element of NAT by ORDINAL1:def 12;



 A39: ((F . mm) . x) = ((M - 1) / (2 |^ m)) by A24, A30, A35, A36, A37, A38;

 per cases ;

 suppose

 A40: n <= (f . x);

 reconsider M1 = (2 |^ mm) as Element of NAT ;

 n < (M / (2 |^ m)) by A38, A40, XXREAL_0: 2;

 then ((2 |^ m) * n) < M by PREPOWER: 6, XREAL_1: 79;

 then ((M1 * n) + 1) <= M by NAT_1: 13;

 then

 A41: (M1 * n) <= (M - 1) by XREAL_1: 19;



 A42: 0 < (2 |^ m) by PREPOWER: 6;

 ((F . n) . x) = nn by A24, A30, A40;

 hence thesis by A39, A42, A41, XREAL_1: 77;

 end;

 suppose

 A43: (f . x) < n;

 consider k be Nat such that

 A44: m = (nn qua Complex + k) by A29, NAT_1: 10;

 reconsider k as Element of NAT by ORDINAL1:def 12;

 reconsider K = (2 |^ k) as Element of NAT ;

 consider N1 be Nat such that

 A45: 1 <= N1 and

 A46: N1 <= ((2 |^ n) * n) and

 A47: ((N1 - 1) / (2 |^ n)) <= (f . x) and

 A48: (f . x) < (N1 / (2 |^ n)) by A34, A43, Th4;

 reconsider N1 as Element of NAT by ORDINAL1:def 12;



 A49: ((F . nn) . x) = ((N1 - 1) / (2 |^ nn)) by A24, A30, A45, A46, A47, A48;

 ((N1 - 1) / (2 |^ n)) < (M / (2 |^ (n + k))) by A38, A47, A44, XXREAL_0: 2;

 then ((N1 - 1) / (2 |^ n)) < (M / ((2 |^ n) * (2 |^ k))) by NEWTON: 8;

 then ((N1 - 1) / (2 |^ n)) < ((M / (2 |^ k)) / (2 |^ n)) by XCMPLX_1: 78;

 then (N1 - 1) < (M / (2 |^ k)) by XREAL_1: 72;

 then (K * (N1 - 1)) < M by PREPOWER: 6, XREAL_1: 79;

 then ((K * (N1 - 1)) + 1) <= M by INT_1: 7;

 then (K * (N1 - 1)) <= (M - 1) by XREAL_1: 19;

 then ((K * (N1 - 1)) / (2 |^ (n + k))) <= ((M - 1) / (2 |^ (n + k))) by XREAL_1: 72;

 then

 A50: ((K * (N1 - 1)) / ((2 |^ n) * (2 |^ k))) <= ((M - 1) / (2 |^ (n + k))) by NEWTON: 8;

 (2 |^ k) > 0 by PREPOWER: 6;

 hence thesis by A39, A49, A44, A50, XCMPLX_1: 91;

 end;

 end;

 end;



 A51: for n be Nat holds (F . n) is_simple_func_in S

 proof

 let n be Nat;

 reconsider nn = n as Element of NAT by ORDINAL1:def 12;

 reconsider N = (2 |^ nn) as Element of NAT ;

 defpred PG[ Nat, set] means ($1 <= (N * n) implies $2 = ((A /\ ( great_eq_dom (f,(($1 - 1) / (2 |^ n))))) /\ ( less_dom (f,($1 / (2 |^ n)))))) & ($1 = ((N * n) + 1) implies $2 = (A /\ ( great_eq_dom (f,n))));

 now

 let x be Element of X;

 assume x in ( dom (F . n));

 then

 A52: x in ( dom f) by A25;

 per cases ;

 suppose

 A53: n <= (f . x);

 then ((F . nn) . x) = n by A24, A52;

 then ((F . n) . x) in REAL by XREAL_0:def 1;

 then

 A54: ((F . n) . x) < +infty by XXREAL_0: 9;

 ( - +infty ) < ((F . nn) . x) by A24, A52, A53;

 hence |.((F . n) . x).| < +infty by A54, EXTREAL1: 22;

 end;

 suppose

 A55: (f . x) < n;



 A56: 0 <= (f . x) by A2, SUPINF_2: 51;

 nn in REAL by XREAL_0:def 1;

 then nn < +infty by XXREAL_0: 9;

 then

 reconsider y = (f . x) as Element of REAL by A55, A56, XXREAL_0: 14;

 set k = ( [\((2 |^ n) * y)/] + 1);



 A57: [\((2 |^ n) * y)/] <= ((2 |^ n) * y) by INT_1:def 6;

 (((2 |^ n) * y) - 1) < [\((2 |^ n) * y)/] by INT_1:def 6;

 then

 A58: ((2 |^ n) * y) < k by XREAL_1: 19;



 A59: 0 < (2 |^ n) by PREPOWER: 6;

 then ((2 |^ n) * y) < ((2 |^ n) * n) by A55, XREAL_1: 68;

 then [\((2 |^ n) * y)/] < ((2 |^ n) * n) by A57, XXREAL_0: 2;

 then

 A60: k <= ((2 |^ n) * n) by INT_1: 7;



 A61: 0 <= ((2 |^ n) * y) by A56;

 then

 A62: ( 0 + 1) <= k by A58, INT_1: 7;

 reconsider k as Element of NAT by A61, A58, INT_1: 3;

 reconsider k as Nat;

 (k - 1) <= ((2 |^ n) * y) by INT_1:def 6;

 then

 A63: ((k - 1) / (2 |^ nn)) <= y by PREPOWER: 6, XREAL_1: 79;



 A64: ((k - 1) / (2 |^ nn)) in REAL by XREAL_0:def 1;

 y < (k / (2 |^ nn)) by A59, INT_1: 29, XREAL_1: 81;

 then

 A65: ((F . nn) . x) = ((k - 1) / (2 |^ nn)) by A24, A52, A62, A60, A63;

 then -infty < ((F . n) . x) by XXREAL_0: 12, A64;

 then

 A66: ( - +infty ) < ((F . n) . x) by XXREAL_3:def 3;

 ((F . n) . x) < +infty by A65, XXREAL_0: 9, A64;

 hence |.((F . n) . x).| < +infty by A66, EXTREAL1: 22;

 end;

 end;

 then

 A67: (F . n) is real-valued by MESFUNC2:def 1;

 A68:

 now

 let k be Nat;

 assume k in ( Seg ((N * n) + 1));

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

 per cases ;

 suppose

 A69: k <> ((N * n) + 1);

 set B = ((A /\ ( great_eq_dom (f,((k1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(k1 / (2 |^ n)))));

 reconsider B as Element of S by A26, A27, Th33;

 take B;

 thus PG[k, B] by A69;

 end;

 suppose

 A70: k = ((N * n) + 1);

 set B = (A /\ ( great_eq_dom (f,n)));

 reconsider B as Element of S by A26, A27, MESFUNC1: 27;

 take B;

 thus PG[k, B] by A70, NAT_1: 13;

 end;

 end;

 consider G be FinSequence of S such that

 A71: ( dom G) = ( Seg ((N * n) + 1)) & for k be Nat st k in ( Seg ((N * n) + 1)) holds PG[k, (G . k)] from FINSEQ_1:sch 5( A68);

 A72:

 now

 let k be Nat;

 assume that

 A73: 1 <= k and

 A74: k <= ((2 |^ n) * n);

 k <= ((N * n) + 1) by A74, NAT_1: 12;

 then k in ( Seg ((N * n) + 1)) by A73;

 hence (G . k) = ((A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) /\ ( less_dom (f,(k / (2 |^ n))))) by A71, A74;

 end;



 A75: ( len G) = (((2 |^ n) * n) + 1) by A71, FINSEQ_1:def 3;

 now

 let x,y be object;

 assume

 A76: x <> y;

 per cases ;

 suppose not x in ( dom G) or not y in ( dom G);

 then (G . x) = {} or (G . y) = {} by FUNCT_1:def 2;

 hence (G . x) misses (G . y);

 end;

 suppose

 A77: x in ( dom G) & y in ( dom G);

 then

 reconsider x1 = x, y1 = y as Nat;



 A78: x1 in ( Seg ( len G)) by A77, FINSEQ_1:def 3;

 then

 A79: 1 <= x1 by FINSEQ_1: 1;



 A80: y1 in ( Seg ( len G)) by A77, FINSEQ_1:def 3;

 then

 A81: 1 <= y1 by FINSEQ_1: 1;



 A82: y1 <= (((2 |^ n) * n) + 1) by A75, A80, FINSEQ_1: 1;



 A83: x1 <= (((2 |^ n) * n) + 1) by A75, A78, FINSEQ_1: 1;

 now

 per cases by A76, XXREAL_0: 1;

 case

 A84: x1 < y1;

 hereby

 assume

 A85: y1 = (((2 |^ n) * n) + 1);

 then

 A86: (G . y) = (A /\ ( great_eq_dom (f,n))) by A71, A77;



 A87: x1 <= (N * n) by A84, A85, NAT_1: 13;

 then

 A88: (G . x) = ((A /\ ( great_eq_dom (f,((x1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(x1 / (2 |^ n))))) by A72, A79;

 now

 given v be object such that

 A89: v in ((G . x) /\ (G . y));

 v in (G . y) by A89, XBOOLE_0:def 4;

 then v in ( great_eq_dom (f,n)) by A86, XBOOLE_0:def 4;

 then

 A90: n <= (f . v) by MESFUNC1:def 14;

 v in (G . x) by A89, XBOOLE_0:def 4;

 then v in ( less_dom (f,(x1 / (2 |^ n)))) by A88, XBOOLE_0:def 4;

 then (f . v) < (x1 / (2 |^ n)) by MESFUNC1:def 11;

 then n < (x1 / (2 |^ n)) by A90, XXREAL_0: 2;

 hence contradiction by A87, PREPOWER: 6, XREAL_1: 79;

 end;

 then ((G . x) /\ (G . y)) = {} by XBOOLE_0:def 1;

 hence (G . x) misses (G . y);

 end;

 assume y1 <> (((2 |^ n) * n) + 1);

 then y1 < ((N * n) + 1) by A82, XXREAL_0: 1;

 then

 A91: y1 <= (N * n) by NAT_1: 13;

 then x1 <= ((2 |^ n) * n) by A84, XXREAL_0: 2;

 then

 A92: (G . x) = ((A /\ ( great_eq_dom (f,((x1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(x1 / (2 |^ n))))) by A72, A79;



 A93: (G . y) = ((A /\ ( great_eq_dom (f,((y1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(y1 / (2 |^ n))))) by A72, A81, A91;

 now

 given v be object such that

 A94: v in ((G . x) /\ (G . y));

 v in (G . y) by A94, XBOOLE_0:def 4;

 then v in (A /\ ( great_eq_dom (f,((y1 - 1) / (2 |^ n))))) by A93, XBOOLE_0:def 4;

 then v in ( great_eq_dom (f,((y1 - 1) / (2 |^ n)))) by XBOOLE_0:def 4;

 then

 A95: ((y1 - 1) / (2 |^ n)) <= (f . v) by MESFUNC1:def 14;

 v in (G . x) by A94, XBOOLE_0:def 4;

 then v in ( less_dom (f,(x1 / (2 |^ n)))) by A92, XBOOLE_0:def 4;

 then (f . v) < (x1 / (2 |^ n)) by MESFUNC1:def 11;

 then ((y1 - 1) / (2 |^ n)) < (x1 / (2 |^ n)) by A95, XXREAL_0: 2;

 then (y1 - 1) < x1 by XREAL_1: 72;

 then y1 < (x1 + 1) by XREAL_1: 19;

 hence contradiction by A84, NAT_1: 13;

 end;

 then ((G . x) /\ (G . y)) = {} by XBOOLE_0:def 1;

 hence (G . x) misses (G . y);

 end;

 case

 A96: y1 < x1;

 hereby

 assume x1 <> (((2 |^ n) * n) + 1);

 then x1 < ((N * n) + 1) by A83, XXREAL_0: 1;

 then

 A97: x1 <= (N * n) by NAT_1: 13;

 then y1 <= ((2 |^ n) * n) by A96, XXREAL_0: 2;

 then

 A98: (G . y) = ((A /\ ( great_eq_dom (f,((y1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(y1 / (2 |^ n))))) by A72, A81;



 A99: (G . x) = ((A /\ ( great_eq_dom (f,((x1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(x1 / (2 |^ n))))) by A72, A79, A97;

 now

 given v be object such that

 A100: v in ((G . x) /\ (G . y));

 v in (G . x) by A100, XBOOLE_0:def 4;

 then v in (A /\ ( great_eq_dom (f,((x1 - 1) / (2 |^ n))))) by A99, XBOOLE_0:def 4;

 then v in ( great_eq_dom (f,((x1 - 1) / (2 |^ n)))) by XBOOLE_0:def 4;

 then

 A101: ((x1 - 1) / (2 |^ n)) <= (f . v) by MESFUNC1:def 14;

 v in (G . y) by A100, XBOOLE_0:def 4;

 then v in ( less_dom (f,(y1 / (2 |^ n)))) by A98, XBOOLE_0:def 4;

 then (f . v) < (y1 / (2 |^ n)) by MESFUNC1:def 11;

 then ((x1 - 1) / (2 |^ n)) < (y1 / (2 |^ n)) by A101, XXREAL_0: 2;

 then (x1 - 1) < y1 by XREAL_1: 72;

 then x1 < (y1 + 1) by XREAL_1: 19;

 hence contradiction by A96, NAT_1: 13;

 end;

 then ((G . x) /\ (G . y)) = {} by XBOOLE_0:def 1;

 hence (G . x) misses (G . y);

 end;

 assume

 A102: x1 = (((2 |^ n) * n) + 1);

 then

 A103: (G . x) = (A /\ ( great_eq_dom (f,n))) by A71, A77;



 A104: y1 <= (N * n) by A96, A102, NAT_1: 13;

 then

 A105: (G . y) = ((A /\ ( great_eq_dom (f,((y1 - 1) / (2 |^ n))))) /\ ( less_dom (f,(y1 / (2 |^ n))))) by A72, A81;

 now

 given v be object such that

 A106: v in ((G . x) /\ (G . y));

 v in (G . y) by A106, XBOOLE_0:def 4;

 then v in ( less_dom (f,(y1 / (2 |^ n)))) by A105, XBOOLE_0:def 4;

 then

 A107: (f . v) < (y1 / (2 |^ n)) by MESFUNC1:def 11;

 v in (G . x) by A106, XBOOLE_0:def 4;

 then v in ( great_eq_dom (f,n)) by A103, XBOOLE_0:def 4;

 then n <= (f . v) by MESFUNC1:def 14;

 then n < (y1 / (2 |^ n)) by A107, XXREAL_0: 2;

 hence contradiction by A104, PREPOWER: 6, XREAL_1: 79;

 end;

 then ((G . x) /\ (G . y)) = {} by XBOOLE_0:def 1;

 hence (G . x) misses (G . y);

 end;

 end;

 hence (G . x) misses (G . y);

 end;

 end;

 then

 reconsider G as Finite_Sep_Sequence of S by PROB_2:def 2;



 A108: for k be Nat, x,y be Element of X st k in ( dom G) & x in (G . k) & y in (G . k) holds ((F . n) . x) = ((F . n) . y)

 proof

 let k be Nat, x,y be Element of X;

 assume that

 A109: k in ( dom G) and

 A110: x in (G . k) and

 A111: y in (G . k);



 A112: 1 <= k by A71, A109, FINSEQ_1: 1;



 A113: k <= ((N * n) + 1) by A71, A109, FINSEQ_1: 1;

 now

 per cases ;

 suppose k = ((N * n) + 1);

 then

 A114: (G . k) = (A /\ ( great_eq_dom (f,n))) by A71, A109;

 then x in ( great_eq_dom (f,n)) by A110, XBOOLE_0:def 4;

 then

 A115: n <= (f . x) by MESFUNC1:def 14;

 y in ( great_eq_dom (f,n)) by A111, A114, XBOOLE_0:def 4;

 then

 A116: n <= (f . y) by MESFUNC1:def 14;

 x in A by A110, A114, XBOOLE_0:def 4;

 then

 A117: ((F . nn) . x) = nn by A26, A24, A115;

 y in A by A111, A114, XBOOLE_0:def 4;

 hence thesis by A26, A24, A117, A116;

 end;

 suppose k <> ((N * n) + 1);

 then k < ((N * n) + 1) by A113, XXREAL_0: 1;

 then

 A118: k <= (N * n) by NAT_1: 13;

 then

 A119: (G . k) = ((A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) /\ ( less_dom (f,(k / (2 |^ n))))) by A71, A109;

 then x in ( less_dom (f,(k / (2 |^ n)))) by A110, XBOOLE_0:def 4;

 then

 A120: (f . x) < (k / (2 |^ n)) by MESFUNC1:def 11;



 A121: x in (A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) by A110, A119, XBOOLE_0:def 4;

 then x in ( great_eq_dom (f,((k - 1) / (2 |^ n)))) by XBOOLE_0:def 4;

 then

 A122: ((k - 1) / (2 |^ n)) <= (f . x) by MESFUNC1:def 14;

 x in A by A121, XBOOLE_0:def 4;

 then

 A123: ((F . n) . x) = ((k - 1) / (2 |^ n)) by A26, A24, A112, A118, A120, A122;

 y in ( less_dom (f,(k / (2 |^ n)))) by A111, A119, XBOOLE_0:def 4;

 then

 A124: (f . y) < (k / (2 |^ n)) by MESFUNC1:def 11;



 A125: y in (A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) by A111, A119, XBOOLE_0:def 4;

 then y in ( great_eq_dom (f,((k - 1) / (2 |^ n)))) by XBOOLE_0:def 4;

 then

 A126: ((k - 1) / (2 |^ n)) <= (f . y) by MESFUNC1:def 14;

 y in A by A125, XBOOLE_0:def 4;

 hence thesis by A26, A24, A112, A118, A124, A126, A123;

 end;

 end;

 hence thesis;

 end;

 for v be object st v in ( dom f) holds v in ( union ( rng G))

 proof

 let v be object;

 reconsider vv = v as set by TARSKI: 1;

 assume

 A127: v in ( dom f);

 ex B be set st v in B & B in ( rng G)

 proof

 per cases ;

 suppose

 A128: (f . v) < n;

 0 <= (f . v) by A2, SUPINF_2: 51;

 then

 consider k be Nat such that

 A129: 1 <= k and

 A130: k <= ((2 |^ n) * n) and

 A131: ((k - 1) / (2 |^ n)) <= (f . vv) and

 A132: (f . vv) < (k / (2 |^ n)) by A128, Th4;

 v in ( great_eq_dom (f,((k - 1) / (2 |^ n)))) by A127, A131, MESFUNC1:def 14;

 then

 A133: v in (A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) by A26, A127, XBOOLE_0:def 4;

 v in ( less_dom (f,(k / (2 |^ n)))) by A127, A132, MESFUNC1:def 11;

 then

 A134: v in ((A /\ ( great_eq_dom (f,((k - 1) / (2 |^ n))))) /\ ( less_dom (f,(k / (2 |^ n))))) by A133, XBOOLE_0:def 4;

 take (G . k);

 (N * n) <= ((N * n) + 1) by NAT_1: 11;

 then k <= ((N * n) + 1) by A130, XXREAL_0: 2;

 then k in ( Seg ((N * n) + 1)) by A129;

 hence thesis by A71, A130, A134, FUNCT_1: 3;

 end;

 suppose

 A135: n <= (f . v);

 set k = ((N * n) + 1);

 take (G . k);

 1 <= k by NAT_1: 11;

 then

 A136: k in ( Seg ((N * n) + 1));

 v in ( great_eq_dom (f,n)) by A127, A135, MESFUNC1:def 14;

 then v in (A /\ ( great_eq_dom (f,n))) by A26, A127, XBOOLE_0:def 4;

 hence thesis by A71, A136, FUNCT_1: 3;

 end;

 end;

 hence thesis by TARSKI:def 4;

 end;

 then

 A137: ( dom f) c= ( union ( rng G));

 for v be object st v in ( union ( rng G)) holds v in ( dom f)

 proof

 let v be object;

 assume v in ( union ( rng G));

 then

 consider B be set such that

 A138: v in B and

 A139: B in ( rng G) by TARSKI:def 4;

 consider m be object such that

 A140: m in ( dom G) and

 A141: B = (G . m) by A139, FUNCT_1:def 3;

 reconsider m as Element of NAT by A140;

 reconsider mm = m as Nat;



 A142: m <= ((N * n) + 1) by A71, A140, FINSEQ_1: 1;

 now

 per cases ;

 suppose m = ((N * n) + 1);

 then B = (A /\ ( great_eq_dom (f,n))) by A71, A140, A141;

 hence v in A by A138, XBOOLE_0:def 4;

 end;

 suppose m <> ((N * n) + 1);

 then m < ((N * n) + 1) by A142, XXREAL_0: 1;

 then m <= (N * n) by NAT_1: 13;

 then B = ((A /\ ( great_eq_dom (f,((mm - 1) / (2 |^ n))))) /\ ( less_dom (f,(mm / (2 |^ n))))) by A71, A140, A141;

 then v in (A /\ ( great_eq_dom (f,((m - 1) / (2 |^ n))))) by A138, XBOOLE_0:def 4;

 hence v in A by XBOOLE_0:def 4;

 end;

 end;

 hence thesis by A26;

 end;

 then ( union ( rng G)) c= ( dom f);

 then ( union ( rng G)) = ( dom f) by A137;

 then ( dom (F . n)) = ( union ( rng G)) by A25;

 hence thesis by A67, A108, MESFUNC2:def 4;

 end;

 A143:

 now

 let x be Element of X such that

 A144: x in ( dom f);

 per cases ;

 suppose

 A145: |.(f . x).| = +infty ;

 now

 assume ( - (f . x)) = +infty ;

 then (f . x) < 0 ;

 hence contradiction by A2, SUPINF_2: 51;

 end;

 then

 A146: (f . x) = +infty by A145, EXTREAL1: 13;

 for g be Real st 0 < g holds ex n be Nat st for m be Nat st n <= m holds g <= ((F # x) . m)

 proof

 let g be Real;

 assume 0 < g;

 then

 reconsider n = [/g\] as Nat by INT_1: 53;



 A147: g <= n by INT_1:def 7;

 for m be Nat st n <= m holds g <= ((F # x) . m)

 proof

 let m be Nat;

 assume n <= m;

 then

 A148: g <= m by A147, XXREAL_0: 2;

 reconsider m as Element of NAT by ORDINAL1:def 12;

 m <= (f . x) by A146, XXREAL_0: 4;

 then ((F . m) . x) = m by A24, A144;

 hence thesis by A148, Def13;

 end;

 hence thesis;

 end;

 then

 A149: (F # x) is convergent_to_+infty;

 then (F # x) is convergent;

 hence (F # x) is convergent & ( lim (F # x)) = (f . x) by A146, A149, Def12;

 end;

 suppose |.(f . x).| <> +infty ;

 then

 reconsider g = (f . x) as Element of REAL by EXTREAL1: 30, XXREAL_0: 14;



 A150: for p be Real st 0 = k holds |.(((F # x) . j) - (f . x)).| < p

 proof

 set N2 = ( [/g\] + 1);

 let p be Real;



 A151: g <= [/g\] by INT_1:def 7;

 [/g\] < ( [/g\] + 1) by XREAL_1: 29;

 then

 A152: g < N2 by A151, XXREAL_0: 2;

 0 <= g by A2, SUPINF_2: 51;

 then

 reconsider N2 as Element of NAT by A151, INT_1: 3;



 A153: for N be Nat st N >= N2 holds |.(((F # x) . N) - (f . x)).| < (1 / (2 |^ N))

 proof

 let N be Nat;

 assume

 A154: N >= N2;

 reconsider NN = N as Element of NAT by ORDINAL1:def 12;



 A155: 0 <= (f . x) by A2, SUPINF_2: 51;

 (f . x) < N by A152, A154, XXREAL_0: 2;

 then

 consider m be Nat such that

 A156: 1 <= m and

 A157: m <= ((2 |^ N) * N) and

 A158: ((m - 1) / (2 |^ N)) <= (f . x) and

 A159: (f . x) < (m / (2 |^ N)) by A155, Th4;

 reconsider m as Element of NAT by ORDINAL1:def 12;



 A160: ((F # x) . N) = ((F . NN) . x) by Def13

 .= ((m - 1) / (2 |^ NN)) by A24, A144, A156, A157, A158, A159;

 then

 A161: ((F # x) . N) in REAL by XREAL_0:def 1;

 ((m / (2 |^ N)) - ((m - 1) / (2 |^ N))) = ((m / (2 |^ N)) - ((m - 1) / (2 |^ N)))

 .= ((m / (2 |^ N)) + ( - ((m - 1) / (2 |^ N))))

 .= ((m / (2 |^ N)) + (( - (m - 1)) / (2 |^ N)))

 .= ((m + ( - (m - 1))) / (2 |^ N));

 then

 A162: ((f . x) - ((F # x) . N)) < (1 / (2 |^ N)) by A159, A160, XXREAL_3: 43, A161;

 ( - (((F # x) . N) - (f . x))) = ((f . x) - ((F # x) . N)) by XXREAL_3: 26;

 then

 A163: |.(((F # x) . N) - (f . x)).| = |.((f . x) - ((F # x) . N)).| by EXTREAL1: 29;

 (2 |^ N) > 0 by PREPOWER: 6;

 then

 A164: ( - (1 / (2 |^ N))) < 0 ;

 0 <= ((f . x) - ((F # x) . N)) by A158, A160, XXREAL_3: 40;

 hence thesis by A163, A162, A164, EXTREAL1: 22;

 end;

 assume 0 < p;

 then

 consider N1 be Nat such that

 A165: (1 qua Complex / (2 |^ N1)) <= p by PREPOWER: 92;

 reconsider k = ( max (N2,N1)) as Element of NAT by ORDINAL1:def 12;



 A166: for k be Nat st k >= N1 holds (1 / (2 |^ k)) <= p

 proof

 let k be Nat;

 assume k >= N1;

 then

 consider i be Nat such that

 A167: k = (N1 qua Complex + i) by NAT_1: 10;

 ((2 |^ N1) * (2 |^ i)) >= (2 |^ N1) by PREPOWER: 11, XREAL_1: 151;

 then

 A168: (2 |^ k) >= (2 |^ N1) by A167, NEWTON: 8;

 (2 |^ N1) > 0 by PREPOWER: 11;

 then ((2 |^ k) " ) <= ((2 |^ N1) " ) by A168, XREAL_1: 85;

 then (1 / (2 |^ k)) <= ((2 |^ N1) " );

 then (1 / (2 |^ k)) <= (1 / (2 |^ N1));

 hence thesis by A165, XXREAL_0: 2;

 end;

 now

 let j be Nat;

 assume

 A169: j >= k;

 k >= N2 by XXREAL_0: 25;

 then j >= N2 by A169, XXREAL_0: 2;

 then

 A170: |.(((F # x) . j) - (f . x)).| < (1 / (2 |^ j)) by A153;

 k >= N1 by XXREAL_0: 25;

 then j >= N1 by A169, XXREAL_0: 2;

 then (1 / (2 |^ j)) <= p by A166;

 hence |.(((F # x) . j) - (f . x)).| < p by A170, XXREAL_0: 2;

 end;

 hence thesis;

 end;



 A171: (f . x) = g;

 then

 A172: (F # x) is convergent_to_finite_number by A150;

 then (F # x) is convergent;

 hence (F # x) is convergent & ( lim (F # x)) = (f . x) by A171, A150, A172, Def12;

 end;

 end;

 for n be Nat holds (F . n) is nonnegative

 proof

 let n be Nat;

 reconsider nn = n as Element of NAT by ORDINAL1:def 12;

 now

 let x be object;

 assume x in ( dom (F . n));

 then

 A173: x in ( dom f) by A25;

 per cases ;

 suppose n <= (f . x);

 hence 0 <= ((F . nn) . x) by A24, A173;

 end;

 suppose

 A174: (f . x) < n;

 0 <= (f . x) by A2, SUPINF_2: 51;

 then

 consider k be Nat such that

 A175: 1 <= k and

 A176: k <= ((2 |^ n) * n) and

 A177: ((k - 1) / (2 |^ n)) <= (f . x) and

 A178: (f . x) < (k / (2 |^ n)) by A174, Th4;

 thus 0 <= ((F . nn) . x) by A24, A173, A175, A176, A177, A178;

 end;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 hence thesis by A25, A51, A28, A143;

 end;

 begin

 definition

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 :: MESFUNC5:def14

 func integral' (M,f) -> Element of ExtREAL equals

 : Def14: ( integral (M,f)) if ( dom f) <> {}

 otherwise 0. ;

 correctness ;

 end

 theorem :: MESFUNC5:65



 Th65: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & g is_simple_func_in S & f is nonnegative & g is nonnegative holds ( dom (f + g)) = (( dom f) /\ ( dom g)) & ( integral' (M,(f + g))) = (( integral' (M,(f | ( dom (f + g))))) + ( integral' (M,(g | ( dom (f + g))))))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: g is_simple_func_in S and

 A3: f is nonnegative and

 A4: g is nonnegative;



 A5: (g | ( dom (f + g))) is nonnegative by A4, Th15;



 A: (f | ( dom (f + g))) is nonnegative by A3, Th15;

 not -infty in ( rng g) by A4, Def3;

 then

 A7: (g " { -infty }) = {} by FUNCT_1: 72;

 not -infty in ( rng f) by A3, Def3;

 then

 A8: (f " { -infty }) = {} by FUNCT_1: 72;

 then

 A9: ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) = (( dom f) /\ ( dom g)) by A7;

 hence

 A10: ( dom (f + g)) = (( dom f) /\ ( dom g)) by MESFUNC1:def 3;



 A11: ( dom (f + g)) is Element of S by A1, A2, Th37, Th38;

 then

 A12: (f | ( dom (f + g))) is_simple_func_in S by A1, Th34;



 A13: (g | ( dom (f + g))) is_simple_func_in S by A2, A11, Th34;

 ( dom (f | ( dom (f + g)))) = (( dom f) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A14: ( dom (f | ( dom (f + g)))) = ((( dom f) /\ ( dom f)) /\ ( dom g)) by A10, XBOOLE_1: 16;

 ( dom (g | ( dom (f + g)))) = (( dom g) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A15: ( dom (g | ( dom (f + g)))) = ((( dom g) /\ ( dom g)) /\ ( dom f)) by A10, XBOOLE_1: 16;

 per cases ;

 suppose

 A16: ( dom (f + g)) = {} ;

 ( dom (g | ( dom (f + g)))) = (( dom g) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A17: ( integral' (M,(g | ( dom (f + g))))) = 0 by A16, Def14;

 ( dom (f | ( dom (f + g)))) = (( dom f) /\ ( dom (f + g))) by RELAT_1: 61;

 then

 A18: ( integral' (M,(f | ( dom (f + g))))) = 0 by A16, Def14;

 ( integral' (M,(f + g))) = 0 by A16, Def14;

 hence thesis by A18, A17;

 end;

 suppose

 A19: ( dom (f + g)) <> {} ;



 A20: ((g | ( dom (f + g))) " { -infty }) = (( dom (f + g)) /\ (g " { -infty })) by FUNCT_1: 70

 .= {} by A7;

 ((f | ( dom (f + g))) " { -infty }) = (( dom (f + g)) /\ (f " { -infty })) by FUNCT_1: 70

 .= {} by A8;

 then ((( dom (f | ( dom (f + g)))) /\ ( dom (g | ( dom (f + g))))) \ ((((f | ( dom (f + g))) " { -infty }) /\ ((g | ( dom (f + g))) " { +infty })) \/ (((f | ( dom (f + g))) " { +infty }) /\ ((g | ( dom (f + g))) " { -infty })))) = ( dom (f + g)) by A9, A14, A15, A20, MESFUNC1:def 3;

 then

 A21: ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g))))) = ( dom (f + g)) by MESFUNC1:def 3;



 A22: for x be Element of X st x in ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g))))) holds (((f | ( dom (f + g))) + (g | ( dom (f + g)))) . x) = ((f + g) . x)

 proof

 let x be Element of X;

 assume

 A23: x in ( dom ((f | ( dom (f + g))) + (g | ( dom (f + g)))));



 then (((f | ( dom (f + g))) + (g | ( dom (f + g)))) . x) = (((f | ( dom (f + g))) . x) + ((g | ( dom (f + g))) . x)) by MESFUNC1:def 3

 .= ((f . x) + ((g | ( dom (f + g))) . x)) by A21, A23, FUNCT_1: 49

 .= ((f . x) + (g . x)) by A21, A23, FUNCT_1: 49;

 hence thesis by A21, A23, MESFUNC1:def 3;

 end;

 ( integral (M,((f | ( dom (f + g))) + (g | ( dom (f + g)))))) = (( integral (M,(f | ( dom (f + g))))) + ( integral (M,(g | ( dom (f + g)))))) by A10, A12, A13, A14, A15, A19, MESFUNC4: 5, A, A5;

 then

 A24: ( integral (M,(f + g))) = (( integral (M,(f | ( dom (f + g))))) + ( integral (M,(g | ( dom (f + g)))))) by A21, A22, PARTFUN1: 5;



 A25: ( integral (M,(g | ( dom (f + g))))) = ( integral' (M,(g | ( dom (f + g))))) by A10, A15, A19, Def14;

 ( integral (M,(f | ( dom (f + g))))) = ( integral' (M,(f | ( dom (f + g))))) by A10, A14, A19, Def14;

 hence thesis by A19, A24, A25, Def14;

 end;

 end;

 theorem :: MESFUNC5:66



 Th66: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real st f is_simple_func_in S & f is nonnegative & 0 <= c holds ( integral' (M,(c (#) f))) = (c * ( integral' (M,f)))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let c be Real;

 assume that

 A1: f is_simple_func_in S and

 A2: f is nonnegative and

 A3: 0 <= c;

 set g = (c (#) f);



 A5: ( dom g) = ( dom f) by MESFUNC1:def 6;



 A6: for x be set st x in ( dom g) holds (g . x) = (c * (f . x)) by MESFUNC1:def 6;

 per cases ;

 suppose

 A7: ( dom g) = {} ;

 then ( integral' (M,f)) = 0 by A5, Def14;

 then (c * ( integral' (M,f))) = 0 ;

 hence thesis by A7, Def14;

 end;

 suppose

 A8: ( dom g) <> {} ;

 then

 A9: ( integral' (M,f)) = ( integral (M,f)) by A5, Def14;

 reconsider cc = c as R_eal by XXREAL_0:def 1;

 c in REAL by XREAL_0:def 1;

 then c < +infty by XXREAL_0: 9;

 then ( integral (M,g)) = (cc * ( integral' (M,f))) by A1, A3, A5, A2, A6, A8, MESFUNC4: 6, A9;

 hence thesis by A8, Def14;

 end;

 end;

 theorem :: MESFUNC5:67



 Th67: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st f is_simple_func_in S & f is nonnegative & A misses B holds ( integral' (M,(f | (A \/ B)))) = (( integral' (M,(f | A))) + ( integral' (M,(f | B))))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let A,B be Element of S;

 assume that

 A1: f is_simple_func_in S and

 A2: f is nonnegative and

 A3: A misses B;

 set g2 = (f | B);

 set g1 = (f | A);

 set g = (f | (A \/ B));



 a4: g is nonnegative by A2, Th15;

 consider G be Finite_Sep_Sequence of S, b be FinSequence of ExtREAL such that

 A5: (G,b) are_Re-presentation_of g and

 A6: (b . 1) = 0 and

 A7: for n be Nat st 2 <= n & n in ( dom b) holds 0 < (b . n) & (b . n) < +infty by A1, Th34, MESFUNC3: 14, a4;

 deffunc G1( Nat) = ((G . $1) /\ A);

 consider G1 be FinSequence such that

 A8: ( len G1) = ( len G) & for k be Nat st k in ( dom G1) holds (G1 . k) = G1(k) from FINSEQ_1:sch 2;



 A9: ( dom G1) = ( Seg ( len G)) by A8, FINSEQ_1:def 3;



 A10: for k be Nat st k in ( dom G) holds (G1 . k) = ((G . k) /\ A)

 proof

 let k be Nat;

 assume k in ( dom G);

 then k in ( Seg ( len G)) by FINSEQ_1:def 3;

 hence thesis by A8, A9;

 end;

 deffunc G2( Nat) = ((G . $1) /\ B);

 consider G2 be FinSequence such that

 A11: ( len G2) = ( len G) & for k be Nat st k in ( dom G2) holds (G2 . k) = G2(k) from FINSEQ_1:sch 2;



 A12: ( dom G2) = ( Seg ( len G)) by A11, FINSEQ_1:def 3;



 A13: for k be Nat st k in ( dom G) holds (G2 . k) = ((G . k) /\ B)

 proof

 let k be Nat;

 assume k in ( dom G);

 then k in ( Seg ( len G)) by FINSEQ_1:def 3;

 hence thesis by A11, A12;

 end;



 A14: ( dom G) = ( Seg ( len G2)) by A11, FINSEQ_1:def 3

 .= ( dom G2) by FINSEQ_1:def 3;

 then

 reconsider G2 as Finite_Sep_Sequence of S by A13, Th35;



 A15: ( dom (g | B)) = (( dom g) /\ B) by RELAT_1: 61

 .= ((( dom f) /\ (A \/ B)) /\ B) by RELAT_1: 61

 .= (( dom f) /\ ((A \/ B) /\ B)) by XBOOLE_1: 16

 .= (( dom f) /\ B) by XBOOLE_1: 21

 .= ( dom g2) by RELAT_1: 61;

 for x be object st x in ( dom (g | B)) holds ((g | B) . x) = (g2 . x)

 proof

 let x be object;

 assume

 A16: x in ( dom (g | B));

 then x in (( dom g) /\ B) by RELAT_1: 61;

 then

 A17: x in ( dom g) by XBOOLE_0:def 4;

 ((g | B) . x) = (g . x) by A16, FUNCT_1: 47

 .= (f . x) by A17, FUNCT_1: 47;

 hence thesis by A15, A16, FUNCT_1: 47;

 end;

 then (g | B) = g2 by A15, FUNCT_1: 2;

 then

 A18: (G2,b) are_Re-presentation_of g2 by A5, A14, A13, Th36;



 A19: ( dom G) = ( Seg ( len G1)) by A8, FINSEQ_1:def 3

 .= ( dom G1) by FINSEQ_1:def 3;

 then

 reconsider G1 as Finite_Sep_Sequence of S by A10, Th35;



 A20: ( dom (g | A)) = (( dom g) /\ A) by RELAT_1: 61

 .= ((( dom f) /\ (A \/ B)) /\ A) by RELAT_1: 61

 .= (( dom f) /\ ((A \/ B) /\ A)) by XBOOLE_1: 16

 .= (( dom f) /\ A) by XBOOLE_1: 21

 .= ( dom g1) by RELAT_1: 61;

 for x be object st x in ( dom (g | A)) holds ((g | A) . x) = (g1 . x)

 proof

 let x be object;

 assume

 A21: x in ( dom (g | A));

 then x in (( dom g) /\ A) by RELAT_1: 61;

 then

 A22: x in ( dom g) by XBOOLE_0:def 4;

 ((g | A) . x) = (g . x) by A21, FUNCT_1: 47

 .= (f . x) by A22, FUNCT_1: 47;

 hence thesis by A20, A21, FUNCT_1: 47;

 end;

 then (g | A) = g1 by A20, FUNCT_1: 2;

 then

 A23: (G1,b) are_Re-presentation_of g1 by A5, A19, A10, Th36;

 deffunc Fy( Nat) = ((b . $1) * ((M * G) . $1));

 consider y be FinSequence of ExtREAL such that

 A24: ( len y) = ( len G) & for j be Nat st j in ( dom y) holds (y . j) = Fy(j) from FINSEQ_2:sch 1;



 A25: ( dom y) = ( Seg ( len y)) by FINSEQ_1:def 3

 .= ( dom G) by A24, FINSEQ_1:def 3;

 per cases ;

 suppose

 A26: ( dom (f | (A \/ B))) = {} ;

 (( dom f) /\ B) c= (( dom f) /\ (A \/ B)) by XBOOLE_1: 7, XBOOLE_1: 26;

 then ( dom (f | B)) c= (( dom f) /\ (A \/ B)) by RELAT_1: 61;

 then ( dom (f | B)) c= ( dom (f | (A \/ B))) by RELAT_1: 61;

 then ( dom (f | B)) = {} by A26;

 then

 A27: ( integral' (M,g2)) = 0 by Def14;

 (( dom f) /\ A) c= (( dom f) /\ (A \/ B)) by XBOOLE_1: 7, XBOOLE_1: 26;

 then ( dom (f | A)) c= (( dom f) /\ (A \/ B)) by RELAT_1: 61;

 then ( dom (f | A)) c= ( dom (f | (A \/ B))) by RELAT_1: 61;

 then ( dom (f | A)) = {} by A26;

 then

 A28: ( integral' (M,g1)) = 0 by Def14;

 ( integral' (M,g)) = 0 by A26, Def14;

 hence thesis by A28, A27;

 end;

 suppose

 A29: ( dom (f | (A \/ B))) <> {} ;

 then ( integral (M,g)) = ( Sum y) by A1, a4, A5, A24, A25, Th34, MESFUNC4: 3;

 then

 A30: ( integral' (M,g)) = ( Sum y) by A29, Def14;

 per cases ;

 suppose

 A31: ( dom (f | A)) = {} ;



 A32: ( dom (f | (A \/ B))) = (( dom f) /\ (A \/ B)) by RELAT_1: 61

 .= ((( dom f) /\ A) \/ (( dom f) /\ B)) by XBOOLE_1: 23

 .= (( dom (f | A)) \/ (( dom f) /\ B)) by RELAT_1: 61

 .= ( dom (f | B)) by A31, RELAT_1: 61;

 now

 let x be object;

 assume

 A33: x in ( dom g);

 then (g . x) = (f . x) by FUNCT_1: 47;

 hence (g . x) = (g2 . x) by A32, A33, FUNCT_1: 47;

 end;

 then

 A34: g = g2 by A32, FUNCT_1: 2;

 ( integral' (M,g1)) = 0 by A31, Def14;

 hence thesis by A34, XXREAL_3: 4;

 end;

 suppose

 A35: ( dom (f | A)) <> {} ;

 per cases ;

 suppose

 A36: ( dom (f | B)) = {} ;



 A37: ( dom (f | (A \/ B))) = (( dom f) /\ (A \/ B)) by RELAT_1: 61

 .= ((( dom f) /\ B) \/ (( dom f) /\ A)) by XBOOLE_1: 23

 .= (( dom (f | B)) \/ (( dom f) /\ A)) by RELAT_1: 61

 .= ( dom (f | A)) by A36, RELAT_1: 61;

 now

 let x be object;

 assume

 A38: x in ( dom g);

 then (g . x) = (f . x) by FUNCT_1: 47;

 hence (g . x) = (g1 . x) by A37, A38, FUNCT_1: 47;

 end;

 then

 A39: g = g1 by A37, FUNCT_1: 2;

 ( integral' (M,g2)) = 0 by A36, Def14;

 hence thesis by A39, XXREAL_3: 4;

 end;

 suppose

 A40: ( dom (f | B)) <> {} ;



 aa: g2 is nonnegative by A2, Th15;

 deffunc Fy2( Nat) = ((b . $1) * ((M * G2) . $1));

 consider y2 be FinSequence of ExtREAL such that

 A42: ( len y2) = ( len G2) & for j be Nat st j in ( dom y2) holds (y2 . j) = Fy2(j) from FINSEQ_2:sch 1;



 A43: for k be Nat st k in ( dom y2) holds 0 <= (y2 . k)

 proof

 let k be Nat;

 assume

 A44: k in ( dom y2);

 then k in ( Seg ( len y2)) by FINSEQ_1:def 3;

 then

 A45: 1 <= k by FINSEQ_1: 1;



 A46: ( dom b) = ( dom G) by A5, MESFUNC3:def 1

 .= ( Seg ( len y2)) by A11, A42, FINSEQ_1:def 3

 .= ( dom y2) by FINSEQ_1:def 3;

 A47:

 now

 per cases ;

 suppose k = 1;

 hence 0 <= (b . k) by A6;

 end;

 suppose k <> 1;

 then 1 < k by A45, XXREAL_0: 1;

 then (1 + 1) <= k by NAT_1: 13;

 hence 0 <= (b . k) by A7, A44, A46;

 end;

 end;

 k in ( Seg ( len G2)) by A42, A44, FINSEQ_1:def 3;

 then

 A48: k in ( dom G2) by FINSEQ_1:def 3;

 then

 A49: ((M * G2) . k) = (M . (G2 . k)) by FUNCT_1: 13;

 (G2 . k) in ( rng G2) by A48, FUNCT_1: 3;

 then

 reconsider G2k = (G2 . k) as Element of S;



 A50: 0 <= (M . G2k) by SUPINF_2: 39;

 (y2 . k) = ((b . k) * ((M * G2) . k)) by A42, A44;

 hence thesis by A47, A49, A50;

 end;

 then for k be object st k in ( dom y2) holds 0 <= (y2 . k);

 then

 cc: y2 is nonnegative by SUPINF_2: 52;



 A51: for x be set st x in ( dom y2) holds not (y2 . x) in { -infty }

 proof

 let x be set;

 assume

 A52: x in ( dom y2);

 then

 reconsider x as Element of NAT ;

 0 <= (y2 . x) by A43, A52;

 hence thesis by TARSKI:def 1;

 end;

 for x be object holds not x in (y2 " { -infty })

 proof

 let x be object;

 not (x in ( dom y2) & (y2 . x) in { -infty }) by A51;

 hence thesis by FUNCT_1:def 7;

 end;

 then

 A53: (y2 " { -infty }) = {} by XBOOLE_0:def 1;

 ( dom y2) = ( Seg ( len G2)) by A42, FINSEQ_1:def 3

 .= ( dom G2) by FINSEQ_1:def 3;

 then ( integral (M,g2)) = ( Sum y2) by A1, A18, A40, A42, Th34, MESFUNC4: 3, aa;

 then

 A54: ( integral' (M,g2)) = ( Sum y2) by A40, Def14;



 ac: g1 is nonnegative by A2, Th15;

 deffunc Fy1( Nat) = ((b . $1) * ((M * G1) . $1));

 consider y1 be FinSequence of ExtREAL such that

 A56: ( len y1) = ( len G1) & for j be Nat st j in ( dom y1) holds (y1 . j) = Fy1(j) from FINSEQ_2:sch 1;



 A57: ( dom y) = (( Seg ( len G)) /\ ( Seg ( len G))) by A25, FINSEQ_1:def 3

 .= (( dom y1) /\ ( Seg ( len G2))) by A8, A11, A56, FINSEQ_1:def 3

 .= (( dom y1) /\ ( dom y2)) by A42, FINSEQ_1:def 3;



 A58: for n be Element of NAT st n in ( dom y) holds (y . n) = ((y1 . n) + (y2 . n))

 proof

 let n be Element of NAT ;

 assume

 A59: n in ( dom y);

 then n in ( Seg ( len G)) by A24, FINSEQ_1:def 3;

 then

 A60: n in ( dom G) by FINSEQ_1:def 3;

 then

 A61: (G2 . n) = ((G . n) /\ B) by A13;

 now

 let v be object;

 assume

 A62: v in (G . n);

 (G . n) in ( rng G) by A60, FUNCT_1: 3;

 then v in ( union ( rng G)) by A62, TARSKI:def 4;

 then v in ( dom g) by A5, MESFUNC3:def 1;

 then v in (( dom f) /\ (A \/ B)) by RELAT_1: 61;

 hence v in (A \/ B) by XBOOLE_0:def 4;

 end;

 then (G . n) c= (A \/ B);



 then

 A63: (G . n) = ((G . n) /\ (A \/ B)) by XBOOLE_1: 28

 .= (((G . n) /\ A) \/ ((G . n) /\ B)) by XBOOLE_1: 23

 .= ((G1 . n) \/ (G2 . n)) by A10, A60, A61;



 A64: n in ( dom y2) by A57, A59, XBOOLE_0:def 4;

 then n in ( Seg ( len G2)) by A42, FINSEQ_1:def 3;

 then

 A65: n in ( dom G2) by FINSEQ_1:def 3;

 then (G2 . n) in ( rng G2) by FUNCT_1: 3;

 then

 reconsider G2n = (G2 . n) as Element of S;

 0 <= (M . G2n) by MEASURE1:def 2;

 then

 A66: 0 = ((M * G2) . n) or 0 < ((M * G2) . n) by A65, FUNCT_1: 13;

 A67:

 now

 assume (G1 . n) meets (G2 . n);

 then

 consider v be object such that

 A68: v in (G1 . n) and

 A69: v in (G2 . n) by XBOOLE_0: 3;

 v in ((G . n) /\ B) by A13, A60, A69;

 then

 A70: v in B by XBOOLE_0:def 4;

 v in ((G . n) /\ A) by A10, A60, A68;

 then v in A by XBOOLE_0:def 4;

 hence contradiction by A3, A70, XBOOLE_0: 3;

 end;



 A71: n in ( dom y1) by A57, A59, XBOOLE_0:def 4;

 then n in ( Seg ( len G1)) by A56, FINSEQ_1:def 3;

 then

 A72: n in ( dom G1) by FINSEQ_1:def 3;

 then (G1 . n) in ( rng G1) by FUNCT_1: 3;

 then

 reconsider G1n = (G1 . n) as Element of S;

 0 <= (M . G1n) by MEASURE1:def 2;

 then

 A73: 0 = ((M * G1) . n) or 0 < ((M * G1) . n) by A72, FUNCT_1: 13;

 ((M * G) . n) = (M . (G . n)) by A60, FUNCT_1: 13

 .= ((M . G1n) + (M . G2n)) by A63, A67, MEASURE1: 30

 .= (((M * G1) . n) + (M . (G2 . n))) by A72, FUNCT_1: 13

 .= (((M * G1) . n) + ((M * G2) . n)) by A65, FUNCT_1: 13;

 then ((b . n) * ((M * G) . n)) = (((b . n) * ((M * G1) . n)) + ((b . n) * ((M * G2) . n))) by A73, A66, XXREAL_3: 96;

 then (y . n) = (((b . n) * ((M * G1) . n)) + ((b . n) * ((M * G2) . n))) by A24, A59;

 then (y . n) = ((y1 . n) + ((b . n) * ((M * G2) . n))) by A56, A71;

 hence thesis by A42, A64;

 end;



 A74: for k be Nat st k in ( dom y1) holds 0 <= (y1 . k)

 proof

 let k be Nat;

 assume

 A75: k in ( dom y1);

 then k in ( Seg ( len y1)) by FINSEQ_1:def 3;

 then

 A76: 1 <= k by FINSEQ_1: 1;



 A77: ( dom b) = ( dom G) by A5, MESFUNC3:def 1

 .= ( Seg ( len y1)) by A8, A56, FINSEQ_1:def 3

 .= ( dom y1) by FINSEQ_1:def 3;

 A78:

 now

 per cases ;

 suppose k = 1;

 hence 0 <= (b . k) by A6;

 end;

 suppose k <> 1;

 then 1 < k by A76, XXREAL_0: 1;

 then (1 + 1) <= k by NAT_1: 13;

 hence 0 <= (b . k) by A7, A75, A77;

 end;

 end;

 k in ( Seg ( len G1)) by A56, A75, FINSEQ_1:def 3;

 then

 A79: k in ( dom G1) by FINSEQ_1:def 3;

 then

 A80: ((M * G1) . k) = (M . (G1 . k)) by FUNCT_1: 13;

 (G1 . k) in ( rng G1) by A79, FUNCT_1: 3;

 then

 reconsider G1k = (G1 . k) as Element of S;



 A81: 0 <= (M . G1k) by SUPINF_2: 39;

 (y1 . k) = ((b . k) * ((M * G1) . k)) by A56, A75;

 hence thesis by A78, A80, A81;

 end;

 then for x be object st x in ( dom y1) holds 0. <= (y1 . x);

 then

 ab: y1 is nonnegative by SUPINF_2: 52;



 A82: for x be set st x in ( dom y1) holds not (y1 . x) in { -infty }

 proof

 let x be set;

 assume

 A83: x in ( dom y1);

 then

 reconsider x as Element of NAT ;

 0 <= (y1 . x) by A74, A83;

 hence thesis by TARSKI:def 1;

 end;

 for x be object holds not x in (y1 " { -infty })

 proof

 let x be object;

 not (x in ( dom y1) & (y1 . x) in { -infty }) by A82;

 hence thesis by FUNCT_1:def 7;

 end;

 then (y1 " { -infty }) = {} by XBOOLE_0:def 1;

 then ( dom y) = ((( dom y1) /\ ( dom y2)) \ (((y1 " { -infty }) /\ (y2 " { +infty })) \/ ((y1 " { +infty }) /\ (y2 " { -infty })))) by A53, A57;

 then

 A84: y = (y1 + y2) by A58, MESFUNC1:def 3;

 ( dom y1) = ( Seg ( len G1)) by A56, FINSEQ_1:def 3

 .= ( dom G1) by FINSEQ_1:def 3;

 then ( integral (M,g1)) = ( Sum y1) by A1, A23, A35, A56, Th34, MESFUNC4: 3, ac;

 then

 A85: ( integral' (M,g1)) = ( Sum y1) by A35, Def14;

 ( dom y1) = ( Seg ( len y2)) by A8, A11, A56, A42, FINSEQ_1:def 3

 .= ( dom y2) by FINSEQ_1:def 3;

 hence thesis by A30, A85, A54, A84, MESFUNC4: 1, ab, cc;

 end;

 end;

 end;

 end;

 theorem :: MESFUNC5:68



 Th68: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & f is nonnegative holds 0 <= ( integral' (M,f))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: f is nonnegative;

 per cases ;

 suppose ( dom f) = {} ;

 hence thesis by Def14;

 end;

 suppose

 A4: ( dom f) <> {} ;

 then

 consider F be Finite_Sep_Sequence of S, a,x be FinSequence of ExtREAL such that

 A5: (F,a) are_Re-presentation_of f and

 A6: ( dom x) = ( dom F) and

 A7: for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n)) and

 A8: ( integral (M,f)) = ( Sum x) by A1, A2, MESFUNC4: 4;



 A9: for n be Nat st n in ( dom x) holds 0 <= (x . n)

 proof

 let n be Nat;

 assume

 A10: n in ( dom x);

 per cases ;

 suppose (F . n) = {} ;

 then (M . (F . n)) = 0 by VALUED_0:def 19;

 then ((M * F) . n) = 0 by A6, A10, FUNCT_1: 13;

 then ((a . n) * ((M * F) . n)) = 0 ;

 hence thesis by A7, A10;

 end;

 suppose (F . n) <> {} ;

 then

 consider v be object such that

 A11: v in (F . n) by XBOOLE_0:def 1;

 (F . n) in ( rng F) by A6, A10, FUNCT_1: 3;

 then

 reconsider Fn = (F . n) as Element of S;

 0 <= (M . Fn) by MEASURE1:def 2;

 then

 A12: 0 <= ((M * F) . n) by A6, A10, FUNCT_1: 13;

 (f . v) = (a . n) by A5, A6, A10, A11, MESFUNC3:def 1;

 then 0 <= (a . n) by A2, SUPINF_2: 51;

 then 0 <= ((a . n) * ((M * F) . n)) by A12;

 hence thesis by A7, A10;

 end;

 end;

 ( integral' (M,f)) = ( integral (M,f)) by A4, Def14;

 hence thesis by A8, A9, Th53;

 end;

 end;



 Lm9: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & ( dom f) <> {} & f is nonnegative & g is_simple_func_in S & ( dom g) = ( dom f) & g is nonnegative & (for x be set st x in ( dom f) holds (g . x) <= (f . x)) holds (f - g) is_simple_func_in S & ( dom (f - g)) <> {} & (f - g) is nonnegative & ( integral (M,f)) = (( integral (M,(f - g))) + ( integral (M,g)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: f is_simple_func_in S and

 A2: ( dom f) <> {} and

 A3: f is nonnegative and

 A4: g is_simple_func_in S and

 A5: ( dom g) = ( dom f) and

 A6: g is nonnegative and

 A7: for x be set st x in ( dom f) holds (g . x) <= (f . x);

 consider G be Finite_Sep_Sequence of S, b,y be FinSequence of ExtREAL such that

 A9: (G,b) are_Re-presentation_of g and ( dom y) = ( dom G) and for n be Nat st n in ( dom y) holds (y . n) = ((b . n) * ((M * G) . n)) and ( integral (M,g)) = ( Sum y) by A2, A4, A5, A6, MESFUNC4: 4;

 g is real-valued by A4, MESFUNC2:def 4;

 then

 A10: ( dom (f - g)) = (( dom f) /\ ( dom g)) by MESFUNC2: 2;

 consider F be Finite_Sep_Sequence of S, a,x be FinSequence of ExtREAL such that

 A12: (F,a) are_Re-presentation_of f and ( dom x) = ( dom F) and for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n)) and ( integral (M,f)) = ( Sum x) by A1, A2, MESFUNC4: 4, A3;

 set la = ( len a);



 A13: ( dom F) = ( dom a) by A12, MESFUNC3:def 1;

 set lb = ( len b);

 deffunc FG1( Nat) = ((F . ((($1 -' 1) div lb) + 1)) /\ (G . ((($1 -' 1) mod lb) + 1)));

 consider FG be FinSequence such that

 A14: ( len FG) = (la * lb) and

 A15: for k be Nat st k in ( dom FG) holds (FG . k) = FG1(k) from FINSEQ_1:sch 2;



 A16: ( dom FG) = ( Seg (la * lb)) by A14, FINSEQ_1:def 3;



 A17: ( dom G) = ( dom b) by A9, MESFUNC3:def 1;

 FG is FinSequence of S

 proof

 let b be object;

 A18:

 now

 let k be Element of NAT ;

 set i = (((k -' 1) div lb) + 1);

 set j = (((k -' 1) mod lb) + 1);



 A19: lb divides (la * lb) by NAT_D:def 3;

 assume

 A20: k in ( dom FG);

 then

 A21: k in ( Seg (la * lb)) by A14, FINSEQ_1:def 3;

 then

 A22: k <= (la * lb) by FINSEQ_1: 1;

 then (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 then

 A23: ((k -' 1) div lb) <= (((la * lb) -' 1) div lb) by NAT_2: 24;

 1 <= k by A21, FINSEQ_1: 1;

 then

 A24: 1 <= (la * lb) by A22, XXREAL_0: 2;



 A25: lb <> 0 by A21;

 then ((k -' 1) mod lb) < lb by NAT_D: 1;

 then

 A26: j <= lb by NAT_1: 13;

 lb >= ( 0 + 1) by A25, NAT_1: 13;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A19, A24, NAT_2: 15;

 then (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by A23, XREAL_1: 19;

 then

 A27: i <= la by A25, NAT_D: 18;

 i >= ( 0 + 1) by XREAL_1: 6;

 then i in ( Seg la) by A27;

 then i in ( dom F) by A13, FINSEQ_1:def 3;

 then

 A28: (F . i) in ( rng F) by FUNCT_1: 3;

 j >= ( 0 + 1) by XREAL_1: 6;

 then j in ( Seg lb) by A26;

 then j in ( dom G) by A17, FINSEQ_1:def 3;

 then

 A29: (G . j) in ( rng G) by FUNCT_1: 3;

 (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A15, A20;

 hence (FG . k) in S by A28, A29, MEASURE1: 34;

 end;

 assume b in ( rng FG);

 then ex a be object st a in ( dom FG) & b = (FG . a) by FUNCT_1:def 3;

 hence thesis by A18;

 end;

 then

 reconsider FG as FinSequence of S;

 for k,l be Nat st k in ( dom FG) & l in ( dom FG) & k <> l holds (FG . k) misses (FG . l)

 proof

 let k,l be Nat;

 assume that

 A30: k in ( dom FG) and

 A31: l in ( dom FG) and

 A32: k <> l;



 A33: k in ( Seg (la * lb)) by A14, A30, FINSEQ_1:def 3;

 then

 A34: 1 <= k by FINSEQ_1: 1;

 set m = (((l -' 1) mod lb) + 1);

 set n = (((l -' 1) div lb) + 1);

 set j = (((k -' 1) mod lb) + 1);

 set i = (((k -' 1) div lb) + 1);



 A35: lb divides (la * lb) by NAT_D:def 3;

 (FG . k) = ((F . i) /\ (G . j)) by A15, A30;



 then

 A36: ((FG . k) /\ (FG . l)) = (((F . i) /\ (G . j)) /\ ((F . n) /\ (G . m))) by A15, A31

 .= ((F . i) /\ ((G . j) /\ ((F . n) /\ (G . m)))) by XBOOLE_1: 16

 .= ((F . i) /\ ((F . n) /\ ((G . j) /\ (G . m)))) by XBOOLE_1: 16

 .= (((F . i) /\ (F . n)) /\ ((G . j) /\ (G . m))) by XBOOLE_1: 16;



 A37: k <= (la * lb) by A33, FINSEQ_1: 1;

 then

 A38: 1 <= (la * lb) by A34, XXREAL_0: 2;



 A39: lb <> 0 by A33;

 then lb >= ( 0 + 1) by NAT_1: 13;

 then

 A40: (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A35, A38, NAT_2: 15;

 (k -' 1) <= ((la * lb) -' 1) by A37, NAT_D: 42;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A40, NAT_2: 24;

 then (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 then

 A41: i <= la by A39, NAT_D: 18;

 i >= ( 0 + 1) by XREAL_1: 6;

 then i in ( Seg la) by A41;

 then

 A42: i in ( dom F) by A13, FINSEQ_1:def 3;

 ((l -' 1) mod lb) < lb by A39, NAT_D: 1;

 then

 A43: m <= lb by NAT_1: 13;

 m >= ( 0 + 1) by XREAL_1: 6;

 then m in ( Seg lb) by A43;

 then

 A44: m in ( dom G) by A17, FINSEQ_1:def 3;

 ((k -' 1) mod lb) < lb by A39, NAT_D: 1;

 then

 A45: j <= lb by NAT_1: 13;

 j >= ( 0 + 1) by XREAL_1: 6;

 then j in ( Seg lb) by A45;

 then

 A46: j in ( dom G) by A17, FINSEQ_1:def 3;



 A47: l in ( Seg (la * lb)) by A14, A31, FINSEQ_1:def 3;

 then

 A48: 1 <= l by FINSEQ_1: 1;

 A49:

 now

 ((l -' 1) + 1) = ((((n - 1) * lb) + (m - 1)) + 1) by A39, NAT_D: 2;

 then

 A50: ((l - 1) + 1) = (((n - 1) * lb) + m) by A48, XREAL_1: 233;

 assume that

 A51: i = n and

 A52: j = m;

 ((k -' 1) + 1) = ((((i - 1) * lb) + (j - 1)) + 1) by A39, NAT_D: 2;

 then ((k - 1) + 1) = (((i - 1) * lb) + j) by A34, XREAL_1: 233;

 hence contradiction by A32, A51, A52, A50;

 end;

 l <= (la * lb) by A47, FINSEQ_1: 1;

 then (l -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 then ((l -' 1) div lb) <= (((la * lb) div lb) - 1) by A40, NAT_2: 24;

 then (((l -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 then

 A53: n <= la by A39, NAT_D: 18;

 n >= ( 0 + 1) by XREAL_1: 6;

 then n in ( Seg la) by A53;

 then

 A54: n in ( dom F) by A13, FINSEQ_1:def 3;

 per cases by A49;

 suppose i <> n;

 then (F . i) misses (F . n) by A42, A54, MESFUNC3: 4;

 then ((FG . k) /\ (FG . l)) = ( {} /\ ((G . j) /\ (G . m))) by A36;

 hence thesis;

 end;

 suppose j <> m;

 then (G . j) misses (G . m) by A46, A44, MESFUNC3: 4;

 then ((FG . k) /\ (FG . l)) = (((F . i) /\ (F . n)) /\ {} ) by A36;

 hence thesis;

 end;

 end;

 then

 reconsider FG as Finite_Sep_Sequence of S by MESFUNC3: 4;



 A55: ( dom f) = ( union ( rng F)) by A12, MESFUNC3:def 1;

 defpred PB[ Nat, set] means ((G . ((($1 -' 1) mod lb) + 1)) = {} implies $2 = 0 ) & ((G . ((($1 -' 1) mod lb) + 1)) <> {} implies $2 = (b . ((($1 -' 1) mod lb) + 1)));

 defpred PA[ Nat, set] means ((F . ((($1 -' 1) div lb) + 1)) = {} implies $2 = 0 ) & ((F . ((($1 -' 1) div lb) + 1)) <> {} implies $2 = (a . ((($1 -' 1) div lb) + 1)));



 A56: for k be Nat st k in ( Seg ( len FG)) holds ex v be Element of ExtREAL st PA[k, v]

 proof

 let k be Nat;

 assume k in ( Seg ( len FG));

 per cases ;

 suppose

 A57: (F . (((k -' 1) div lb) + 1)) = {} ;

 take 0. ;

 thus thesis by A57;

 end;

 suppose

 A58: (F . (((k -' 1) div lb) + 1)) <> {} ;

 take (a . (((k -' 1) div lb) + 1));

 thus thesis by A58;

 end;

 end;

 consider a1 be FinSequence of ExtREAL such that

 A59: ( dom a1) = ( Seg ( len FG)) & for k be Nat st k in ( Seg ( len FG)) holds PA[k, (a1 . k)] from FINSEQ_1:sch 5( A56);



 A60: ( dom g) = ( union ( rng G)) by A9, MESFUNC3:def 1;



 A61: ( dom f) = ( union ( rng FG))

 proof

 thus ( dom f) c= ( union ( rng FG))

 proof

 let z be object;

 assume

 A62: z in ( dom f);

 then

 consider Y be set such that

 A63: z in Y and

 A64: Y in ( rng F) by A55, TARSKI:def 4;

 consider i be Nat such that

 A65: i in ( dom F) and

 A66: Y = (F . i) by A64, FINSEQ_2: 10;



 A67: i in ( Seg ( len a)) by A13, A65, FINSEQ_1:def 3;

 then 1 <= i by FINSEQ_1: 1;

 then

 consider i9 be Nat such that

 A68: i = (1 qua Complex + i9) by NAT_1: 10;

 consider Z be set such that

 A69: z in Z and

 A70: Z in ( rng G) by A5, A60, A62, TARSKI:def 4;

 consider j be Nat such that

 A71: j in ( dom G) and

 A72: Z = (G . j) by A70, FINSEQ_2: 10;



 A73: j in ( Seg ( len b)) by A17, A71, FINSEQ_1:def 3;

 then

 A74: 1 <= j by FINSEQ_1: 1;

 then

 consider j9 be Nat such that

 A75: j = (1 qua Complex + j9) by NAT_1: 10;

 ((i9 * lb) + j) in NAT by ORDINAL1:def 12;

 then

 reconsider k = (((i - 1) * lb) + j) as Element of NAT by A68;

 i <= la by A67, FINSEQ_1: 1;

 then (i - 1) <= (la - 1) by XREAL_1: 9;

 then ((i - 1) * lb) <= ((la - 1) * lb) by XREAL_1: 64;

 then

 A76: k <= (((la - 1) * lb) + j) by XREAL_1: 6;



 A77: k >= ( 0 + j) by A68, XREAL_1: 6;

 then (k -' 1) = (k - 1) by A74, XREAL_1: 233, XXREAL_0: 2;

 then

 A78: (k -' 1) = ((i9 * lb) + j9) by A68, A75;



 A79: j <= lb by A73, FINSEQ_1: 1;

 then (((la - 1) * lb) + j) <= (((la - 1) * lb) + lb) by XREAL_1: 6;

 then

 A80: k <= (la * lb) by A76, XXREAL_0: 2;

 k >= 1 by A74, A77, XXREAL_0: 2;

 then

 A81: k in ( Seg (la * lb)) by A80;

 then k in ( dom FG) by A14, FINSEQ_1:def 3;

 then

 A82: (FG . k) in ( rng FG) by FUNCT_1:def 3;



 A83: j9 < lb by A79, A75, NAT_1: 13;

 then

 A84: j = (((k -' 1) mod lb) + 1) by A75, A78, NAT_D:def 2;



 A85: i = (((k -' 1) div lb) + 1) by A68, A78, A83, NAT_D:def 1;

 z in ((F . i) /\ (G . j)) by A63, A66, A69, A72, XBOOLE_0:def 4;

 then z in (FG . k) by A15, A16, A85, A84, A81;

 hence thesis by A82, TARSKI:def 4;

 end;

 let z be object;



 A86: lb divides (la * lb) by NAT_D:def 3;

 assume z in ( union ( rng FG));

 then

 consider Y be set such that

 A87: z in Y and

 A88: Y in ( rng FG) by TARSKI:def 4;

 consider k be Nat such that

 A89: k in ( dom FG) and

 A90: Y = (FG . k) by A88, FINSEQ_2: 10;

 set i = (((k -' 1) div lb) + 1);



 A91: k in ( Seg ( len FG)) by A89, FINSEQ_1:def 3;

 then

 A92: k <= (la * lb) by A14, FINSEQ_1: 1;

 then

 A93: (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 1 <= k by A91, FINSEQ_1: 1;

 then

 A94: 1 <= (la * lb) by A92, XXREAL_0: 2;



 A95: lb <> 0 by A14, A91;

 then lb >= ( 0 + 1) by NAT_1: 13;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A86, A94, NAT_2: 15;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A93, NAT_2: 24;

 then

 A96: i <= ((la * lb) div lb) by XREAL_1: 19;

 set j = (((k -' 1) mod lb) + 1);



 A97: i >= ( 0 + 1) by XREAL_1: 6;

 ((la * lb) div lb) = la by A95, NAT_D: 18;

 then i in ( Seg la) by A97, A96;

 then i in ( dom F) by A13, FINSEQ_1:def 3;

 then

 A98: (F . i) in ( rng F) by FUNCT_1:def 3;

 (FG . k) = ((F . i) /\ (G . j)) by A15, A89;

 then z in (F . i) by A87, A90, XBOOLE_0:def 4;

 hence thesis by A55, A98, TARSKI:def 4;

 end;



 A99: for k be Nat, x,y be Element of X st k in ( dom FG) & x in (FG . k) & y in (FG . k) holds ((f - g) . x) = ((f - g) . y)

 proof

 let k be Nat;

 let x,y be Element of X;

 assume that

 A100: k in ( dom FG) and

 A101: x in (FG . k) and

 A102: y in (FG . k);

 set j = (((k -' 1) mod lb) + 1);



 A103: (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A15, A100;

 then

 A104: x in (G . j) by A101, XBOOLE_0:def 4;

 set i = (((k -' 1) div lb) + 1);



 A105: i >= ( 0 + 1) by XREAL_1: 6;



 A106: k in ( Seg ( len FG)) by A100, FINSEQ_1:def 3;

 then

 A107: 1 <= k by FINSEQ_1: 1;



 A108: lb > 0 by A14, A106;

 then

 A109: lb >= ( 0 + 1) by NAT_1: 13;



 A110: y in (G . j) by A102, A103, XBOOLE_0:def 4;



 A111: lb divides (la * lb) by NAT_D:def 3;



 A112: k <= (la * lb) by A14, A106, FINSEQ_1: 1;

 then

 A113: (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 1 <= (la * lb) by A107, A112, XXREAL_0: 2;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A109, A111, NAT_2: 15;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A113, NAT_2: 24;

 then

 A114: (((k -' 1) div lb) + 1) <= ((la * lb) div lb) by XREAL_1: 19;

 lb <> 0 by A14, A106;

 then i <= la by A114, NAT_D: 18;

 then i in ( Seg la) by A105;

 then

 A115: i in ( dom F) by A13, FINSEQ_1:def 3;

 ((k -' 1) mod lb) < lb by A108, NAT_D: 1;

 then

 A116: j <= lb by NAT_1: 13;

 j >= ( 0 + 1) by XREAL_1: 6;

 then j in ( Seg lb) by A116;

 then

 A117: j in ( dom G) by A17, FINSEQ_1:def 3;

 y in (F . i) by A102, A103, XBOOLE_0:def 4;

 then

 A118: (f . y) = (a . i) by A12, A115, MESFUNC3:def 1;

 x in (F . i) by A101, A103, XBOOLE_0:def 4;

 then

 A119: (f . x) = (a . i) by A12, A115, MESFUNC3:def 1;



 A120: (FG . k) in ( rng FG) by A100, FUNCT_1:def 3;

 then

 A121: y in ( dom (f - g)) by A5, A61, A10, A102, TARSKI:def 4;

 x in ( dom (f - g)) by A5, A61, A10, A101, A120, TARSKI:def 4;



 then ((f - g) . x) = ((f . x) - (g . x)) by MESFUNC1:def 4

 .= ((a . i) - (b . j)) by A9, A117, A104, A119, MESFUNC3:def 1

 .= ((f . y) - (g . y)) by A9, A117, A110, A118, MESFUNC3:def 1;

 hence thesis by A121, MESFUNC1:def 4;

 end;

 deffunc X1( Nat) = ((a1 . $1) * ((M * FG) . $1));

 consider x1 be FinSequence of ExtREAL such that

 A122: ( len x1) = ( len FG) & for k be Nat st k in ( dom x1) holds (x1 . k) = X1(k) from FINSEQ_2:sch 1;



 A123: for k be Nat st k in ( dom FG) holds for x be object st x in (FG . k) holds (f . x) = (a1 . k)

 proof

 let k be Nat;

 set i = (((k -' 1) div lb) + 1);



 A124: i >= ( 0 + 1) by XREAL_1: 6;

 assume

 A125: k in ( dom FG);

 then

 A126: k in ( Seg ( len FG)) by FINSEQ_1:def 3;

 let x be object;

 assume

 A127: x in (FG . k);

 (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A15, A125;

 then

 A128: x in (F . i) by A127, XBOOLE_0:def 4;



 A129: k in ( Seg ( len FG)) by A125, FINSEQ_1:def 3;

 then

 A130: k <= (la * lb) by A14, FINSEQ_1: 1;

 then (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 then

 A131: ((k -' 1) div lb) <= (((la * lb) -' 1) div lb) by NAT_2: 24;



 A132: lb divides (la * lb) by NAT_D:def 3;

 1 <= k by A129, FINSEQ_1: 1;

 then

 A133: 1 <= (la * lb) by A130, XXREAL_0: 2;



 A134: lb <> 0 by A14, A129;

 then lb >= ( 0 + 1) by NAT_1: 13;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A132, A133, NAT_2: 15;

 then

 A135: i <= ((la * lb) div lb) by A131, XREAL_1: 19;

 ((la * lb) div lb) = la by A134, NAT_D: 18;

 then i in ( Seg la) by A124, A135;

 then i in ( dom F) by A13, FINSEQ_1:def 3;

 then (f . x) = (a . i) by A12, A128, MESFUNC3:def 1;

 hence thesis by A59, A126, A128;

 end;



 A136: for k be Nat st k in ( Seg ( len FG)) holds ex v be Element of ExtREAL st PB[k, v]

 proof

 let k be Nat;

 assume k in ( Seg ( len FG));

 per cases ;

 suppose

 A137: (G . (((k -' 1) mod lb) + 1)) = {} ;

 reconsider z = 0 as R_eal by XXREAL_0:def 1;

 take z;

 thus thesis by A137;

 end;

 suppose

 A138: (G . (((k -' 1) mod lb) + 1)) <> {} ;

 take (b . (((k -' 1) mod lb) + 1));

 thus thesis by A138;

 end;

 end;

 consider b1 be FinSequence of ExtREAL such that

 A139: ( dom b1) = ( Seg ( len FG)) & for k be Nat st k in ( Seg ( len FG)) holds PB[k, (b1 . k)] from FINSEQ_1:sch 5( A136);

 deffunc C1( Nat) = ((a1 . $1) - (b1 . $1));

 consider c1 be FinSequence of ExtREAL such that

 A140: ( len c1) = ( len FG) and

 A141: for k be Nat st k in ( dom c1) holds (c1 . k) = C1(k) from FINSEQ_2:sch 1;



 A142: ( dom c1) = ( Seg ( len FG)) by A140, FINSEQ_1:def 3;



 A143: for k be Nat st k in ( dom FG) holds for x be object st x in (FG . k) holds (g . x) = (b1 . k)

 proof

 let k be Nat;

 set j = (((k -' 1) mod lb) + 1);

 assume

 A144: k in ( dom FG);

 then

 A145: k in ( Seg ( len FG)) by FINSEQ_1:def 3;

 k in ( Seg ( len FG)) by A144, FINSEQ_1:def 3;

 then lb <> 0 by A14;

 then ((k -' 1) mod lb) < lb by NAT_D: 1;

 then

 A146: j <= lb by NAT_1: 13;

 let x be object;

 assume

 A147: x in (FG . k);

 (FG . k) = ((F . (((k -' 1) div lb) + 1)) /\ (G . (((k -' 1) mod lb) + 1))) by A15, A144;

 then

 A148: x in (G . j) by A147, XBOOLE_0:def 4;

 j >= ( 0 + 1) by XREAL_1: 6;

 then j in ( Seg lb) by A146;

 then j in ( dom G) by A17, FINSEQ_1:def 3;



 hence (g . x) = (b . j) by A9, A148, MESFUNC3:def 1

 .= (b1 . k) by A139, A148, A145;

 end;



 A149: for k be Nat st k in ( dom FG) holds for x be object st x in (FG . k) holds ((f - g) . x) = (c1 . k)

 proof

 let k be Nat;

 assume

 A150: k in ( dom FG);

 let x be object;

 assume

 A151: x in (FG . k);

 (FG . k) in ( rng FG) by A150, FUNCT_1:def 3;

 then x in ( dom (f - g)) by A5, A61, A10, A151, TARSKI:def 4;

 then

 A152: ((f - g) . x) = ((f . x) - (g . x)) by MESFUNC1:def 4;

 k in ( Seg ( len FG)) by A150, FINSEQ_1:def 3;

 then ((a1 . k) - (b1 . k)) = (c1 . k) by A141, A142;

 then ((a1 . k) - (g . x)) = (c1 . k) by A143, A150, A151;

 hence thesis by A123, A150, A151, A152;

 end;

 deffunc Z1( Nat) = ((c1 . $1) * ((M * FG) . $1));

 consider z1 be FinSequence of ExtREAL such that

 A153: ( len z1) = ( len FG) & for k be Nat st k in ( dom z1) holds (z1 . k) = Z1(k) from FINSEQ_2:sch 1;

 deffunc Y1( Nat) = ((b1 . $1) * ((M * FG) . $1));

 consider y1 be FinSequence of ExtREAL such that

 A154: ( len y1) = ( len FG) & for k be Nat st k in ( dom y1) holds (y1 . k) = Y1(k) from FINSEQ_2:sch 1;



 A155: ( dom x1) = ( dom FG) by A122, FINSEQ_3: 29;



 A156: ( dom z1) = ( dom FG) by A153, FINSEQ_3: 29;



 A157: for i be Nat st i in ( dom x1) holds 0 <= (z1 . i)

 proof

 reconsider EMPTY = {} as Element of S by PROB_1: 4;

 let i be Nat;

 assume

 A158: i in ( dom x1);

 then

 A159: ((M * FG) . i) = (M . (FG . i)) by A155, FUNCT_1: 13;

 (FG . i) in ( rng FG) by A155, A158, FUNCT_1: 3;

 then

 reconsider V = (FG . i) as Element of S;

 (M . EMPTY) <= (M . V) by MEASURE1: 31, XBOOLE_1: 2;

 then

 A160: 0 <= ((M * FG) . i) by A159, VALUED_0:def 19;



 A161: i in ( Seg ( len FG)) by A122, A158, FINSEQ_1:def 3;

 per cases ;

 suppose (FG . i) <> {} ;

 then

 consider x be object such that

 A162: x in (FG . i) by XBOOLE_0:def 1;

 (FG . i) in ( rng FG) by A155, A158, FUNCT_1: 3;

 then x in ( union ( rng FG)) by A162, TARSKI:def 4;

 then (g . x) <= (f . x) by A7, A61;

 then (g . x) <= (a1 . i) by A155, A123, A158, A162;

 then (b1 . i) <= (a1 . i) by A155, A143, A158, A162;

 then 0 <= ((a1 . i) - (b1 . i)) by XXREAL_3: 40;

 then 0 <= (c1 . i) by A141, A142, A161;

 then 0 <= ((c1 . i) * ((M * FG) . i)) by A160;

 hence thesis by A155, A153, A156, A158;

 end;

 suppose (FG . i) = {} ;

 then ((M * FG) . i) = 0 by A159, VALUED_0:def 19;

 then ((c1 . i) * ((M * FG) . i)) = 0 ;

 hence thesis by A155, A153, A156, A158;

 end;

 end;

 then for i be object st i in ( dom z1) holds 0 <= (z1 . i) by A156, A155;

 then

 cd: z1 is nonnegative by SUPINF_2: 52;

 not -infty in ( rng z1)

 proof

 assume -infty in ( rng z1);

 then ex i be object st i in ( dom z1) & (z1 . i) = -infty by FUNCT_1:def 3;

 hence contradiction by A155, A156, A157;

 end;



 then

 A163: ((z1 " { -infty }) /\ (y1 " { +infty })) = ( {} /\ (y1 " { +infty })) by FUNCT_1: 72

 .= {} ;



 A164: ( dom y1) = ( dom FG) by A154, FINSEQ_3: 29;



 A165: for i be Nat st i in ( dom y1) holds 0 <= (y1 . i)

 proof

 let i be Nat;

 set i9 = (((i -' 1) mod lb) + 1);



 A166: i9 >= ( 0 + 1) by XREAL_1: 6;

 assume

 A167: i in ( dom y1);

 then

 A168: (y1 . i) = ((b1 . i) * ((M * FG) . i)) by A154;



 A169: i in ( Seg ( len FG)) by A154, A167, FINSEQ_1:def 3;

 then lb <> 0 by A14;

 then ((i -' 1) mod lb) < lb by NAT_D: 1;

 then i9 <= lb by NAT_1: 13;

 then i9 in ( Seg lb) by A166;

 then

 A170: i9 in ( dom G) by A17, FINSEQ_1:def 3;

 per cases ;

 suppose

 A171: (G . i9) <> {} ;

 (FG . i) in ( rng FG) by A164, A167, FUNCT_1: 3;

 then

 reconsider FGi = (FG . i) as Element of S;

 reconsider EMPTY = {} as Element of S by MEASURE1: 7;

 EMPTY c= FGi;

 then

 A172: (M . {} ) <= (M . FGi) by MEASURE1: 31;

 consider x9 be object such that

 A173: x9 in (G . i9) by A171, XBOOLE_0:def 1;

 (g . x9) = (b . i9) by A9, A170, A173, MESFUNC3:def 1

 .= (b1 . i) by A139, A169, A171;

 then

 A174: 0 <= (b1 . i) by A6, SUPINF_2: 51;

 (M . {} ) = 0 by VALUED_0:def 19;

 then 0 <= ((M * FG) . i) by A164, A167, A172, FUNCT_1: 13;

 hence thesis by A168, A174;

 end;

 suppose

 A175: (G . i9) = {} ;

 (FG . i) = ((F . (((i -' 1) div lb) + 1)) /\ (G . i9)) by A14, A15, A16, A169;

 then (M . (FG . i)) = 0 by A175, VALUED_0:def 19;

 then ((M * FG) . i) = 0 by A164, A167, FUNCT_1: 13;

 hence thesis by A168;

 end;

 end;

 then for i be object st i in ( dom y1) holds 0 <= (y1 . i);

 then

 ag: y1 is nonnegative by SUPINF_2: 52;

 not -infty in ( rng y1)

 proof

 assume -infty in ( rng y1);

 then ex i be object st i in ( dom y1) & (y1 . i) = -infty by FUNCT_1:def 3;

 hence contradiction by A165;

 end;



 then ((z1 " { +infty }) /\ (y1 " { -infty })) = ((z1 " { +infty }) /\ {} ) by FUNCT_1: 72

 .= {} ;



 then

 A176: ( dom (z1 + y1)) = ((( dom z1) /\ ( dom y1)) \ ( {} \/ {} )) by A163, MESFUNC1:def 3

 .= ( dom x1) by A122, A164, A156, FINSEQ_3: 29;



 A177: for k be Nat st k in ( dom x1) holds (x1 . k) = ((z1 + y1) . k)

 proof



 A178: lb divides (la * lb) by NAT_D:def 3;

 let k be Nat;

 set p = (((k -' 1) div lb) + 1);

 set q = (((k -' 1) mod lb) + 1);



 A179: p >= ( 0 + 1) by XREAL_1: 6;

 assume

 A180: k in ( dom x1);

 then

 A181: k in ( Seg ( len FG)) by A122, FINSEQ_1:def 3;

 then

 A182: 1 <= k by FINSEQ_1: 1;



 A183: lb > 0 by A14, A181;

 then

 A184: lb >= ( 0 + 1) by NAT_1: 13;



 A185: k <= (la * lb) by A14, A181, FINSEQ_1: 1;

 then

 A186: (k -' 1) <= ((la * lb) -' 1) by NAT_D: 42;

 1 <= (la * lb) by A182, A185, XXREAL_0: 2;

 then (((la * lb) -' 1) div lb) = (((la * lb) div lb) - 1) by A184, A178, NAT_2: 15;

 then ((k -' 1) div lb) <= (((la * lb) div lb) - 1) by A186, NAT_2: 24;

 then

 A187: p <= ((la * lb) div lb) by XREAL_1: 19;

 lb <> 0 by A14, A181;

 then p <= la by A187, NAT_D: 18;

 then p in ( Seg la) by A179;

 then

 A188: p in ( dom F) by A13, FINSEQ_1:def 3;



 A189: q >= ( 0 + 1) by XREAL_1: 6;

 ((k -' 1) mod lb) < lb by A183, NAT_D: 1;

 then q <= lb by NAT_1: 13;

 then q in ( Seg lb) by A189;

 then

 A190: q in ( dom G) by A17, FINSEQ_1:def 3;



 A191: (((c1 . k) + (b1 . k)) * ((M * FG) . k)) = (((c1 . k) * ((M * FG) . k)) + ((b1 . k) * ((M * FG) . k)))

 proof

 per cases ;

 suppose (FG . k) <> {} ;

 then ((F . p) /\ (G . q)) <> {} by A14, A15, A16, A181;

 then

 consider v be object such that

 A192: v in ((F . p) /\ (G . q)) by XBOOLE_0:def 1;



 A193: (G . q) <> {} by A192;



 A194: v in (F . p) by A192, XBOOLE_0:def 4;

 v in (G . q) by A192, XBOOLE_0:def 4;

 then

 A195: (b . q) = (g . v) by A9, A190, MESFUNC3:def 1;

 (F . p) in ( rng F) by A188, FUNCT_1: 3;

 then

 A196: v in ( dom f) by A55, A194, TARSKI:def 4;

 (a . p) = (f . v) by A12, A188, A194, MESFUNC3:def 1;

 then (b . q) <= (a . p) by A7, A195, A196;

 then

 A197: (b1 . k) <= (a . p) by A139, A181, A193;

 (F . p) <> {} by A192;

 then (b1 . k) <= (a1 . k) by A59, A181, A197;

 then 0 <= ((a1 . k) - (b1 . k)) by XXREAL_3: 40;

 then

 A198: 0 = (c1 . k) or 0 < (c1 . k) by A141, A142, A181;

 0 <= (b . q) by A6, A195, SUPINF_2: 51;

 then 0 = (b1 . k) or 0 < (b1 . k) by A139, A181, A192;

 hence thesis by A198, XXREAL_3: 96;

 end;

 suppose (FG . k) = {} ;

 then (M . (FG . k)) = 0 by VALUED_0:def 19;

 then

 A199: ((M * FG) . k) = 0 by A155, A180, FUNCT_1: 13;



 hence (((c1 . k) + (b1 . k)) * ((M * FG) . k)) = 0

 .= (((c1 . k) * ((M * FG) . k)) + ((b1 . k) * ((M * FG) . k))) by A199;

 end;

 end;



 A200: (a1 . k) <> +infty & (a1 . k) <> -infty & (b1 . k) <> +infty & (b1 . k) <> -infty

 proof

 now

 per cases ;

 suppose

 A201: (F . p) <> {} ;

 then

 consider v be object such that

 A202: v in (F . p) by XBOOLE_0:def 1;



 A203: f is real-valued by A1, MESFUNC2:def 4;

 (a1 . k) = (a . p) by A59, A181, A201;

 then (a1 . k) = (f . v) by A12, A188, A202, MESFUNC3:def 1;

 hence (a1 . k) <> +infty & -infty <> (a1 . k) by A203;

 end;

 suppose (F . p) = {} ;

 hence (a1 . k) <> +infty & -infty <> (a1 . k) by A59, A181;

 end;

 end;

 hence +infty <> (a1 . k) & (a1 . k) <> -infty ;

 now

 per cases ;

 suppose

 A204: (G . q) <> {} ;

 then

 consider v be object such that

 A205: v in (G . q) by XBOOLE_0:def 1;



 A206: g is real-valued by A4, MESFUNC2:def 4;

 (b1 . k) = (b . q) by A139, A181, A204;

 then (b1 . k) = (g . v) by A9, A190, A205, MESFUNC3:def 1;

 hence thesis by A206;

 end;

 suppose (G . q) = {} ;

 hence thesis by A139, A181;

 end;

 end;

 hence thesis;

 end;



 A207: ((b1 . k) - (b1 . k)) = ( - 0 ) by XXREAL_3: 7;

 (c1 . k) = ((a1 . k) - (b1 . k)) by A141, A142, A181;



 then ((c1 . k) + (b1 . k)) = ((a1 . k) - ((b1 . k) - (b1 . k))) by A200, XXREAL_3: 32

 .= ((a1 . k) + ( - 0 )) by A207

 .= (a1 . k) by XXREAL_3: 4;



 hence (x1 . k) = (((c1 . k) + (b1 . k)) * ((M * FG) . k)) by A122, A180

 .= ((z1 . k) + ((b1 . k) * ((M * FG) . k))) by A155, A153, A156, A180, A191

 .= ((z1 . k) + (y1 . k)) by A155, A154, A164, A180

 .= ((z1 + y1) . k) by A176, A180, MESFUNC1:def 3;

 end;

 now

 let x be Element of X;

 assume

 A208: x in ( dom (f - g));

 g is real-valued by A4, MESFUNC2:def 4;

 then

 A209: |.(g . x).| < +infty by A5, A10, A208, MESFUNC2:def 1;

 f is real-valued by A1, MESFUNC2:def 4;

 then |.(f . x).| < +infty by A5, A10, A208, MESFUNC2:def 1;

 then

 A210: ( |.(f . x).| + |.(g . x).|) <> +infty by A209, XXREAL_3: 16;

 |.((f - g) . x).| = |.((f . x) - (g . x)).| by A208, MESFUNC1:def 4;

 then |.((f - g) . x).| <= ( |.(f . x).| + |.(g . x).|) by EXTREAL1: 32;

 hence |.((f - g) . x).| < +infty by A210, XXREAL_0: 2, XXREAL_0: 4;

 end;

 then (f - g) is real-valued by MESFUNC2:def 1;

 hence

 A211: (f - g) is_simple_func_in S by A5, A61, A10, A99, MESFUNC2:def 4;

 ( dom FG) = ( dom a1) by A59, FINSEQ_1:def 3;

 then (FG,a1) are_Re-presentation_of f by A61, A123, MESFUNC3:def 1;

 then

 A212: ( integral (M,f)) = ( Sum x1) by A1, A2, A3, A122, A155, MESFUNC4: 3;

 ( dom (z1 + y1)) = ( Seg ( len x1)) by A176, FINSEQ_1:def 3;

 then (z1 + y1) is FinSequence by FINSEQ_1:def 2;

 then

 A213: x1 = (z1 + y1) by A176, A177, FINSEQ_1: 13;

 ( dom FG) = ( dom b1) by A139, FINSEQ_1:def 3;

 then (FG,b1) are_Re-presentation_of g by A5, A61, A143, MESFUNC3:def 1;

 then

 A214: ( integral (M,g)) = ( Sum y1) by A2, A4, A5, A6, A154, A164, MESFUNC4: 3;

 thus ( dom (f - g)) <> {} by A2, A5, A10;

 for x be object st x in ( dom (f - g)) holds 0 <= ((f - g) . x)

 proof

 let x be object;

 assume

 A216: x in ( dom (f - g));

 then 0 <= ((f . x) - (g . x)) by A5, A7, A10, XXREAL_3: 40;

 hence thesis by A216, MESFUNC1:def 4;

 end;

 hence

 aa: (f - g) is nonnegative by SUPINF_2: 52;

 ( dom FG) = ( dom c1) by A140, FINSEQ_3: 29;

 then (FG,c1) are_Re-presentation_of (f - g) by A5, A61, A10, A149, MESFUNC3:def 1;

 then ( integral (M,(f - g))) = ( Sum z1) by aa, A2, A5, A153, A156, A10, A211, MESFUNC4: 3;

 hence thesis by A164, A156, A212, A214, A213, MESFUNC4: 1, ag, cd;

 end;

 theorem :: MESFUNC5:69



 Th69: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & f is nonnegative & g is_simple_func_in S & g is nonnegative & (for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x)) holds ( dom (f - g)) = (( dom f) /\ ( dom g)) & ( integral' (M,(f | ( dom (f - g))))) = (( integral' (M,(f - g))) + ( integral' (M,(g | ( dom (f - g))))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: f is_simple_func_in S and

 A2: f is nonnegative and

 A3: g is_simple_func_in S and

 A4: g is nonnegative and

 A5: for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x);



 A6: (f | ( dom (f - g))) is nonnegative by A2, Th15;

 (( - jj) (#) g) is_simple_func_in S by A3, Th39;

 then ( - g) is_simple_func_in S by MESFUNC2: 9;

 then (f + ( - g)) is_simple_func_in S by A1, Th38;

 then (f - g) is_simple_func_in S by MESFUNC2: 8;

 then

 A7: ( dom (f - g)) is Element of S by Th37;

 then

 A8: (g | ( dom (f - g))) is_simple_func_in S by A3, Th34;



 A9: (g | ( dom (f - g))) is nonnegative by A4, Th15;

 g is without-infty by A3, Th14;

 then not -infty in ( rng g);

 then

 A10: (g " { -infty }) = {} by FUNCT_1: 72;

 f is without+infty by A1, Th14;

 then not +infty in ( rng f);

 then

 A11: (f " { +infty }) = {} by FUNCT_1: 72;

 then

 A12: ((( dom f) /\ ( dom g)) \ (((f " { +infty }) /\ (g " { +infty })) \/ ((f " { -infty }) /\ (g " { -infty })))) = (( dom f) /\ ( dom g)) by A10;

 hence

 A13: ( dom (f - g)) = (( dom f) /\ ( dom g)) by MESFUNC1:def 4;

 ( dom (f | ( dom (f - g)))) = (( dom f) /\ ( dom (f - g))) by RELAT_1: 61;

 then

 A14: ( dom (f | ( dom (f - g)))) = ((( dom f) /\ ( dom f)) /\ ( dom g)) by A13, XBOOLE_1: 16;



 A15: for x be set st x in ( dom (f | ( dom (f - g)))) holds ((g | ( dom (f - g))) . x) <= ((f | ( dom (f - g))) . x)

 proof

 let x be set;

 assume

 A16: x in ( dom (f | ( dom (f - g))));

 then (g . x) <= (f . x) by A5, A13, A14;

 then ((g | ( dom (f - g))) . x) <= (f . x) by A13, A14, A16, FUNCT_1: 49;

 hence thesis by A13, A14, A16, FUNCT_1: 49;

 end;

 ( dom (g | ( dom (f - g)))) = (( dom g) /\ ( dom (f - g))) by RELAT_1: 61;

 then

 A17: ( dom (g | ( dom (f - g)))) = ((( dom g) /\ ( dom g)) /\ ( dom f)) by A13, XBOOLE_1: 16;



 A18: (f | ( dom (f - g))) is_simple_func_in S by A1, A7, Th34;

 thus ( integral' (M,(f | ( dom (f - g))))) = (( integral' (M,(f - g))) + ( integral' (M,(g | ( dom (f - g))))))

 proof

 per cases ;

 suppose

 A19: ( dom (f - g)) = {} ;

 ( dom (g | ( dom (f - g)))) = (( dom g) /\ ( dom (f - g))) by RELAT_1: 61;

 then

 A20: ( integral' (M,(g | ( dom (f - g))))) = 0 by A19, Def14;

 ( dom (f | ( dom (f - g)))) = (( dom f) /\ ( dom (f - g))) by RELAT_1: 61;

 then

 A21: ( integral' (M,(f | ( dom (f - g))))) = 0 by A19, Def14;

 ( integral' (M,(f - g))) = 0 by A19, Def14;

 hence thesis by A21, A20;

 end;

 suppose

 A22: ( dom (f - g)) <> {} ;



 A23: ((g | ( dom (f - g))) " { -infty }) = (( dom (f - g)) /\ (g " { -infty })) by FUNCT_1: 70;

 ((f | ( dom (f - g))) " { +infty }) = (( dom (f - g)) /\ (f " { +infty })) by FUNCT_1: 70;

 then ((( dom (f | ( dom (f - g)))) /\ ( dom (g | ( dom (f - g))))) \ ((((f | ( dom (f - g))) " { +infty }) /\ ((g | ( dom (f - g))) " { +infty })) \/ (((f | ( dom (f - g))) " { -infty }) /\ ((g | ( dom (f - g))) " { -infty })))) = ( dom (f - g)) by A11, A10, A12, A14, A17, A23, MESFUNC1:def 4;

 then

 A24: ( dom ((f | ( dom (f - g))) - (g | ( dom (f - g))))) = ( dom (f - g)) by MESFUNC1:def 4;



 A25: for x be Element of X st x in ( dom ((f | ( dom (f - g))) - (g | ( dom (f - g))))) holds (((f | ( dom (f - g))) - (g | ( dom (f - g)))) . x) = ((f - g) . x)

 proof

 let x be Element of X;

 assume

 A26: x in ( dom ((f | ( dom (f - g))) - (g | ( dom (f - g)))));



 then (((f | ( dom (f - g))) - (g | ( dom (f - g)))) . x) = (((f | ( dom (f - g))) . x) - ((g | ( dom (f - g))) . x)) by MESFUNC1:def 4

 .= ((f . x) - ((g | ( dom (f - g))) . x)) by A24, A26, FUNCT_1: 49

 .= ((f . x) - (g . x)) by A24, A26, FUNCT_1: 49;

 hence thesis by A24, A26, MESFUNC1:def 4;

 end;

 ( integral (M,(f | ( dom (f - g))))) = (( integral (M,((f | ( dom (f - g))) - (g | ( dom (f - g)))))) + ( integral (M,(g | ( dom (f - g)))))) by A13, A18, A8, A6, A9, A14, A17, A15, A22, Lm9;

 then

 A27: ( integral (M,(f | ( dom (f - g))))) = (( integral (M,(f - g))) + ( integral (M,(g | ( dom (f - g)))))) by A24, A25, PARTFUN1: 5;



 A28: ( integral (M,(g | ( dom (f - g))))) = ( integral' (M,(g | ( dom (f - g))))) by A13, A17, A22, Def14;

 ( integral (M,(f | ( dom (f - g))))) = ( integral' (M,(f | ( dom (f - g))))) by A13, A14, A22, Def14;

 hence thesis by A22, A27, A28, Def14;

 end;

 end;

 end;

 theorem :: MESFUNC5:70



 Th70: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_simple_func_in S & g is_simple_func_in S & f is nonnegative & g is nonnegative & (for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x)) holds ( integral' (M,(g | ( dom (f - g))))) <= ( integral' (M,(f | ( dom (f - g)))))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: g is_simple_func_in S and

 A3: f is nonnegative and

 A4: g is nonnegative and

 A5: for x be object st x in ( dom (f - g)) holds (g . x) <= (f . x);

 (( - jj) (#) g) is_simple_func_in S by A2, Th39;

 then ( - g) is_simple_func_in S by MESFUNC2: 9;

 then (f + ( - g)) is_simple_func_in S by A1, Th38;

 then

 A6: (f - g) is_simple_func_in S by MESFUNC2: 8;



 A7: ( integral' (M,(f | ( dom (f - g))))) = (( integral' (M,(f - g))) + ( integral' (M,(g | ( dom (f - g)))))) by A1, A2, A3, A4, A5, Th69;

 now

 assume ( integral' (M,(f | ( dom (f - g))))) <> +infty ;

 0 <= ( integral' (M,(f - g))) by A1, A2, A5, A6, Th40, Th68;

 hence thesis by A7, XXREAL_3: 39;

 end;

 hence thesis by XXREAL_0: 4;

 end;

 theorem :: MESFUNC5:71



 Th71: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be R_eal st 0 <= c & f is_simple_func_in S & (for x be object st x in ( dom f) holds (f . x) = c) holds ( integral' (M,f)) = (c * (M . ( dom f)))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let c be R_eal;

 assume that

 A1: 0 <= c and

 A2: f is_simple_func_in S and

 A3: for x be object st x in ( dom f) holds (f . x) = c;

 for x be object st x in ( dom f) holds 0 <= (f . x) by A1, A3;

 then

 a4: f is nonnegative by SUPINF_2: 52;

 reconsider A = ( dom f) as Element of S by A2, Th37;

 per cases ;

 suppose

 A5: ( dom f) = {} ;

 then

 A6: (M . A) = 0 by VALUED_0:def 19;

 ( integral' (M,f)) = 0 by A5, Def14;

 hence thesis by A6;

 end;

 suppose

 A7: ( dom f) <> {} ;

 set x = <*(c * (M . A))*>;

 reconsider a = <*c*> as FinSequence of ExtREAL ;

 set F = <*( dom f)*>;

 reconsider x as FinSequence of ExtREAL ;



 A8: ( rng F) = {A} by FINSEQ_1: 38;

 ( rng F) c= S

 proof

 let z be object;

 assume z in ( rng F);

 then z = A by A8, TARSKI:def 1;

 hence thesis;

 end;

 then

 reconsider F as FinSequence of S by FINSEQ_1:def 4;

 for i,j be Nat st i in ( dom F) & j in ( dom F) & i <> j holds (F . i) misses (F . j)

 proof

 let i,j be Nat;

 assume that

 A9: i in ( dom F) and

 A10: j in ( dom F) and

 A11: i <> j;



 A12: ( dom F) = {1} by FINSEQ_1: 2, FINSEQ_1: 38;

 then i = 1 by A9, TARSKI:def 1;

 hence thesis by A10, A11, A12, TARSKI:def 1;

 end;

 then

 reconsider F as Finite_Sep_Sequence of S by MESFUNC3: 4;



 A13: ( dom F) = ( Seg 1) by FINSEQ_1: 38

 .= ( dom a) by FINSEQ_1: 38;



 A14: for n be Nat st n in ( dom F) holds for x be object st x in (F . n) holds (f . x) = (a . n)

 proof

 let n be Nat;

 assume n in ( dom F);

 then n in {1} by FINSEQ_1: 2, FINSEQ_1: 38;

 then

 A15: n = 1 by TARSKI:def 1;

 let x be object;

 assume x in (F . n);

 then x in ( dom f) by A15, FINSEQ_1: 40;

 then (f . x) = c by A3;

 hence thesis by A15, FINSEQ_1: 40;

 end;



 A16: for n be Nat st n in ( dom x) holds (x . n) = (c * (M . A))

 proof

 let n be Nat;

 assume n in ( dom x);

 then n in {1} by FINSEQ_1: 2, FINSEQ_1: 38;

 then n = 1 by TARSKI:def 1;

 hence thesis by FINSEQ_1: 40;

 end;



 A17: ( dom x) = ( Seg 1) by FINSEQ_1: 38

 .= ( dom F) by FINSEQ_1: 38;



 A18: for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n))

 proof

 let n be Nat;

 assume

 A19: n in ( dom x);

 then n in {1} by FINSEQ_1: 2, FINSEQ_1: 38;

 then

 A20: n = 1 by TARSKI:def 1;

 then

 A21: (x . n) = (c * (M . A)) by FINSEQ_1: 40;

 ((M * F) . n) = (M . (F . n)) by A17, A19, FUNCT_1: 13

 .= (M . A) by A20, FINSEQ_1: 40;

 hence thesis by A20, A21, FINSEQ_1: 40;

 end;

 ( dom f) = ( union ( rng F)) by A8, ZFMISC_1: 25;

 then (F,a) are_Re-presentation_of f by A13, A14, MESFUNC3:def 1;

 then ( integral (M,f)) = ( Sum x) by A2, a4, A7, A17, A18, MESFUNC4: 3;

 then

 A22: ( integral' (M,f)) = ( Sum x) by A7, Def14;

 reconsider j = 1 as R_eal by XXREAL_0:def 1;

 1 = ( len x) by FINSEQ_1: 40;

 then ( Sum x) = (j * (c * (M . A))) by A16, MESFUNC3: 18;

 hence thesis by A22, XXREAL_3: 81;

 end;

 end;

 theorem :: MESFUNC5:72



 Th72: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & f is nonnegative holds ( integral' (M,(f | ( eq_dom (f, 0 ))))) = 0

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: f is nonnegative;

 set A = ( dom f);

 set g = (f | (A /\ ( eq_dom (f, 0 ))));

 for x be object st x in ( eq_dom (f, 0 )) holds x in A by MESFUNC1:def 15;

 then ( eq_dom (f, 0 )) c= A;

 then

 A3: (f | (A /\ ( eq_dom (f, 0 )))) = (f | ( eq_dom (f, 0 ))) by XBOOLE_1: 28;



 A4: ex G be Finite_Sep_Sequence of S st (( dom g) = ( union ( rng G)) & for n be Nat, x,y be Element of X st n in ( dom G) & x in (G . n) & y in (G . n) holds (g . x) = (g . y))

 proof

 consider F be Finite_Sep_Sequence of S such that

 A5: ( dom f) = ( union ( rng F)) and

 A6: for n be Nat, x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y) by A1, MESFUNC2:def 4;

 deffunc G( Nat) = ((F . $1) /\ (A /\ ( eq_dom (f, 0 ))));

 reconsider A as Element of S by A5, MESFUNC2: 31;

 consider G be FinSequence such that

 A7: ( len G) = ( len F) & for n be Nat st n in ( dom G) holds (G . n) = G(n) from FINSEQ_1:sch 2;

 f is A -measurable by A1, MESFUNC2: 34;

 then (A /\ ( less_dom (f, 0 ))) in S by MESFUNC1:def 16;

 then (A \ (A /\ ( less_dom (f, 0 )))) in S by PROB_1: 6;

 then

 reconsider A1 = (A /\ ( great_eq_dom (f, 0. ))) as Element of S by MESFUNC1: 14;

 f is A1 -measurable by A1, MESFUNC2: 34;

 then ((A /\ ( great_eq_dom (f, 0 ))) /\ ( less_eq_dom (f, 0 ))) in S by MESFUNC1: 28;

 then

 reconsider A2 = (A /\ ( eq_dom (f, 0 ))) as Element of S by MESFUNC1: 18;



 A8: ( dom F) = ( Seg ( len F)) by FINSEQ_1:def 3;

 ( dom G) = ( Seg ( len F)) by A7, FINSEQ_1:def 3;

 then

 A9: for i be Nat st i in ( dom F) holds (G . i) = ((F . i) /\ A2) by A7, A8;

 ( dom G) = ( Seg ( len F)) by A7, FINSEQ_1:def 3;

 then

 A10: ( dom G) = ( dom F) by FINSEQ_1:def 3;

 then

 reconsider G as Finite_Sep_Sequence of S by A9, Th35;

 take G;

 for i be Nat st i in ( dom G) holds (G . i) = (A2 /\ (F . i)) by A7;



 then

 A11: ( union ( rng G)) = (A2 /\ ( dom f)) by A5, A10, MESFUNC3: 6

 .= ( dom g) by RELAT_1: 61;

 for i be Nat, x,y be Element of X st i in ( dom G) & x in (G . i) & y in (G . i) holds (g . x) = (g . y)

 proof

 let i be Nat;

 let x,y be Element of X;

 assume that

 A12: i in ( dom G) and

 A13: x in (G . i) and

 A14: y in (G . i);



 A15: (G . i) = ((F . i) /\ A2) by A7, A12;

 then

 A16: y in (F . i) by A14, XBOOLE_0:def 4;



 A17: (G . i) in ( rng G) by A12, FUNCT_1: 3;

 then x in ( dom g) by A11, A13, TARSKI:def 4;

 then

 A18: (g . x) = (f . x) by FUNCT_1: 47;

 y in ( dom g) by A11, A14, A17, TARSKI:def 4;

 then

 A19: (g . y) = (f . y) by FUNCT_1: 47;

 x in (F . i) by A13, A15, XBOOLE_0:def 4;

 hence thesis by A6, A10, A12, A16, A18, A19;

 end;

 hence thesis by A11;

 end;

 for x be object st x in ( dom g) holds 0 <= (g . x)

 proof

 let x be object;

 assume

 A21: x in ( dom g);

 0 <= (f . x) by A2, SUPINF_2: 51;

 hence thesis by A21, FUNCT_1: 47;

 end;

 then

 a2: g is nonnegative by SUPINF_2: 52;

 f is real-valued by A1, MESFUNC2:def 4;

 then

 A22: g is_simple_func_in S by A4, MESFUNC2:def 4;

 now

 consider F be Finite_Sep_Sequence of S, a,x be FinSequence of ExtREAL such that

 A23: (F,a) are_Re-presentation_of g and (a . 1) = 0 and for n be Nat st 2 <= n & n in ( dom a) holds 0 < (a . n) & (a . n) < +infty and

 A24: ( dom x) = ( dom F) and

 A25: for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n)) and

 A26: ( integral (M,g)) = ( Sum x) by a2, A22, MESFUNC3:def 2;



 A27: for x be set st x in ( dom g) holds (g . x) = 0

 proof

 let x be set;

 assume

 A28: x in ( dom g);

 then x in (( dom f) /\ (A /\ ( eq_dom (f, 0 )))) by RELAT_1: 61;

 then x in (A /\ ( eq_dom (f, 0 ))) by XBOOLE_0:def 4;

 then x in ( eq_dom (f, 0 )) by XBOOLE_0:def 4;

 then 0 = (f . x) by MESFUNC1:def 15;

 hence thesis by A28, FUNCT_1: 47;

 end;



 A29: for n be Nat st n in ( dom F) holds (a . n) = 0 or (F . n) = {}

 proof

 let n be Nat;

 assume

 A30: n in ( dom F);

 now

 assume (F . n) <> {} ;

 then

 consider x be object such that

 A31: x in (F . n) by XBOOLE_0:def 1;

 (F . n) in ( rng F) by A30, FUNCT_1: 3;

 then x in ( union ( rng F)) by A31, TARSKI:def 4;

 then x in ( dom g) by A23, MESFUNC3:def 1;

 then (g . x) = 0 by A27;

 hence thesis by A23, A30, A31, MESFUNC3:def 1;

 end;

 hence thesis;

 end;



 A32: for n be Nat st n in ( dom x) holds (x . n) = 0

 proof

 let n be Nat;

 assume

 A33: n in ( dom x);

 per cases by A24, A29, A33;

 suppose (a . n) = 0 ;

 then ((a . n) * ((M * F) . n)) = 0 ;

 hence thesis by A25, A33;

 end;

 suppose (F . n) = {} ;

 then (M . (F . n)) = 0 by VALUED_0:def 19;

 then ((M * F) . n) = 0 by A24, A33, FUNCT_1: 13;

 then ((a . n) * ((M * F) . n)) = 0 ;

 hence thesis by A25, A33;

 end;

 end;



 A34: ( Sum x) = 0

 proof

 consider sumx be sequence of ExtREAL such that

 A35: ( Sum x) = (sumx . ( len x)) and

 A36: (sumx . 0 ) = 0 and

 A37: for i be Nat st i < ( len x) holds (sumx . (i + 1)) = ((sumx . i) + (x . (i + 1))) by EXTREAL1:def 2;

 now

 defpred P[ Nat] means $1 <= ( len x) implies (sumx . $1) = 0 ;

 assume x <> {} ;



 A38: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A39: P[k];

 assume

 A40: (k + 1) <= ( len x);

 reconsider k as Element of NAT by ORDINAL1:def 12;

 1 <= (k + 1) by NAT_1: 11;

 then (k + 1) in ( Seg ( len x)) by A40;

 then (k + 1) in ( dom x) by FINSEQ_1:def 3;

 then

 A41: (x . (k + 1)) = 0 by A32;

 k < ( len x) by A40, NAT_1: 13;

 then (sumx . (k + 1)) = ((sumx . k) + (x . (k + 1))) by A37;

 hence thesis by A39, A40, A41, NAT_1: 13;

 end;



 A42: P[ 0 ] by A36;

 for i be Nat holds P[i] from NAT_1:sch 2( A42, A38);

 hence thesis by A35;

 end;

 hence thesis by A35, A36, CARD_1: 27;

 end;

 assume ( dom g) <> {} ;

 hence thesis by A3, A26, A34, Def14;

 end;

 hence thesis by A3, Def14;

 end;

 theorem :: MESFUNC5:73



 Th73: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, B be Element of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & (M . B) = 0 & f is nonnegative holds ( integral' (M,(f | B))) = 0

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let B be Element of S;

 let f be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: (M . B) = 0 and

 A3: f is nonnegative;

 set A = ( dom f);

 set g = (f | (A /\ B));

 for x be object st x in ( dom g) holds 0 <= (g . x)

 proof

 let x be object;

 assume

 A5: x in ( dom g);

 0 <= (f . x) by A3, SUPINF_2: 51;

 hence thesis by A5, FUNCT_1: 47;

 end;

 then

 a4: g is nonnegative by SUPINF_2: 52;



 A6: ex G be Finite_Sep_Sequence of S st (( dom g) = ( union ( rng G)) & for n be Nat, x,y be Element of X st n in ( dom G) & x in (G . n) & y in (G . n) holds (g . x) = (g . y))

 proof

 consider F be Finite_Sep_Sequence of S such that

 A7: ( dom f) = ( union ( rng F)) and

 A8: for n be Nat, x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y) by A1, MESFUNC2:def 4;

 deffunc G( Nat) = ((F . $1) /\ (A /\ B));

 reconsider A as Element of S by A7, MESFUNC2: 31;

 reconsider A2 = (A /\ B) as Element of S;

 consider G be FinSequence such that

 A9: ( len G) = ( len F) & for n be Nat st n in ( dom G) holds (G . n) = G(n) from FINSEQ_1:sch 2;



 A10: ( dom F) = ( Seg ( len F)) by FINSEQ_1:def 3;

 ( dom G) = ( Seg ( len F)) by A9, FINSEQ_1:def 3;

 then

 A11: for i be Nat st i in ( dom F) holds (G . i) = ((F . i) /\ A2) by A9, A10;

 ( dom G) = ( Seg ( len F)) by A9, FINSEQ_1:def 3;

 then

 A12: ( dom G) = ( dom F) by FINSEQ_1:def 3;

 then

 reconsider G as Finite_Sep_Sequence of S by A11, Th35;

 take G;

 for i be Nat st i in ( dom G) holds (G . i) = (A2 /\ (F . i)) by A9;



 then

 A13: ( union ( rng G)) = (A2 /\ ( dom f)) by A7, A12, MESFUNC3: 6

 .= ( dom g) by RELAT_1: 61;

 for i be Nat, x,y be Element of X st i in ( dom G) & x in (G . i) & y in (G . i) holds (g . x) = (g . y)

 proof

 let i be Nat;

 let x,y be Element of X;

 assume that

 A14: i in ( dom G) and

 A15: x in (G . i) and

 A16: y in (G . i);



 A17: (G . i) = ((F . i) /\ A2) by A9, A14;

 then

 A18: y in (F . i) by A16, XBOOLE_0:def 4;



 A19: (G . i) in ( rng G) by A14, FUNCT_1: 3;

 then x in ( dom g) by A13, A15, TARSKI:def 4;

 then

 A20: (g . x) = (f . x) by FUNCT_1: 47;

 y in ( dom g) by A13, A16, A19, TARSKI:def 4;

 then

 A21: (g . y) = (f . y) by FUNCT_1: 47;

 x in (F . i) by A15, A17, XBOOLE_0:def 4;

 hence thesis by A8, A12, A14, A18, A20, A21;

 end;

 hence thesis by A13;

 end;

 ( dom (f | (A /\ B))) = (A /\ (A /\ B)) by RELAT_1: 61;

 then

 A22: ( dom (f | (A /\ B))) = ((A /\ A) /\ B) by XBOOLE_1: 16;

 then

 A23: ( dom (f | (A /\ B))) = ( dom (f | B)) by RELAT_1: 61;

 for x be object st x in ( dom (f | (A /\ B))) holds ((f | (A /\ B)) . x) = ((f | B) . x)

 proof

 let x be object;

 assume

 A24: x in ( dom (f | (A /\ B)));

 then ((f | (A /\ B)) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A23, A24, FUNCT_1: 47;

 end;

 then

 A25: (f | (A /\ B)) = (f | B) by A23, FUNCT_1: 2;

 f is real-valued by A1, MESFUNC2:def 4;

 then

 A26: g is_simple_func_in S by A6, MESFUNC2:def 4;

 now

 per cases ;

 suppose ( dom g) = {} ;

 hence thesis by A23, Def14;

 end;

 suppose

 A27: ( dom g) <> {} ;

 consider F be Finite_Sep_Sequence of S, a,x be FinSequence of ExtREAL such that

 A28: (F,a) are_Re-presentation_of g and (a . 1) = 0 and for n be Nat st 2 <= n & n in ( dom a) holds 0 < (a . n) & (a . n) < +infty and

 A29: ( dom x) = ( dom F) and

 A30: for n be Nat st n in ( dom x) holds (x . n) = ((a . n) * ((M * F) . n)) and

 A31: ( integral (M,g)) = ( Sum x) by A26, MESFUNC3:def 2, a4;



 A32: for n be Nat st n in ( dom F) holds (M . (F . n)) = 0

 proof

 reconsider BB = B as measure_zero of M by A2, MEASURE1:def 7;

 let n be Nat;



 A33: ( dom g) c= B by A22, XBOOLE_1: 17;

 assume

 A34: n in ( dom F);

 then (F . n) in ( rng F) by FUNCT_1: 3;

 then

 reconsider FF = (F . n) as Element of S;

 for v be object st v in (F . n) holds v in ( union ( rng F))

 proof

 let v be object;

 assume

 A35: v in (F . n);

 (F . n) in ( rng F) by A34, FUNCT_1: 3;

 hence thesis by A35, TARSKI:def 4;

 end;

 then

 A36: (F . n) c= ( union ( rng F));

 ( union ( rng F)) = ( dom g) by A28, MESFUNC3:def 1;

 then FF c= BB by A36, A33;

 then (F . n) is measure_zero of M by MEASURE1: 36;

 hence thesis by MEASURE1:def 7;

 end;



 A37: for n be Nat st n in ( dom x) holds (x . n) = 0

 proof

 let n be Nat;

 assume

 A38: n in ( dom x);

 then (M . (F . n)) = 0 by A29, A32;

 then ((M * F) . n) = 0 by A29, A38, FUNCT_1: 13;

 then ((a . n) * ((M * F) . n)) = 0 ;

 hence thesis by A30, A38;

 end;

 ( Sum x) = 0

 proof

 consider sumx be sequence of ExtREAL such that

 A39: ( Sum x) = (sumx . ( len x)) and

 A40: (sumx . 0 ) = 0 and

 A41: for i be Nat st i < ( len x) holds (sumx . (i + 1)) = ((sumx . i) + (x . (i + 1))) by EXTREAL1:def 2;

 now

 defpred P[ Nat] means $1 <= ( len x) implies (sumx . $1) = 0 ;

 assume x <> {} ;



 A42: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A43: P[k];

 assume

 A44: (k + 1) <= ( len x);

 reconsider k as Element of NAT by ORDINAL1:def 12;

 1 <= (k + 1) by NAT_1: 11;

 then (k + 1) in ( Seg ( len x)) by A44;

 then (k + 1) in ( dom x) by FINSEQ_1:def 3;

 then

 A45: (x . (k + 1)) = 0 by A37;

 k < ( len x) by A44, NAT_1: 13;

 then (sumx . (k + 1)) = ((sumx . k) + (x . (k + 1))) by A41;

 hence thesis by A43, A44, A45, NAT_1: 13;

 end;



 A46: P[ 0 ] by A40;

 for i be Nat holds P[i] from NAT_1:sch 2( A46, A42);

 hence thesis by A39;

 end;

 hence thesis by A39, A40, CARD_1: 27;

 end;

 hence thesis by A25, A27, A31, Def14;

 end;

 end;

 hence thesis;

 end;

 theorem :: MESFUNC5:74



 Th74: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, g be PartFunc of X, ExtREAL , F be Functional_Sequence of X, ExtREAL , L be ExtREAL_sequence st g is_simple_func_in S & (for x be object st x in ( dom g) holds 0 < (g . x)) & (for n be Nat holds (F . n) is_simple_func_in S) & (for n be Nat holds ( dom (F . n)) = ( dom g)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((F . n) . x) <= ((F . m) . x)) & (for x be Element of X st x in ( dom g) holds (F # x) is convergent & (g . x) <= ( lim (F # x))) & (for n be Nat holds (L . n) = ( integral' (M,(F . n)))) holds L is convergent & ( integral' (M,g)) <= ( lim L)

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let g be PartFunc of X, ExtREAL ;

 let F be Functional_Sequence of X, ExtREAL ;

 let L be ExtREAL_sequence;

 assume that

 A1: g is_simple_func_in S and

 A2: for x be object st x in ( dom g) holds 0 < (g . x) and

 A3: for n be Nat holds (F . n) is_simple_func_in S and

 A4: for n be Nat holds ( dom (F . n)) = ( dom g) and

 A5: for n be Nat holds (F . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((F . n) . x) <= ((F . m) . x) and

 A7: for x be Element of X st x in ( dom g) holds (F # x) is convergent & (g . x) <= ( lim (F # x)) and

 A8: for n be Nat holds (L . n) = ( integral' (M,(F . n)));

 per cases ;

 suppose

 A9: ( dom g) = {} ;

 A10:

 now

 let n be Nat;

 ( dom (F . n)) = {} by A4, A9;

 then ( integral' (M,(F . n))) = 0 by Def14;

 hence (L . n) = 0 by A8;

 end;

 then L is convergent_to_finite_number by Th52;

 hence L is convergent;

 ( lim L) = 0 by A10, Th52;

 hence thesis by A9, Def14;

 end;

 suppose

 A11: ( dom g) <> {} ;

 for v be object st v in ( dom g) holds 0 <= (g . v) by A2;

 then g is nonnegative by SUPINF_2: 52;

 then

 consider G be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL such that

 A12: (G,a) are_Re-presentation_of g and

 A13: (a . 1) = 0 and

 A14: for n be Nat st 2 <= n & n in ( dom a) holds 0 < (a . n) & (a . n) < +infty by A1, MESFUNC3: 14;

 defpred PP1[ Nat, set] means $2 = (a . $1);



 A15: for k be Nat st k in ( Seg ( len a)) holds ex x be Element of REAL st PP1[k, x]

 proof

 let k be Nat;

 assume

 A16: k in ( Seg ( len a));

 then

 A17: 1 <= k by FINSEQ_1: 1;



 A18: k in ( dom a) by A16, FINSEQ_1:def 3;

 per cases ;

 suppose

 A19: k = 1;

 take ( In ( 0 , REAL ));

 thus thesis by A13, A19;

 end;

 suppose k <> 1;

 then k > 1 by A17, XXREAL_0: 1;

 then

 A20: k >= (1 + 1) by NAT_1: 13;

 then

 A21: (a . k) < +infty by A14, A18;

 0 < (a . k) by A14, A18, A20;

 then

 reconsider x = (a . k) as Element of REAL by A21, XXREAL_0: 14;

 take x;

 thus thesis;

 end;

 end;

 consider a1 be FinSequence of REAL such that

 A22: ( dom a1) = ( Seg ( len a)) & for k be Nat st k in ( Seg ( len a)) holds PP1[k, (a1 . k)] from FINSEQ_1:sch 5( A15);



 A23: ( len a) <> 0

 proof

 assume ( len a) = 0 ;

 then

 A24: ( dom a) = ( Seg 0 ) by FINSEQ_1:def 3;



 A25: ( rng G) = {}

 proof

 assume ( rng G) <> {} ;

 then

 consider y be object such that

 A26: y in ( rng G) by XBOOLE_0:def 1;

 ex x be object st x in ( dom G) & y = (G . x) by A26, FUNCT_1:def 3;

 hence contradiction by A12, A24, MESFUNC3:def 1;

 end;

 ( union ( rng G)) <> {} by A11, A12, MESFUNC3:def 1;

 then

 consider x be object such that

 A27: x in ( union ( rng G)) by XBOOLE_0:def 1;

 ex Y be set st x in Y & Y in ( rng G) by A27, TARSKI:def 4;

 hence contradiction by A25;

 end;



 A28: 2 <= ( len a)

 proof

 assume not 2 <= ( len a);

 then ( len a) = 1 by A23, NAT_1: 23;

 then ( dom a) = {1} by FINSEQ_1: 2, FINSEQ_1:def 3;

 then

 A29: ( dom G) = {1} by A12, MESFUNC3:def 1;



 A30: ( dom g) = ( union ( rng G)) by A12, MESFUNC3:def 1

 .= ( union {(G . 1)}) by A29, FUNCT_1: 4

 .= (G . 1) by ZFMISC_1: 25;

 then

 consider x be object such that

 A31: x in (G . 1) by A11, XBOOLE_0:def 1;

 1 in ( dom G) by A29, TARSKI:def 1;

 then (g . x) = 0 by A12, A13, A31, MESFUNC3:def 1;

 hence contradiction by A2, A30, A31;

 end;

 then 1 <= ( len a) by XXREAL_0: 2;

 then 1 in ( Seg ( len a));

 then

 A32: (a . 1) = (a1 . 1) by A22;



 A33: 2 in ( dom a1) by A22, A28;

 then

 A34: 2 in ( dom a) by A22, FINSEQ_1:def 3;

 (a1 . 2) = (a . 2) by A22, A33;

 then (a1 . 2) <> (a . 1) by A13, A14, A34;

 then

 A35: not (a1 . 2) in {(a1 . 1)} by A32, TARSKI:def 1;

 (a1 . 2) in ( rng a1) by A33, FUNCT_1: 3;

 then

 reconsider RINGA = (( rng a1) \ {(a1 . 1)}) as finite non empty real-membered set by A35, XBOOLE_0:def 5;

 reconsider alpha = ( min RINGA) as R_eal by XXREAL_0:def 1;

 reconsider beta1 = ( max RINGA) as Element of REAL by XREAL_0:def 1;



 A36: ( min RINGA) in RINGA by XXREAL_2:def 7;

 then ( min RINGA) in ( rng a1) by XBOOLE_0:def 5;

 then

 consider i be object such that

 A37: i in ( dom a1) and

 A38: ( min RINGA) = (a1 . i) by FUNCT_1:def 3;

 reconsider i as Element of NAT by A37;



 A39: (a . i) = (a1 . i) by A22, A37;

 i in ( Seg ( len a1)) by A37, FINSEQ_1:def 3;

 then

 A40: 1 <= i by FINSEQ_1: 1;

 not ( min RINGA) in {(a1 . 1)} by A36, XBOOLE_0:def 5;

 then i <> 1 by A38, TARSKI:def 1;

 then 1 < i by A40, XXREAL_0: 1;

 then

 A41: (1 + 1) <= i by NAT_1: 13;



 A42: i in ( dom a) by A22, A37, FINSEQ_1:def 3;

 then

 A43: 0 < alpha by A14, A38, A41, A39;

 reconsider beta = ( max RINGA) as R_eal by XXREAL_0:def 1;



 A44: for x be set st x in ( dom g) holds alpha <= (g . x) & (g . x) <= beta

 proof

 let x be set;

 assume

 A45: x in ( dom g);

 then x in ( union ( rng G)) by A12, MESFUNC3:def 1;

 then

 consider Y be set such that

 A46: x in Y and

 A47: Y in ( rng G) by TARSKI:def 4;

 consider k be object such that

 A48: k in ( dom G) and

 A49: Y = (G . k) by A47, FUNCT_1:def 3;

 reconsider k as Element of NAT by A48;

 k in ( dom a) by A12, A48, MESFUNC3:def 1;

 then

 A50: k in ( Seg ( len a)) by FINSEQ_1:def 3;

 now

 1 <= ( len a) by A28, XXREAL_0: 2;

 then

 A51: 1 in ( dom a1) by A22;



 A52: (g . x) = (a . k) by A12, A46, A48, A49, MESFUNC3:def 1;

 assume

 A53: (a1 . k) = (a1 . 1);

 (a . k) = (a1 . k) by A22, A50;

 then (a . k) = (a . 1) by A22, A53, A51;

 hence contradiction by A2, A13, A45, A52;

 end;

 then

 A54: not (a1 . k) in {(a1 . 1)} by TARSKI:def 1;

 (a1 . k) in ( rng a1) by A22, A50, FUNCT_1: 3;

 then

 A55: (a1 . k) in RINGA by A54, XBOOLE_0:def 5;

 (g . x) = (a . k) by A12, A46, A48, A49, MESFUNC3:def 1

 .= (a1 . k) by A22, A50;

 hence thesis by A55, XXREAL_2:def 7, XXREAL_2:def 8;

 end;



 A56: for n be Nat holds ( dom (g - (F . n))) = ( dom g)

 proof

 g is without-infty by A1, Th14;

 then not -infty in ( rng g);

 then

 A57: (g " { -infty }) = {} by FUNCT_1: 72;

 g is without+infty by A1, Th14;

 then not +infty in ( rng g);

 then

 A58: (g " { +infty }) = {} by FUNCT_1: 72;

 let n be Nat;



 A59: ( dom (g - (F . n))) = ((( dom (F . n)) /\ ( dom g)) \ ((((F . n) " { +infty }) /\ (g " { +infty })) \/ (((F . n) " { -infty }) /\ (g " { -infty })))) by MESFUNC1:def 4;

 ( dom (F . n)) = ( dom g) by A4;

 hence thesis by A58, A57, A59;

 end;



 A60: g is real-valued by A1, MESFUNC2:def 4;



 A61: for e be R_eal st 0 < e & e < alpha holds ex H be SetSequence of X, MF be ExtREAL_sequence st (for n be Nat holds (H . n) = ( less_dom ((g - (F . n)),e))) & (for n,m be Nat st n <= m holds (H . n) c= (H . m)) & (for n be Nat holds (H . n) c= ( dom g)) & (for n be Nat holds (MF . n) = (M . (H . n))) & (M . ( dom g)) = ( sup ( rng MF)) & for n be Nat holds (H . n) in S

 proof

 let e be R_eal;

 assume that

 A62: 0 < e and

 A63: e < alpha;

 deffunc FFH( Nat) = ( less_dom ((g - (F . $1)),e));

 consider H be SetSequence of X such that

 A64: for n be Element of NAT holds (H . n) = FFH(n) from FUNCT_2:sch 4;

 A65:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (H . n) = FFH(n) by A64;

 end;



 A66: for n be Nat holds (H . n) c= ( dom g)

 proof

 let n be Nat;

 now

 let x be object;

 assume x in (H . n);

 then x in ( less_dom ((g - (F . n)),e)) by A65;

 then x in ( dom (g - (F . n))) by MESFUNC1:def 11;

 hence x in ( dom g) by A56;

 end;

 hence thesis;

 end;



 A67: ( Union H) c= ( dom g)

 proof

 let x be object;

 assume x in ( Union H);

 then

 consider n be Nat such that

 A68: x in (H . n) by PROB_1: 12;

 (H . n) c= ( dom g) by A66;

 hence thesis by A68;

 end;

 now

 let x be object;

 assume

 A69: x in ( dom g);

 then

 reconsider x1 = x as Element of X;



 A70: (F # x1) is convergent by A7, A69;

 A71:

 now

 reconsider E = e as Element of REAL by A62, A63, XXREAL_0: 48;

 assume (F # x1) is convergent_to_-infty;

 then

 consider N be Nat such that

 A72: for m be Nat st N <= m holds ((F # x1) . m) <= ( - E) by A62;

 (F . N) is nonnegative by A5;

 then

 A73: 0 <= ((F . N) . x) by SUPINF_2: 51;

 ((F # x1) . N) < 0 by A62, A72;

 hence contradiction by A73, Def13;

 end;

 now

 per cases by A70, A71;

 suppose

 A74: (F # x1) is convergent_to_finite_number;

 reconsider E = e as Element of REAL by A62, A63, XXREAL_0: 48;



 A75: (ex limFx be Real st ( lim (F # x1)) = limFx & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.(((F # x1) . m) - ( lim (F # x1))).| < p) & (F # x1) is convergent_to_finite_number) or ( lim (F # x1)) = +infty & (F # x1) is convergent_to_+infty or ( lim (F # x1)) = -infty & (F # x1) is convergent_to_-infty by A70, Def12;

 then

 consider N be Nat such that

 A76: for m be Nat st N <= m holds |.(((F # x1) . m) - ( lim (F # x1))).| < (E / 2) by A62, A74, Th50, Th51;

 reconsider N as Element of NAT by ORDINAL1:def 12;

 (g . x) <= ( lim (F # x1)) by A7, A69;

 then ((g . x) - (E / 2)) <= (( lim (F # x1)) - 0. ) by A62, XXREAL_3: 37;

 then

 A77: ((g . x) - (E / 2)) <= ( lim (F # x1)) by XXREAL_3: 4;

 now

 let k be Nat;

 set m = (N + k);



 A78: x1 in ( dom (g - (F . m))) by A56, A69;

 now

 let e be Real;

 assume 0 < e;

 then

 consider N0 be Nat such that

 A79: for M be Nat st N0 <= M holds |.(((F # x1) . M) - ( lim (F # x1))).| < e by A74, A75, Th50, Th51;

 reconsider N0, n1 = m as Element of NAT by ORDINAL1:def 12;

 set M = ( max (N0,n1));



 A80: (((F # x1) . M) - ( lim (F # x1))) <= |.(((F # x1) . M) - ( lim (F # x1))).| by EXTREAL1: 20;

 ((F . m) . x1) <= ((F . M) . x1) by A6, A69, XXREAL_0: 25;

 then ((F . m) . x1) <= ((F # x1) . M) by Def13;

 then

 A81: ((F # x1) . m) <= ((F # x1) . M) by Def13;

 |.(((F # x1) . M) - ( lim (F # x1))).| < e by A79, XXREAL_0: 25;

 then (((F # x1) . M) - ( lim (F # x1))) < e by A80, XXREAL_0: 2;

 then ((F # x1) . M) < (e + ( lim (F # x1))) by A74, A75, Th50, Th51, XXREAL_3: 54;

 hence ((F # x1) . m) < (( lim (F # x1)) + e) by A81, XXREAL_0: 2;

 end;

 then ((F # x1) . m) <= ( lim (F # x1)) by XXREAL_3: 61;

 then

 A82: 0 <= (( lim (F # x1)) - ((F # x1) . m)) by XXREAL_3: 40;

 |.(((F # x1) . m) - ( lim (F # x1))).| = |.(( lim (F # x1)) - ((F # x1) . m)).| by Th1

 .= (( lim (F # x1)) - ((F # x1) . m)) by A82, EXTREAL1:def 1;

 then (( lim (F # x1)) - ((F # x1) . m)) < (E / 2) by A76, NAT_1: 11;

 then

 A83: (( lim (F # x1)) - ((F . m) . x1)) < (E / 2) by Def13;



 A84: |.(((F # x1) . m) - ( lim (F # x1))).| < (E / 2) by A76, NAT_1: 11;

 then ((F # x1) . m) <> -infty by A74, A75, Th3, Th51;

 then

 A85: ((F . m) . x) <> -infty by Def13;

 ((F # x1) . m) <> +infty by A74, A75, A84, Th3, Th50;

 then ((F . m) . x) <> +infty by Def13;

 then ( lim (F # x1)) < (((F . m) . x) + (E / 2)) by A85, A83, XXREAL_3: 54;

 then

 A86: (( lim (F # x1)) + (E / 2)) < ((((F . m) . x) + (E / 2)) + (E / 2)) by XXREAL_3: 62;

 (g . x) <= (( lim (F # x1)) + (E / 2)) by A77, XXREAL_3: 41;

 then (g . x) < ((((F . m) . x1) + (E / 2)) + (E / 2)) by A86, XXREAL_0: 2;

 then (g . x) < (((F . m) . x1) + ((E / 2) + (E / 2))) by XXREAL_3: 29;

 then (g . x) < (((F . m) . x1) + ((E / 2) + (E / 2)));

 then ((g . x) - ((F . m) . x1)) < e by XXREAL_3: 55;

 then ((g - (F . m)) . x1) < e by A78, MESFUNC1:def 4;

 then x in ( less_dom ((g - (F . (N + k))),e)) by A78, MESFUNC1:def 11;

 hence x in (H . (N + k qua Complex)) by A65;

 end;

 then

 A87: x in (( inferior_setsequence H) . N) by SETLIM_1: 19;

 ( dom ( inferior_setsequence H)) = NAT by FUNCT_2:def 1;

 hence ex N be Nat st N in ( dom ( inferior_setsequence H)) & x in (( inferior_setsequence H) . N) by A87;

 end;

 suppose

 A88: (F # x1) is convergent_to_+infty;

 ex N be Nat st for m be Nat st N <= m holds ((g . x1) - ((F . m) . x1)) < e

 proof



 A89: e in REAL by A62, A63, XXREAL_0: 48;

 per cases ;

 suppose

 A90: ((g . x1) - e) <= 0 ;

 consider N be Nat such that

 A91: for m be Nat st N <= m holds 1 <= ((F # x1) . m) by A88;

 now

 let m be Nat;

 assume N <= m;

 then ((g . x1) - e) < ((F # x1) . m) by A90, A91;

 then (g . x1) < (((F # x1) . m) + e) by A89, XXREAL_3: 54;

 then ((g . x1) - ((F # x1) . m)) < e by A89, XXREAL_3: 55;

 hence ((g . x1) - ((F . m) . x1)) < e by Def13;

 end;

 hence thesis;

 end;

 suppose

 A92: 0 < ((g . x1) - e);

 reconsider e1 = e as Element of REAL by A62, A63, XXREAL_0: 48;

 reconsider gx1 = (g . x) as Real by A60;

 ((g . x) - e) = (gx1 - e1);

 then

 reconsider ee = ((g . x1) - e) as Real;

 consider N be Nat such that

 A93: for m be Nat st N <= m holds (ee + 1) <= ((F # x1) . m) by A88, A92;



 A94: ee < (ee + 1) by XREAL_1: 29;

 now

 let m be Nat;

 assume N <= m;

 then (ee + 1) <= ((F # x1) . m) by A93;

 then ee < ((F # x1) . m) by A94, XXREAL_0: 2;

 then (g . x1) < (((F # x1) . m) + e) by A89, XXREAL_3: 54;

 then ((g . x1) - ((F # x1) . m)) < e by A89, XXREAL_3: 55;

 hence ((g . x1) - ((F . m) . x1)) < e by Def13;

 end;

 hence thesis;

 end;

 end;

 then

 consider N be Nat such that

 A95: for m be Nat st N <= m holds ((g . x1) - ((F . m) . x1)) < e;

 reconsider N as Element of NAT by ORDINAL1:def 12;

 A96:

 now

 let m be Nat;



 A97: x1 in ( dom (g - (F . m))) by A56, A69;

 assume N <= m;

 then ((g . x1) - ((F . m) . x1)) < e by A95;

 then ((g - (F . m)) . x1) < e by A97, MESFUNC1:def 4;

 hence x1 in ( less_dom ((g - (F . m)),e)) by A97, MESFUNC1:def 11;

 end;

 now

 let k be Nat;

 x in ( less_dom ((g - (F . (N + k))),e)) by A96, NAT_1: 11;

 hence x in (H . (N + k qua Complex)) by A65;

 end;

 then

 A98: x in (( inferior_setsequence H) . N) by SETLIM_1: 19;

 ( dom ( inferior_setsequence H)) = NAT by FUNCT_2:def 1;

 hence ex N be Nat st N in ( dom ( inferior_setsequence H)) & x in (( inferior_setsequence H) . N) by A98;

 end;

 end;

 then

 consider N be Nat such that

 A99: N in ( dom ( inferior_setsequence H)) and

 A100: x in (( inferior_setsequence H) . N);

 (( inferior_setsequence H) . N) in ( rng ( inferior_setsequence H)) by A99, FUNCT_1: 3;

 then x in ( Union ( inferior_setsequence H)) by A100, TARSKI:def 4;

 hence x in ( lim_inf H) by SETLIM_1:def 4;

 end;

 then

 A101: ( dom g) c= ( lim_inf H);

 deffunc U( Nat) = (M . (H . $1));



 A102: ( lim_inf H) c= ( lim_sup H) by KURATO_0: 6;

 consider MF be ExtREAL_sequence such that

 A103: for n be Element of NAT holds (MF . n) = U(n) from FUNCT_2:sch 4;



 A104: for n,m be Nat st n <= m holds (H . n) c= (H . m)

 proof

 let n,m be Nat;

 assume

 A105: n <= m;

 now

 let x be object;

 assume x in (H . n);

 then

 A106: x in ( less_dom ((g - (F . n)),e)) by A65;

 then

 A107: x in ( dom (g - (F . n))) by MESFUNC1:def 11;

 then

 A108: ((g - (F . n)) . x) = ((g . x) - ((F . n) . x)) by MESFUNC1:def 4;



 A109: ((g - (F . n)) . x) < e by A106, MESFUNC1:def 11;



 A110: ( dom (g - (F . n))) = ( dom g) by A56;

 then

 A111: ((F . n) . x) <= ((F . m) . x) by A6, A105, A107;



 A112: ( dom (g - (F . m))) = ( dom g) by A56;

 then ((g - (F . m)) . x) = ((g . x) - ((F . m) . x)) by A107, A110, MESFUNC1:def 4;

 then ((g - (F . m)) . x) <= ((g - (F . n)) . x) by A108, A111, XXREAL_3: 37;

 then ((g - (F . m)) . x) < e by A109, XXREAL_0: 2;

 then x in ( less_dom ((g - (F . m)),e)) by A107, A110, A112, MESFUNC1:def 11;

 hence x in (H . m) by A65;

 end;

 hence thesis;

 end;

 then for n,m be Nat st n <= m holds (H . n) c= (H . m);

 then

 A113: H is non-descending by PROB_1:def 5;

 A114:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (MF . n) = U(n) by A103;

 end;

 now

 let x be object;

 assume x in ( lim_inf H);

 then x in ( Union ( inferior_setsequence H)) by SETLIM_1:def 4;

 then

 consider V be set such that

 A115: x in V and

 A116: V in ( rng ( inferior_setsequence H)) by TARSKI:def 4;

 consider n be object such that

 A117: n in ( dom ( inferior_setsequence H)) and

 A118: V = (( inferior_setsequence H) . n) by A116, FUNCT_1:def 3;

 reconsider n as Element of NAT by A117;

 x in (H . (n + 0 )) by A115, A118, SETLIM_1: 19;

 then x in ( less_dom ((g - (F . n)),e)) by A65;

 then x in ( dom (g - (F . n))) by MESFUNC1:def 11;

 hence x in ( dom g) by A56;

 end;

 then ( lim_inf H) c= ( dom g);

 then

 A119: ( lim_inf H) = ( dom g) by A101;



 A120: (M . ( dom g)) = ( sup ( rng MF)) & for n be Element of NAT holds (H . n) in S

 proof

 A121:

 now

 reconsider E = e as Element of REAL by A62, A63, XXREAL_0: 48;

 let x be object;

 assume x in NAT ;

 then

 reconsider n = x as Element of NAT ;



 A122: ( less_dom ((g - (F . n)),E)) c= ( dom (g - (F . n))) by MESFUNC1:def 11;



 A123: (F . n) is_simple_func_in S by A3;

 then

 consider GF be Finite_Sep_Sequence of S such that

 A124: ( dom (F . n)) = ( union ( rng GF)) and for m be Nat, x,y be Element of X st m in ( dom GF) & x in (GF . m) & y in (GF . m) holds ((F . n) . x) = ((F . n) . y) by MESFUNC2:def 4;



 A125: (F . n) is real-valued by A123, MESFUNC2:def 4;

 reconsider DGH = ( union ( rng GF)) as Element of S by MESFUNC2: 31;

 ( dom (F . n)) = ( dom g) by A4;

 then (DGH /\ ( less_dom ((g - (F . n)),E))) = (( dom (g - (F . n))) /\ ( less_dom ((g - (F . n)),E))) by A56, A124;

 then

 A126: (DGH /\ ( less_dom ((g - (F . n)),E))) = ( less_dom ((g - (F . n)),E)) by A122, XBOOLE_1: 28;



 A127: (F . n) is DGH -measurable by A3, MESFUNC2: 34;



 A128: g is real-valued by A1, MESFUNC2:def 4;

 g is DGH -measurable by A1, MESFUNC2: 34;

 then (g - (F . n)) is DGH -measurable by A124, A128, A125, A127, MESFUNC2: 11;

 then (DGH /\ ( less_dom ((g - (F . n)),E))) in S by MESFUNC1:def 16;

 hence (H . x) in S by A65, A126;

 end;

 ( dom H) = NAT by FUNCT_2:def 1;

 then

 reconsider HH = H as sequence of S by A121, FUNCT_2: 3;



 A129: for n be Nat holds (HH . n) c= (HH . (n + 1)) by A104, NAT_1: 11;

 ( rng HH) c= S by RELAT_1:def 19;

 then

 A130: ( rng H) c= ( dom M) by FUNCT_2:def 1;

 ( lim_sup H) = ( Union H) by A113, SETLIM_1: 59;

 then

 A131: (M . ( union ( rng H))) = (M . ( dom g)) by A119, A67, A102, XBOOLE_0:def 10;



 A132: ( dom H) = NAT by FUNCT_2:def 1;



 A133: ( dom MF) = NAT by FUNCT_2:def 1;



 A134: for x be object holds x in ( dom MF) iff x in ( dom H) & (H . x) in ( dom M)

 proof

 let x be object;

 now

 assume

 A135: x in ( dom MF);

 then (H . x) in ( rng H) by A132, FUNCT_1: 3;

 hence x in ( dom H) & (H . x) in ( dom M) by A132, A130, A135;

 end;

 hence thesis by A133;

 end;

 for x be object st x in ( dom MF) holds (MF . x) = (M . (H . x)) by A103;

 then (M * H) = MF by A134, FUNCT_1: 10;

 hence thesis by A121, A129, A131, MEASURE2: 23;

 end;

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (H . n) in S by A120;

 end;

 hence thesis by A65, A104, A66, A114, A120;

 end;

 per cases ;

 suppose

 A136: (M . ( dom g)) <> +infty ;



 A137: 0 < beta

 proof

 consider x be object such that

 A138: x in ( dom g) by A11, XBOOLE_0:def 1;



 A139: (g . x) <= beta by A44, A138;

 alpha <= (g . x) by A44, A138;

 hence thesis by A14, A38, A41, A42, A39, A139;

 end;



 A140: {} in S by MEASURE1: 34;



 A141: (M . {} ) = 0 by VALUED_0:def 19;

 ( dom g) is Element of S by A1, Th37;

 then

 A142: (M . ( dom g)) <> -infty by A141, A140, MEASURE1: 31, XBOOLE_1: 2;

 then

 reconsider MG = (M . ( dom g)) as Element of REAL by A136, XXREAL_0: 14;

 reconsider DG = ( dom g) as Element of S by A1, Th37;



 A143: for x be object st x in ( dom g) holds 0 <= (g . x) by A2;

 then

 A144: ( integral' (M,g)) <> -infty by A1, Th68, SUPINF_2: 52;



 A145: g is nonnegative by A143, SUPINF_2: 52;



 A146: ( integral' (M,g)) <= (beta * (M . DG))

 proof

 consider GP be PartFunc of X, ExtREAL such that

 A147: GP is_simple_func_in S and

 A148: ( dom GP) = DG and

 A149: for x be object st x in DG holds (GP . x) = beta by Th41;



 A150: for x be object st x in ( dom (GP - g)) holds (g . x) <= (GP . x)

 proof

 let x be object;

 assume x in ( dom (GP - g));

 then x in ((( dom GP) /\ ( dom g)) \ (((GP " { +infty }) /\ (g " { +infty })) \/ ((GP " { -infty }) /\ (g " { -infty })))) by MESFUNC1:def 4;

 then

 A151: x in (( dom GP) /\ ( dom g)) by XBOOLE_0:def 5;

 then (GP . x) = beta by A148, A149;

 hence thesis by A44, A148, A151;

 end;

 for x be object st x in ( dom GP) holds 0 <= (GP . x) by A137, A148, A149;

 then

 A152: GP is nonnegative by SUPINF_2: 52;

 then

 A153: ( dom (GP - g)) = (( dom GP) /\ ( dom g)) by A1, A145, A147, A150, Th69;

 then

 A154: (g | ( dom (GP - g))) = g by A148, GRFUNC_1: 23;



 A155: (GP | ( dom (GP - g))) = GP by A148, A153, GRFUNC_1: 23;

 ( integral' (M,(g | ( dom (GP - g))))) <= ( integral' (M,(GP | ( dom (GP - g))))) by A1, A145, A147, A152, A150, Th70;

 hence thesis by A137, A147, A148, A149, A154, A155, Th71;

 end;

 (beta * (M . DG)) = (beta1 * MG) by EXTREAL1: 1;

 then

 A156: ( integral' (M,g)) <> +infty by A146, XXREAL_0: 9;



 A157: for e be R_eal st 0 < e & e < alpha holds ex N0 be Nat st for n be Nat st N0 <= n holds (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < ( integral' (M,(F . n)))

 proof

 let e be R_eal;

 assume that

 A158: 0 < e and

 A159: e < alpha;



 A160: e <> +infty by A159, XXREAL_0: 4;

 consider H be SetSequence of X, MF be ExtREAL_sequence such that

 A161: for n be Nat holds (H . n) = ( less_dom ((g - (F . n)),e)) and

 A162: for n,m be Nat st n <= m holds (H . n) c= (H . m) and

 A163: for n be Nat holds (H . n) c= ( dom g) and

 A164: for n be Nat holds (MF . n) = (M . (H . n)) and

 A165: (M . ( dom g)) = ( sup ( rng MF)) and

 A166: for n be Nat holds (H . n) in S by A61, A158, A159;

 ( sup ( rng MF)) in REAL by A136, A142, A165, XXREAL_0: 14;

 then

 consider y be ExtReal such that

 A167: y in ( rng MF) and

 A168: (( sup ( rng MF)) - e) < y by A158, MEASURE6: 6;

 consider N0 be object such that

 A169: N0 in ( dom MF) and

 A170: y = (MF . N0) by A167, FUNCT_1:def 3;

 reconsider N0 as Element of NAT by A169;

 reconsider B0 = (H . N0) as Element of S by A166;

 (M . B0) <= (M . DG) by A163, MEASURE1: 31;

 then (M . B0) < +infty by A136, XXREAL_0: 2, XXREAL_0: 4;

 then

 A171: (M . (DG \ B0)) = ((M . DG) - (M . B0)) by A163, MEASURE1: 32;

 take N0;

 ((M . ( dom g)) - e) < (M . (H . N0)) by A164, A165, A168, A170;

 then (M . ( dom g)) < ((M . (H . N0)) + e) by A158, A160, XXREAL_3: 54;

 then

 A172: ((M . ( dom g)) - (M . (H . N0))) < e by A158, A160, XXREAL_3: 55;

 A173:

 now

 let n be Nat;

 reconsider BN = (H . n) as Element of S by A166;

 assume N0 <= n;

 then (H . N0) c= (H . n) by A162;

 then (M . (DG \ BN)) <= (M . (DG \ B0)) by MEASURE1: 31, XBOOLE_1: 34;

 hence (M . (( dom g) \ (H . n))) < e by A172, A171, XXREAL_0: 2;

 end;

 now

 reconsider XSMg = ( integral' (M,g)) as Element of REAL by A156, A144, XXREAL_0: 14;

 let n be Nat;



 A174: for x be object st x in ( dom (F . n)) holds ((F . n) . x) = ((F . n) . x);

 reconsider B = (H . n) as Element of S by A166;

 (H . n) in S by A166;

 then (X \ (H . n)) in S by MEASURE1: 34;

 then

 A175: (DG /\ (X \ (H . n))) in S by MEASURE1: 34;

 (DG /\ (X \ (H . n))) = ((DG /\ X) \ (H . n)) by XBOOLE_1: 49;

 then

 reconsider A = (DG \ (H . n)) as Element of S by A175, XBOOLE_1: 28;

 e <> +infty by A159, XXREAL_0: 4;

 then

 reconsider ee = e as Element of REAL by A158, XXREAL_0: 14;



 A176: A misses B by XBOOLE_1: 79;

 (beta * e) = (beta1 * ee) by EXTREAL1: 1;

 then

 reconsider betae = (beta * e) as Real;



 A177: for x be object st x in ( dom g) holds (g . x) = (g . x);



 A178: (M . B) <= (M . DG) by A163, MEASURE1: 31;

 then (M . ( dom g)) <> -infty by A141, A140, MEASURE1: 31, XBOOLE_1: 2;

 then

 A179: (M . ( dom g)) in REAL by A136, XXREAL_0: 14;



 A180: DG = (DG \/ (H . n)) by A163, XBOOLE_1: 12;

 then

 A181: DG = ((DG \ (H . n)) \/ (H . n)) by XBOOLE_1: 39;

 then ( dom g) = ((A \/ B) /\ ( dom g));

 then g = (g | (A \/ B)) by A177, FUNCT_1: 46;

 then

 A182: ( integral' (M,g)) = (( integral' (M,(g | A))) + ( integral' (M,(g | B)))) by A1, A145, Th67, XBOOLE_1: 79;

 (M . A) <= (M . DG) by A181, MEASURE1: 31, XBOOLE_1: 7;

 then (M . A) < +infty by A136, XXREAL_0: 2, XXREAL_0: 4;

 then (beta * (M . A)) < (beta * +infty ) by A137, XXREAL_3: 72;

 then

 A183: (beta * (M . A)) <> +infty by A137, XXREAL_3:def 5;



 A184: (g | B) is nonnegative by A143, Th15, SUPINF_2: 52;



 A185: ( dom (F . n)) = ( dom g) by A4;

 then ( dom (F . n)) = ((A \/ B) /\ ( dom (F . n))) by A181;

 then

 A186: (F . n) = ((F . n) | (A \/ B)) by A174, FUNCT_1: 46;

 consider GP be PartFunc of X, ExtREAL such that

 A187: GP is_simple_func_in S and

 A188: ( dom GP) = A and

 A189: for x be object st x in A holds (GP . x) = beta by Th41;



 A190: ( integral' (M,GP)) = (beta * (M . A)) by A137, A187, A188, A189, Th71;



 A191: ( dom (g | A)) = A by A181, RELAT_1: 62, XBOOLE_1: 7;



 A192: for x be object st x in ( dom (GP - (g | A))) holds ((g | A) . x) <= (GP . x)

 proof

 let x be object;

 assume x in ( dom (GP - (g | A)));

 then x in ((( dom GP) /\ ( dom (g | A))) \ (((GP " { +infty }) /\ ((g | A) " { +infty })) \/ ((GP " { -infty }) /\ ((g | A) " { -infty })))) by MESFUNC1:def 4;

 then

 A193: x in (( dom GP) /\ ( dom (g | A))) by XBOOLE_0:def 5;

 then

 A194: x in ( dom GP) by XBOOLE_0:def 4;

 x in (( dom g) /\ A) by A191, A188, A193, RELAT_1: 61;

 then x in ( dom g) by XBOOLE_0:def 4;

 then

 A195: (g . x) <= beta by A44;

 ((g | A) . x) = (g . x) by A191, A188, A193, FUNCT_1: 47;

 hence thesis by A188, A189, A194, A195;

 end;

 for x be object st x in ( dom GP) holds 0 <= (GP . x) by A137, A188, A189;

 then

 A196: GP is nonnegative by SUPINF_2: 52;

 0 <= (M . A) by A141, A140, MEASURE1: 31, XBOOLE_1: 2;

 then

 reconsider XSMGP = ( integral' (M,GP)) as Element of REAL by A137, A190, A183, XXREAL_0: 14;



 A197: (( integral' (M,g)) - ( integral' (M,GP))) = (XSMg - XSMGP) by SUPINF_2: 3;



 A198: (g | A) is_simple_func_in S by A1, Th34;

 then

 A199: ( integral' (M,(g | A))) <> -infty by A145, Th15, Th68;



 A200: (g | A) is nonnegative by A143, Th15, SUPINF_2: 52;

 then

 A201: ( dom (GP - (g | A))) = (( dom GP) /\ ( dom (g | A))) by A198, A187, A196, A192, Th69;

 then

 A202: (GP | ( dom (GP - (g | A)))) = GP by A191, A188, GRFUNC_1: 23;

 ((g | A) | ( dom (GP - (g | A)))) = (g | A) by A191, A188, A201, GRFUNC_1: 23;

 then

 A203: ( integral' (M,(g | A))) <= ( integral' (M,GP)) by A198, A200, A187, A196, A192, A202, Th70;

 then

 A204: (( integral' (M,g)) - ( integral' (M,GP))) <= (( integral' (M,g)) - ( integral' (M,(g | A)))) by XXREAL_3: 37;

 assume N0 <= n;

 then (M . A) < e by A173;

 then

 A205: (beta * (M . A)) < (beta * e) by A137, XXREAL_3: 72;

 then

 A206: ( integral' (M,(g | A))) <> +infty by A203, A190, XXREAL_0: 2, XXREAL_0: 4;

 then

 reconsider XSMgA = ( integral' (M,(g | A))) as Element of REAL by A199, XXREAL_0: 14;



 A207: ( integral' (M,(g | A))) is Element of REAL by A199, A206, XXREAL_0: 14;

 (XSMg - XSMgA) = (( integral' (M,g)) - ( integral' (M,(g | A)))) by SUPINF_2: 3

 .= ( integral' (M,(g | B))) by A182, A207, XXREAL_3: 24;

 then

 reconsider XSMgB = ( integral' (M,(g | B))) as Real;



 A208: (H . n) c= DG by A163;

 ( integral' (M,(g | A))) is Element of REAL by A199, A206, XXREAL_0: 14;

 then

 A209: (( integral' (M,g)) - ( integral' (M,(g | A)))) = ( integral' (M,(g | B))) by A182, XXREAL_3: 24;

 (XSMg - betae) < (XSMg - XSMGP) by A190, A205, XREAL_1: 15;

 then

 A210: (XSMg - betae) < XSMgB by A209, A204, A197, XXREAL_0: 2;

 consider EP be PartFunc of X, ExtREAL such that

 A211: EP is_simple_func_in S and

 A212: ( dom EP) = B and

 A213: for x be object st x in B holds (EP . x) = e by A158, A160, Th41;



 A214: ( integral' (M,EP)) = (e * (M . B)) by A158, A211, A212, A213, Th71;

 for x be object st x in ( dom EP) holds 0 <= (EP . x) by A158, A212, A213;

 then

 A215: EP is nonnegative by SUPINF_2: 52;

 (M . B) < +infty by A136, A178, XXREAL_0: 2, XXREAL_0: 4;

 then (e * (M . B)) < (e * +infty ) by A158, A160, XXREAL_3: 72;

 then

 A216: ( integral' (M,EP)) <> +infty by A214, XXREAL_0: 4;



 A217: 0 <= (M . B) by A141, A140, MEASURE1: 31, XBOOLE_1: 2;

 then

 reconsider XSMEP = ( integral' (M,EP)) as Element of REAL by A158, A214, A216, XXREAL_0: 14;



 A218: (F . n) is_simple_func_in S by A3;

 ((F . n) | A) is nonnegative by A5, Th15;

 then

 A219: 0 <= ( integral' (M,((F . n) | A))) by A218, Th34, Th68;

 (F . n) is nonnegative by A5;

 then ( integral' (M,(F . n))) = (( integral' (M,((F . n) | A))) + ( integral' (M,((F . n) | B)))) by A3, A186, A176, Th67;

 then

 A220: ( integral' (M,((F . n) | B))) <= ( integral' (M,(F . n))) by A219, XXREAL_3: 39;



 A221: (M . ( dom g)) < +infty by A136, XXREAL_0: 4;

 (M . B) <> -infty by A141, A140, MEASURE1: 31, XBOOLE_1: 2;

 then (M . B) in REAL by A221, A178, XXREAL_0: 14;

 then

 consider MB,MG be Real such that

 A222: MB = (M . B) and

 A223: MG = (M . ( dom g)) and

 A224: MB <= MG by A208, A179, MEASURE1: 31;



 A225: (g | B) is_simple_func_in S by A1, Th34;

 (ee * MB) <= (ee * MG) by A158, A224, XREAL_1: 64;

 then

 A226: ((XSMg - betae) - (ee * MG)) <= ((XSMg - betae) - (ee * MB)) by XREAL_1: 13;

 XSMEP = (e * (M . B)) by A158, A211, A212, A213, Th71

 .= (ee * MB) by A222;

 then

 A227: ((XSMg - betae) - (ee * MB)) < (XSMgB - XSMEP) by A210, XREAL_1: 14;

 betae = (ee * beta1) by EXTREAL1: 1;

 then

 A228: (XSMg - (ee * (beta1 + MG))) < (XSMgB - XSMEP) by A227, A226, XXREAL_0: 2;

 ( dom ((F . n) | B)) = (( dom (F . n)) /\ B) by RELAT_1: 61;

 then

 A229: ( dom ((F . n) | B)) = B by A163, A185, XBOOLE_1: 28;



 A230: ((F . n) | B) is_simple_func_in S by A3, Th34;

 then

 A231: (((F . n) | B) + EP) is_simple_func_in S by A211, Th38;



 A232: ((F . n) | B) is nonnegative by A5, Th15;

 then

 A233: ( dom (((F . n) | B) + EP)) = (( dom ((F . n) | B)) /\ ( dom EP)) by A230, A211, A215, Th65;



 A234: ( dom (((F . n) | B) + EP)) = (( dom ((F . n) | B)) /\ ( dom EP)) by A232, A230, A211, A215, Th65

 .= B by A229, A212;



 A235: ( dom (g | B)) = B by A180, RELAT_1: 62, XBOOLE_1: 7;



 A236: for x be object st x in ( dom ((((F . n) | B) + EP) - (g | B))) holds ((g | B) . x) <= ((((F . n) | B) + EP) . x)

 proof

 set f = (g - (F . n));

 let x be object;

 assume x in ( dom ((((F . n) | B) + EP) - (g | B)));

 then x in ((( dom (((F . n) | B) + EP)) /\ ( dom (g | B))) \ ((((((F . n) | B) + EP) " { +infty }) /\ ((g | B) " { +infty })) \/ (((((F . n) | B) + EP) " { -infty }) /\ ((g | B) " { -infty })))) by MESFUNC1:def 4;

 then

 A237: x in (( dom (((F . n) | B) + EP)) /\ ( dom (g | B))) by XBOOLE_0:def 5;

 then

 A238: x in ( dom (((F . n) | B) + EP)) by XBOOLE_0:def 4;

 then ((((F . n) | B) + EP) . x) = ((((F . n) | B) . x) + (EP . x)) by MESFUNC1:def 3;

 then ((((F . n) | B) + EP) . x) = (((F . n) . x) + (EP . x)) by A229, A234, A238, FUNCT_1: 47;

 then

 A239: ((((F . n) | B) + EP) . x) = (((F . n) . x) + e) by A213, A234, A238;



 A240: x in ( less_dom ((g - (F . n)),e)) by A161, A234, A238;

 then

 A241: (f . x) < e by MESFUNC1:def 11;

 x in ( dom f) by A240, MESFUNC1:def 11;

 then

 A242: ((g . x) - ((F . n) . x)) <= e by A241, MESFUNC1:def 4;

 ((g | B) . x) = (g . x) by A235, A234, A237, FUNCT_1: 47;

 hence thesis by A158, A160, A239, A242, XXREAL_3: 41;

 end;



 A243: (((F . n) | B) + EP) is nonnegative by A232, A215, Th19;

 then

 A244: ( dom ((((F . n) | B) + EP) - (g | B))) = (( dom (((F . n) | B) + EP)) /\ ( dom (g | B))) by A225, A184, A231, A236, Th69;

 then

 A245: (g | B) = ((g | B) | ( dom ((((F . n) | B) + EP) - (g | B)))) by A229, A212, A235, A233, GRFUNC_1: 23;

 (((F . n) | B) + EP) = ((((F . n) | B) + EP) | ( dom ((((F . n) | B) + EP) - (g | B)))) by A229, A212, A235, A244, A233, GRFUNC_1: 23;

 then

 A246: ( integral' (M,(g | B))) <= ( integral' (M,(((F . n) | B) + EP))) by A225, A184, A231, A243, A236, A245, Th70;

 ( integral' (M,(((F . n) | B) + EP))) = (( integral' (M,(((F . n) | B) | B))) + ( integral' (M,(EP | B)))) by A232, A230, A211, A215, A234, Th65

 .= (( integral' (M,(((F . n) | B) | B))) + ( integral' (M,EP))) by A212, GRFUNC_1: 23

 .= (( integral' (M,((F . n) | B))) + ( integral' (M,EP))) by FUNCT_1: 51;

 then

 A247: (( integral' (M,(g | B))) - ( integral' (M,EP))) <= ( integral' (M,((F . n) | B))) by A158, A214, A217, A216, A246, XXREAL_3: 42;

 (beta1 + MG) = (beta + (M . ( dom g))) by A223;

 then (ee * (beta1 + MG)) = (e * (beta + (M . ( dom g))));

 then

 A248: (XSMg - (ee * (beta1 + MG))) = (( integral' (M,g)) - (e * (beta + (M . ( dom g)))));

 (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < ( integral' (M,((F . n) | B))) by A247, A228, A248, XXREAL_0: 2;

 hence (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < ( integral' (M,(F . n))) by A220, XXREAL_0: 2;

 end;

 hence thesis;

 end;



 A249: for e be R_eal st 0 < e & e < alpha holds ex N0 be Nat st for n be Nat st N0 <= n holds (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < (L . n)

 proof

 let e be R_eal;

 assume that

 A250: 0 < e and

 A251: e < alpha;

 consider N0 be Nat such that

 A252: for n be Nat st N0 <= n holds (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < ( integral' (M,(F . n))) by A157, A250, A251;

 now

 let n be Nat;

 assume N0 <= n;

 then (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < ( integral' (M,(F . n))) by A252;

 hence (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < (L . n) by A8;

 end;

 hence thesis;

 end;



 A253: for e1 be R_eal st 0 < e1 holds ex e be R_eal st 0 < e & e < alpha & (e * (beta + (M . ( dom g)))) <= e1

 proof

 reconsider ralpha = alpha as Real;

 reconsider rdomg = (M . ( dom g)) as Element of REAL by A136, A142, XXREAL_0: 14;

 let e1 be R_eal;

 assume

 A254: 0 < e1;

 {} c= DG;

 then

 A255: 0 <= rdomg by A141, A140, MEASURE1: 31;

 per cases ;

 suppose

 A256: e1 = +infty ;

 reconsider z = (ralpha / 2) as R_eal;

 (z * (beta + (M . ( dom g)))) <= +infty by XXREAL_0: 4;

 hence thesis by A43, A256, XREAL_1: 216;

 end;

 suppose e1 <> +infty ;

 then

 reconsider re1 = e1 as Element of REAL by A254, XXREAL_0: 14;

 set x = (re1 / (beta1 + rdomg));

 set y = (ralpha / 2);

 set z = ( min (x,y));



 A257: z <= y by XXREAL_0: 17;

 y < ralpha by A43, XREAL_1: 216;

 then

 A258: z < ralpha by A257, XXREAL_0: 2;

 (beta1 + rdomg) = (beta1 + rdomg qua ExtReal);

 then

 A259: (z * (beta1 + rdomg)) = (z qua ExtReal * (beta + (M . ( dom g))));



 A260: (z * (beta1 + rdomg)) <= ((re1 / (beta1 + rdomg)) * (beta1 + rdomg)) by A137, A255, XREAL_1: 64, XXREAL_0: 17;

 reconsider z as R_eal;

 take z;

 A261:

 now

 per cases by XXREAL_0: 15;

 suppose ( min (x,y)) = x;

 hence 0 < z by A137, A254, A255;

 end;

 suppose ( min (x,y)) = y;

 hence 0 < z by A43;

 end;

 end;

 (z * (beta1 + rdomg)) <= re1 by A137, A255, XCMPLX_1: 87, A260;

 hence thesis by A261, A258, A259;

 end;

 end;



 A262: for e1 be R_eal st 0 < e1 holds ex N0 be Nat st for n be Nat st N0 <= n holds (( integral' (M,g)) - e1) < (L . n)

 proof

 let e1 be R_eal;

 assume 0 < e1;

 then

 consider e be R_eal such that

 A263: 0 < e and

 A264: e < alpha and

 A265: (e * (beta + (M . ( dom g)))) <= e1 by A253;

 consider N0 be Nat such that

 A266: for n be Nat st N0 <= n holds (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < (L . n) by A249, A263, A264;

 take N0;

 now

 let n be Nat;

 assume N0 <= n;

 then

 A267: (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) < (L . n) by A266;

 (( integral' (M,g)) - e1) <= (( integral' (M,g)) - (e * (beta + (M . ( dom g))))) by A265, XXREAL_3: 37;

 hence (( integral' (M,g)) - e1) < (L . n) by A267, XXREAL_0: 2;

 end;

 hence thesis;

 end;



 A268: for n be Nat holds 0 <= (L . n)

 proof

 let n be Nat;

 (F . n) is nonnegative by A5;

 then 0 <= ( integral' (M,(F . n))) by A3, Th68;

 hence thesis by A8;

 end;



 A269: for n,m be Nat st n <= m holds (L . n) <= (L . m)

 proof

 let n,m be Nat;



 A270: ( dom (F . n)) = ( dom g) by A4;



 A271: (F . m) is_simple_func_in S by A3;



 A272: ( dom (F . m)) = ( dom g) by A4;

 assume

 A273: n <= m;



 A274: for x be object st x in ( dom ((F . m) - (F . n))) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let x be object;

 assume x in ( dom ((F . m) - (F . n)));

 then x in ((( dom (F . m)) /\ ( dom (F . n))) \ ((((F . m) " { +infty }) /\ ((F . n) " { +infty })) \/ (((F . m) " { -infty }) /\ ((F . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F . m)) /\ ( dom (F . n))) by XBOOLE_0:def 5;

 hence thesis by A6, A273, A270, A272;

 end;



 A275: (F . n) is_simple_func_in S by A3;



 A276: (F . m) is nonnegative by A5;



 A277: (F . n) is nonnegative by A5;

 then

 A278: ( dom ((F . m) - (F . n))) = (( dom (F . m)) /\ ( dom (F . n))) by A276, A275, A271, A274, Th69;

 then

 A279: ((F . m) | ( dom ((F . m) - (F . n)))) = (F . m) by A270, A272, GRFUNC_1: 23;



 A280: ((F . n) | ( dom ((F . m) - (F . n)))) = (F . n) by A270, A272, A278, GRFUNC_1: 23;

 ( integral' (M,((F . n) | ( dom ((F . m) - (F . n)))))) <= ( integral' (M,((F . m) | ( dom ((F . m) - (F . n)))))) by A277, A276, A275, A271, A274, Th70;

 then (L . n) <= ( integral' (M,(F . m))) by A8, A279, A280;

 hence thesis by A8;

 end;

 per cases ;

 suppose ex K be Real st 0 < K & for n be Nat holds (L . n) < K;

 then

 consider K be Real such that 0 < K and

 A281: for n be Nat holds (L . n) < K;

 now

 let x be ExtReal;

 assume x in ( rng L);

 then ex z be object st z in ( dom L) & x = (L . z) by FUNCT_1:def 3;

 hence x <= K by A281;

 end;

 then K is UpperBound of ( rng L) by XXREAL_2:def 1;

 then

 A282: ( sup ( rng L)) <= K by XXREAL_2:def 3;

 K in REAL by XREAL_0:def 1;

 then

 A283: ( sup ( rng L)) <> +infty by A282, XXREAL_0: 9;



 A284: for n be Nat holds (L . n) <= ( sup ( rng L))

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 ( dom L) = NAT by FUNCT_2:def 1;

 then

 A285: (L . n) in ( rng L) by FUNCT_1:def 3;

 ( sup ( rng L)) is UpperBound of ( rng L) by XXREAL_2:def 3;

 hence thesis by A285, XXREAL_2:def 1;

 end;

 then (L . 1) <= ( sup ( rng L));

 then

 A286: ( sup ( rng L)) <> -infty by A268;

 then

 reconsider h = ( sup ( rng L)) as Element of REAL by A283, XXREAL_0: 14;



 A287: for p be Real st 0 < p holds ex N0 be Nat st for m be Nat st N0 <= m holds |.((L . m) - ( sup ( rng L))).| (( sup ( rng L)) + p)

 proof

 assume

 A290: ( sup ( rng L)) = (( sup ( rng L)) + p qua ExtReal);

 ((p + ( sup ( rng L))) + ( - ( sup ( rng L)))) = (p + (( sup ( rng L)) + ( - ( sup ( rng L))))) by A286, A283, XXREAL_3: 29

 .= (p + 0 ) by XXREAL_3: 7

 .= p;

 hence contradiction by A288, A290, XXREAL_3: 7;

 end;

 ( sup ( rng L)) in REAL by A286, A283, XXREAL_0: 14;

 then

 consider y be ExtReal such that

 A291: y in ( rng L) and

 A292: (( sup ( rng L)) - p) < y by A288, MEASURE6: 6;

 consider x be object such that

 A293: x in ( dom L) and

 A294: y = (L . x) by A291, FUNCT_1:def 3;

 reconsider N0 = x as Element of NAT by A293;

 take N0;

 let n be Nat;

 assume N0 <= n;

 then (L . N0) <= (L . n) by A269;

 then (( sup ( rng L)) - p) < (L . n) by A292, A294, XXREAL_0: 2;

 then ( sup ( rng L)) < ((L . n) + p) by XXREAL_3: 54;

 then (( sup ( rng L)) - (L . n)) < p by XXREAL_3: 55;

 then ( - p) < ( - (( sup ( rng L)) - (L . n))) by XXREAL_3: 38;

 then

 A295: ( - p) < ((L . n) - ( sup ( rng L))) by XXREAL_3: 26;



 A296: (L . n) <= ( sup ( rng L)) by A284;

 (( sup ( rng L)) + 0. ) <= (( sup ( rng L)) + p) by A288, XXREAL_3: 36;

 then ( sup ( rng L)) <= (( sup ( rng L)) + p) by XXREAL_3: 4;

 then ( sup ( rng L)) < (( sup ( rng L)) + p) by A289, XXREAL_0: 1;

 then (L . n) < (( sup ( rng L)) + p) by A296, XXREAL_0: 2;

 then ((L . n) - ( sup ( rng L))) < p by XXREAL_3: 55;

 hence thesis by A295, EXTREAL1: 22;

 end;



 A297: h = ( sup ( rng L));

 then

 A298: L is convergent_to_finite_number by A287;

 hence L is convergent;

 then

 A299: ( lim L) = ( sup ( rng L)) by A287, A297, A298, Def12;

 now

 let e be Real;

 assume

 A300: 0 < e;

 reconsider ee = e as R_eal by XXREAL_0:def 1;

 consider N0 be Nat such that

 A301: for n be Nat st N0 <= n holds (( integral' (M,g)) - ee) < (L . n) by A262, A300;



 A302: (L . N0) <= ( sup ( rng L)) by A284;

 (( integral' (M,g)) - ee) < (L . N0) by A301;

 then (( integral' (M,g)) - ee) < ( sup ( rng L)) by A302, XXREAL_0: 2;

 hence ( integral' (M,g)) < (( lim L) + e) by A299, XXREAL_3: 54;

 end;

 hence thesis by XXREAL_3: 61;

 end;

 suppose

 A303: not (ex K be Real st 0 < K & for n be Nat holds (L . n) < K);

 now

 let K be Real;

 assume 0 < K;

 then

 consider N0 be Nat such that

 A304: K <= (L . N0) by A303;

 now

 let n be Nat;

 assume N0 <= n;

 then (L . N0) <= (L . n) by A269;

 hence K <= (L . n) by A304, XXREAL_0: 2;

 end;

 hence ex N0 be Nat st for n be Nat st N0 <= n holds K <= (L . n);

 end;

 then

 A305: L is convergent_to_+infty;

 hence L is convergent;

 then ( lim L) = +infty by A305, Def12;

 hence thesis by XXREAL_0: 4;

 end;

 end;

 suppose

 A306: (M . ( dom g)) = +infty ;

 reconsider DG = ( dom g) as Element of S by A1, Th37;



 A307: for e be R_eal st 0 < e & e < alpha holds for n be Nat holds ((alpha - e) * (M . ( less_dom ((g - (F . n)),e)))) <= ( integral' (M,(F . n)))

 proof

 let e be R_eal;

 assume that

 A308: 0 < e and

 A309: e < alpha;



 A310: 0 <= (alpha - e) by A309, XXREAL_3: 40;

 consider H be SetSequence of X, MF be ExtREAL_sequence such that

 A311: for n be Nat holds (H . n) = ( less_dom ((g - (F . n)),e)) and for n,m be Nat st n <= m holds (H . n) c= (H . m) and

 A312: for n be Nat holds (H . n) c= ( dom g) and for n be Nat holds (MF . n) = (M . (H . n)) and (M . ( dom g)) = ( sup ( rng MF)) and

 A313: for n be Nat holds (H . n) in S by A61, A308, A309;



 A314: e <> +infty by A309, XXREAL_0: 4;

 now

 let n be Nat;

 reconsider B = (H . n) as Element of S by A313;



 A315: for x be object st x in ( dom (F . n)) holds ((F . n) . x) = ((F . n) . x);

 (H . n) in S by A313;

 then

 A316: (X \ (H . n)) in S by MEASURE1: 34;

 (DG /\ (X \ (H . n))) = ((DG /\ X) \ (H . n)) by XBOOLE_1: 49

 .= (DG \ (H . n)) by XBOOLE_1: 28;

 then

 reconsider A = (DG \ (H . n)) as Element of S by A316, MEASURE1: 34;



 A317: ( dom (F . n)) = ( dom g) by A4;



 A318: DG = (DG \/ (H . n)) by A312, XBOOLE_1: 12

 .= ((DG \ (H . n)) \/ (H . n)) by XBOOLE_1: 39;

 then ( dom (F . n)) = ((A \/ B) /\ ( dom (F . n))) by A317;

 then

 A319: (F . n) = ((F . n) | (A \/ B)) by A315, FUNCT_1: 46;

 consider EP be PartFunc of X, ExtREAL such that

 A320: EP is_simple_func_in S and

 A321: ( dom EP) = B and

 A322: for x be object st x in B holds (EP . x) = (alpha - e) by A308, A310, Th41, XXREAL_3: 18;

 for x be object st x in ( dom EP) holds 0 <= (EP . x) by A310, A321, A322;

 then

 A323: EP is nonnegative by SUPINF_2: 52;



 A324: ( dom ((F . n) | B)) = (( dom (F . n)) /\ B) by RELAT_1: 61

 .= B by A318, A317, XBOOLE_1: 7, XBOOLE_1: 28;



 A325: for x be object st x in ( dom (((F . n) | B) - EP)) holds (EP . x) <= (((F . n) | B) . x)

 proof

 set f = (g - (F . n));

 let x be object;

 assume x in ( dom (((F . n) | B) - EP));

 then x in ((( dom ((F . n) | B)) /\ ( dom EP)) \ (((((F . n) | B) " { +infty }) /\ (EP " { +infty })) \/ ((((F . n) | B) " { -infty }) /\ (EP " { -infty })))) by MESFUNC1:def 4;

 then

 A326: x in (( dom ((F . n) | B)) /\ ( dom EP)) by XBOOLE_0:def 5;

 then

 A327: x in ( dom ((F . n) | B)) by XBOOLE_0:def 4;

 then

 A328: (((F . n) | B) . x) = ((F . n) . x) by FUNCT_1: 47;



 A329: x in ( less_dom ((g - (F . n)),e)) by A311, A324, A327;

 then

 A330: x in ( dom f) by MESFUNC1:def 11;

 (f . x) < e by A329, MESFUNC1:def 11;

 then ((g . x) - ((F . n) . x)) <= e by A330, MESFUNC1:def 4;

 then (g . x) <= (((F . n) . x) + e) by A308, A314, XXREAL_3: 41;

 then

 A331: ((g . x) - e) <= ((F . n) . x) by A308, A314, XXREAL_3: 42;

 ( dom f) = ( dom g) by A56;

 then alpha <= (g . x) by A44, A330;

 then (alpha - e) <= ((g . x) - e) by XXREAL_3: 37;

 then (alpha - e) <= ((F . n) . x) by A331, XXREAL_0: 2;

 hence thesis by A324, A321, A322, A326, A328;

 end;



 A332: (F . n) is_simple_func_in S by A3;

 ((F . n) | A) is nonnegative by A5, Th15;

 then

 A333: 0 <= ( integral' (M,((F . n) | A))) by A332, Th34, Th68;



 A334: A misses B by XBOOLE_1: 79;

 (F . n) is nonnegative by A5;

 then ( integral' (M,(F . n))) = (( integral' (M,((F . n) | A))) + ( integral' (M,((F . n) | B)))) by A3, A319, A334, Th67;

 then

 A335: ( integral' (M,((F . n) | B))) <= ( integral' (M,(F . n))) by A333, XXREAL_3: 39;



 A336: ((F . n) | B) is_simple_func_in S by A3, Th34;



 A337: ((F . n) | B) is nonnegative by A5, Th15;

 then

 A338: ( dom (((F . n) | B) - EP)) = (( dom ((F . n) | B)) /\ ( dom EP)) by A336, A320, A323, A325, Th69;

 then

 A339: (EP | ( dom (((F . n) | B) - EP))) = EP by A324, A321, GRFUNC_1: 23;



 A340: (((F . n) | B) | ( dom (((F . n) | B) - EP))) = ((F . n) | B) by A324, A321, A338, GRFUNC_1: 23;

 ( integral' (M,(EP | ( dom (((F . n) | B) - EP))))) <= ( integral' (M,(((F . n) | B) | ( dom (((F . n) | B) - EP))))) by A337, A336, A320, A323, A325, Th70;

 then

 A341: ( integral' (M,EP)) <= ( integral' (M,(F . n))) by A335, A339, A340, XXREAL_0: 2;

 ( integral' (M,EP)) = ((alpha - e) * (M . B)) by A309, A320, A321, A322, Th71, XXREAL_3: 40;

 hence ((alpha - e) * (M . ( less_dom ((g - (F . n)),e)))) <= ( integral' (M,(F . n))) by A311, A341;

 end;

 hence thesis;

 end;

 for y be Real st 0 < y holds ex n be Nat st for m be Nat st n <= m holds y <= (L . m)

 proof

 reconsider ralpha = alpha as Real;

 reconsider e = (alpha / 2) as R_eal;

 let y be Real;

 assume 0 < y;

 set a2 = (ralpha / 2);

 reconsider y1 = y as Real;

 y = ((ralpha - a2) * (y1 / (ralpha - a2))) by A43, XCMPLX_1: 87;

 then

 A342: y = ((ralpha - a2) * (y1 / (ralpha - a2)));



 A343: e = a2;

 then

 consider H be SetSequence of X, MF be ExtREAL_sequence such that

 A344: for n be Nat holds (H . n) = ( less_dom ((g - (F . n)),e)) and

 A345: for n,m be Nat st n <= m holds (H . n) c= (H . m) and for n be Nat holds (H . n) c= ( dom g) and

 A346: for n be Nat holds (MF . n) = (M . (H . n)) and

 A347: (M . ( dom g)) = ( sup ( rng MF)) and

 A348: for n be Nat holds (H . n) in S by A61, A43, XREAL_1: 216;



 A349: (y / (ralpha - a2)) in REAL by XREAL_0:def 1;



 A350: (y / (alpha - e)) < +infty by XXREAL_0: 9, A349;

 ex z be ExtReal st z in ( rng MF) & (y / (alpha - e)) <= z

 proof

 assume not (ex z be ExtReal st z in ( rng MF) & (y / (alpha - e)) <= z);

 then for z be ExtReal st z in ( rng MF) holds z <= (y / (alpha - e));

 then (y / (alpha - e)) is UpperBound of ( rng MF) by XXREAL_2:def 1;

 hence contradiction by A306, A350, A347, XXREAL_2:def 3;

 end;

 then

 consider z be R_eal such that

 A351: z in ( rng MF) and

 A352: (y / (alpha - e)) <= z;

 (a2 - a2) < (ralpha - a2) by A43;

 then

 A353: 0 < (alpha - e);

 consider x be object such that

 A354: x in ( dom MF) and

 A355: z = (MF . x) by A351, FUNCT_1:def 3;

 reconsider N0 = x as Element of NAT by A354;

 take N0;



 A356: ((alpha - e) * (y / (alpha - e))) = y by A342;

 thus for m be Nat st N0 <= m holds y <= (L . m)

 proof

 (y / (alpha - e)) <= (M . (H . N0)) by A346, A352, A355;

 then

 A357: y <= ((alpha - e) * (M . (H . N0))) by A353, A356, XXREAL_3: 71;

 let m be Nat;



 A358: (H . m) in S by A348;

 assume N0 <= m;

 then

 A359: (H . N0) c= (H . m) by A345;

 (H . N0) in S by A348;

 then ((alpha - e) * (M . (H . N0))) <= ((alpha - e) * (M . (H . m))) by A353, A359, A358, MEASURE1: 31, XXREAL_3: 71;

 then y <= ((alpha - e) * (M . (H . m))) by A357, XXREAL_0: 2;

 then

 A360: y <= ((alpha - e) * (M . ( less_dom ((g - (F . m)),e)))) by A344;

 ((alpha - e) * (M . ( less_dom ((g - (F . m)),e)))) <= ( integral' (M,(F . m))) by A43, A307, A343, XREAL_1: 216;

 then y <= ( integral' (M,(F . m))) by A360, XXREAL_0: 2;

 hence thesis by A8;

 end;

 end;

 then

 A361: L is convergent_to_+infty;

 hence L is convergent;

 then (ex g be Real st ( lim L) = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((L . m) - ( lim L)).| < p) & L is convergent_to_finite_number) or ( lim L) = +infty & L is convergent_to_+infty or ( lim L) = -infty & L is convergent_to_-infty by Def12;

 hence thesis by A361, Th50, XXREAL_0: 4;

 end;

 end;

 end;

 theorem :: MESFUNC5:75



 Th75: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, g be PartFunc of X, ExtREAL , F be Functional_Sequence of X, ExtREAL st g is_simple_func_in S & g is nonnegative & (for n be Nat holds (F . n) is_simple_func_in S) & (for n be Nat holds ( dom (F . n)) = ( dom g)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((F . n) . x) <= ((F . m) . x)) & (for x be Element of X st x in ( dom g) holds (F # x) is convergent & (g . x) <= ( lim (F # x))) holds ex G be ExtREAL_sequence st (for n be Nat holds (G . n) = ( integral' (M,(F . n)))) & G is convergent & ( sup ( rng G)) = ( lim G) & ( integral' (M,g)) <= ( lim G)

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let g be PartFunc of X, ExtREAL ;

 let F be Functional_Sequence of X, ExtREAL ;

 assume that

 A1: g is_simple_func_in S and

 A2: g is nonnegative and

 A3: for n be Nat holds (F . n) is_simple_func_in S and

 A4: for n be Nat holds ( dom (F . n)) = ( dom g) and

 A5: for n be Nat holds (F . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((F . n) . x) <= ((F . m) . x) and

 A7: for x be Element of X st x in ( dom g) holds (F # x) is convergent & (g . x) <= ( lim (F # x));

 set E0 = ( eq_dom (g, 0 ));

 reconsider DG = ( dom g) as Element of S by A1, Th37;

 g is DG -measurable by A1, MESFUNC2: 34;

 then

 reconsider GG = (DG /\ ( great_eq_dom (g, 0 ))) as Element of S by MESFUNC1: 27;

 for x be object st x in E0 holds x in ( dom g) by MESFUNC1:def 15;

 then

 A8: E0 c= DG;



 then

 A9: DG = (DG \/ E0) by XBOOLE_1: 12

 .= ((DG \ E0) \/ E0) by XBOOLE_1: 39;

 set E9 = (( dom g) \ E0);

 g is GG -measurable by A1, MESFUNC2: 34;

 then (GG /\ ( less_eq_dom (g, 0 ))) in S by MESFUNC1: 28;

 then

 A10: (DG /\ ( eq_dom (g, 0 ))) in S by MESFUNC1: 18;

 then E0 in S by A8, XBOOLE_1: 28;

 then

 A11: (X \ E0) in S by MEASURE1: 34;

 (DG /\ (X \ E0)) = ((DG /\ X) \ E0) by XBOOLE_1: 49

 .= (DG \ E0) by XBOOLE_1: 28;

 then

 reconsider E9 as Element of S by A11, MEASURE1: 34;

 reconsider E0 as Element of S by A8, A10, XBOOLE_1: 28;



 A12: E0 misses E9 by XBOOLE_1: 79;

 thus ex G be ExtREAL_sequence st (for n be Nat holds (G . n) = ( integral' (M,(F . n)))) & G is convergent & ( sup ( rng G)) = ( lim G) & ( integral' (M,g)) <= ( lim G)

 proof



 A13: ( dom (g | E9)) = (( dom g) /\ E9) by RELAT_1: 61

 .= E9 by A9, XBOOLE_1: 7, XBOOLE_1: 28;



 A14: for x be object st x in ( dom (g | E9)) holds 0 < ((g | E9) . x)

 proof

 let x be object;

 assume

 A15: x in ( dom (g | E9));

 then

 A16: not x in E0 by A13, XBOOLE_0:def 5;

 x in DG by A13, A15, XBOOLE_0:def 5;

 then (g . x) <> 0 by A16, MESFUNC1:def 15;

 then 0 < (g . x) by A2, SUPINF_2: 51;

 hence thesis by A15, FUNCT_1: 47;

 end;

 deffunc V( Nat) = ( integral' (M,((F . $1) | E9)));

 deffunc U( Nat) = ( integral' (M,(F . $1)));

 deffunc W( Nat) = ((F . $1) | E9);

 consider F9 be Functional_Sequence of X, ExtREAL such that

 A17: for n be Nat holds (F9 . n) = W(n) from SEQFUNC:sch 1;

 consider L be ExtREAL_sequence such that

 A18: for n be Element of NAT holds (L . n) = V(n) from FUNCT_2:sch 4;

 A19:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (L . n) = V(n) by A18;

 end;



 A20: for n be Nat holds (L . n) = ( integral' (M,(F9 . n)))

 proof

 let n be Nat;



 thus (L . n) = ( integral' (M,((F . n) | E9))) by A19

 .= ( integral' (M,(F9 . n))) by A17;

 end;

 consider G be ExtREAL_sequence such that

 A21: for n be Element of NAT holds (G . n) = U(n) from FUNCT_2:sch 4;

 take G;



 A22: for x be object st x in ( dom g) holds (g . x) = (g . x);

 ( dom g) = ((E0 \/ E9) /\ ( dom g)) by A9;

 then (g | (E0 \/ E9)) = g by A22, FUNCT_1: 46;

 then

 A23: ( integral' (M,g)) = (( integral' (M,(g | E0))) + ( integral' (M,(g | E9)))) by A1, A2, Th67, XBOOLE_1: 79;

 ( integral' (M,(g | E0))) = 0 by A1, A2, Th72;

 then

 A24: ( integral' (M,g)) = ( integral' (M,(g | E9))) by A23, XXREAL_3: 4;



 A25: (g | E9) is_simple_func_in S by A1, Th34;



 A26: for n be Nat holds ((F . n) | E9) is_simple_func_in S & (F9 . n) is_simple_func_in S

 proof

 let n be Nat;

 thus ((F . n) | E9) is_simple_func_in S by A3, Th34;

 hence thesis by A17;

 end;



 A27: for n be Nat holds ( dom ((F . n) | E9)) = ( dom (g | E9)) & ( dom (F9 . n)) = ( dom (g | E9))

 proof

 let n be Nat;



 A28: ( dom (F . n)) = (E9 \/ E0) by A4, A9;



 thus ( dom ((F . n) | E9)) = (( dom (F . n)) /\ E9) by RELAT_1: 61

 .= ( dom (g | E9)) by A13, A28, XBOOLE_1: 7, XBOOLE_1: 28;

 hence thesis by A17;

 end;



 A29: for x be Element of X st x in ( dom (g | E9)) holds (F9 # x) is convergent & ((g | E9) . x) <= ( lim (F9 # x))

 proof

 let x be Element of X;

 assume

 A30: x in ( dom (g | E9));

 now

 let n be Element of NAT ;



 A31: x in ( dom ((F . n) | E9)) by A27, A30;



 thus ((F9 # x) . n) = ((F9 . n) . x) by Def13

 .= (((F . n) | E9) . x) by A17

 .= ((F . n) . x) by A31, FUNCT_1: 47

 .= ((F # x) . n) by Def13;

 end;

 then

 A32: (F9 # x) = (F # x) by FUNCT_2: 63;

 x in (( dom g) /\ E9) by A30, RELAT_1: 61;

 then

 A33: x in ( dom g) by XBOOLE_0:def 4;

 then (g . x) <= ( lim (F # x)) by A7;

 hence thesis by A7, A30, A33, A32, FUNCT_1: 47;

 end;



 A34: for n be Nat holds (F9 . n) is nonnegative

 proof

 let n be Nat;

 ((F . n) | E9) is nonnegative by A5, Th15;

 hence thesis by A17;

 end;



 A35: E9 c= ( dom g) by A9, XBOOLE_1: 7;



 A36: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (g | E9)) holds (((F . n) | E9) . x) <= (((F . m) | E9) . x) & ((F9 . n) . x) <= ((F9 . m) . x)

 proof

 let n,m be Nat;

 assume

 A37: n <= m;

 thus for x be Element of X st x in ( dom (g | E9)) holds (((F . n) | E9) . x) <= (((F . m) | E9) . x) & ((F9 . n) . x) <= ((F9 . m) . x)

 proof

 let x be Element of X;

 assume

 A38: x in ( dom (g | E9));

 then

 A39: x in ( dom ((F . n) | E9)) by A27;

 ((F . n) . x) <= ((F . m) . x) by A6, A35, A13, A37, A38;

 then

 A40: (((F . n) | E9) . x) <= ((F . m) . x) by A39, FUNCT_1: 47;

 x in ( dom ((F . m) | E9)) by A27, A38;

 hence (((F . n) | E9) . x) <= (((F . m) | E9) . x) by A40, FUNCT_1: 47;

 then ((F9 . n) . x) <= (((F . m) | E9) . x) by A17;

 hence thesis by A17;

 end;

 end;

 then for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (g | E9)) holds ((F9 . n) . x) <= ((F9 . m) . x);

 then

 A41: ( integral' (M,(g | E9))) <= ( lim L) by A25, A14, A27, A26, A29, A34, A20, Th74;

 for n,m be Nat st n <= m holds (L . n) <= (L . m)

 proof

 let n,m be Nat;



 A42: (F9 . m) is_simple_func_in S by A26;



 A43: ( dom (F9 . m)) = ( dom (g | E9)) by A27;



 A44: (L . m) = ( integral' (M,(F9 . m))) by A20;



 A45: (L . n) = ( integral' (M,(F9 . n))) by A20;



 A46: ( dom (F9 . n)) = ( dom (g | E9)) by A27;

 assume

 A47: n <= m;



 A48: for x be object st x in ( dom ((F9 . m) - (F9 . n))) holds ((F9 . n) . x) <= ((F9 . m) . x)

 proof

 let x be object;

 assume x in ( dom ((F9 . m) - (F9 . n)));

 then x in ((( dom (F9 . m)) /\ ( dom (F9 . n))) \ ((((F9 . m) " { +infty }) /\ ((F9 . n) " { +infty })) \/ (((F9 . m) " { -infty }) /\ ((F9 . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F9 . m)) /\ ( dom (F9 . n))) by XBOOLE_0:def 5;

 hence thesis by A36, A47, A46, A43;

 end;



 A49: (F9 . m) is nonnegative by A34;



 A50: (F9 . n) is nonnegative by A34;



 A51: (F9 . n) is_simple_func_in S by A26;

 then

 A52: ( dom ((F9 . m) - (F9 . n))) = (( dom (F9 . m)) /\ ( dom (F9 . n))) by A42, A50, A49, A48, Th69;

 then

 A53: ((F9 . m) | ( dom ((F9 . m) - (F9 . n)))) = (F9 . m) by A46, A43, GRFUNC_1: 23;

 ((F9 . n) | ( dom ((F9 . m) - (F9 . n)))) = (F9 . n) by A46, A43, A52, GRFUNC_1: 23;

 hence thesis by A51, A42, A50, A49, A48, A53, A45, A44, Th70;

 end;

 then

 A54: ( lim L) = ( sup ( rng L)) by Th54;

 A55:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (G . n) = U(n) by A21;

 end;

 for n be Nat holds (L . n) <= (G . n)

 proof

 let n be Nat;



 A56: (F . n) is_simple_func_in S by A3;

 ( dom (F . n)) = (E9 \/ E0) by A4, A9;

 then

 A57: ( dom (F . n)) = ((E0 \/ E9) /\ ( dom (F . n)));

 for x be object st x in ( dom (F . n)) holds ((F . n) . x) = ((F . n) . x);

 then

 A58: (F . n) = ((F . n) | (E0 \/ E9)) by A57, FUNCT_1: 46;

 then ((F . n) | (E0 \/ E9)) is nonnegative by A5;

 then

 A59: ( integral' (M,(F . n))) = (( integral' (M,((F . n) | E0))) + ( integral' (M,((F . n) | E9)))) by A3, A12, A58, Th67;

 ((F . n) | E0) is nonnegative by A5, Th15;

 then 0 <= ( integral' (M,((F . n) | E0))) by A56, Th34, Th68;

 then

 A60: ( integral' (M,((F . n) | E9))) <= ( integral' (M,(F . n))) by A59, XXREAL_3: 39;

 (G . n) = ( integral' (M,(F . n))) by A55;

 hence thesis by A19, A60;

 end;

 then

 A61: ( sup ( rng L)) <= ( sup ( rng G)) by Th55;



 A62: for n,m be Nat st n <= m holds (G . n) <= (G . m)

 proof

 let n,m be Nat;



 A63: (F . m) is_simple_func_in S by A3;



 A64: ( dom (F . m)) = ( dom g) by A4;



 A65: (G . m) = ( integral' (M,(F . m))) by A55;



 A66: (G . n) = ( integral' (M,(F . n))) by A55;



 A67: ( dom (F . n)) = ( dom g) by A4;

 assume

 A68: n <= m;



 A69: for x be object st x in ( dom ((F . m) - (F . n))) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let x be object;

 assume x in ( dom ((F . m) - (F . n)));

 then x in ((( dom (F . m)) /\ ( dom (F . n))) \ ((((F . m) " { +infty }) /\ ((F . n) " { +infty })) \/ (((F . m) " { -infty }) /\ ((F . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F . m)) /\ ( dom (F . n))) by XBOOLE_0:def 5;

 hence thesis by A6, A68, A67, A64;

 end;



 A70: (F . m) is nonnegative by A5;



 A71: (F . n) is nonnegative by A5;



 A72: (F . n) is_simple_func_in S by A3;

 then

 A73: ( dom ((F . m) - (F . n))) = (( dom (F . m)) /\ ( dom (F . n))) by A63, A71, A70, A69, Th69;

 then

 A74: ((F . m) | ( dom ((F . m) - (F . n)))) = (F . m) by A67, A64, GRFUNC_1: 23;

 ((F . n) | ( dom ((F . m) - (F . n)))) = (F . n) by A67, A64, A73, GRFUNC_1: 23;

 hence thesis by A72, A63, A71, A70, A69, A74, A66, A65, Th70;

 end;

 then ( lim G) = ( sup ( rng G)) by Th54;

 hence thesis by A24, A55, A62, A41, A54, A61, Th54, XXREAL_0: 2;

 end;

 end;

 theorem :: MESFUNC5:76



 Th76: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, F,G be Functional_Sequence of X, ExtREAL , K,L be ExtREAL_sequence st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = A) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in A holds ((F . n) . x) <= ((F . m) . x)) & (for n be Nat holds (G . n) is_simple_func_in S & ( dom (G . n)) = A) & (for n be Nat holds (G . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in A holds ((G . n) . x) <= ((G . m) . x)) & (for x be Element of X st x in A holds (F # x) is convergent & (G # x) is convergent & ( lim (F # x)) = ( lim (G # x))) & (for n be Nat holds (K . n) = ( integral' (M,(F . n))) & (L . n) = ( integral' (M,(G . n)))) holds K is convergent & L is convergent & ( lim K) = ( lim L)

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, F,G be Functional_Sequence of X, ExtREAL , K,L be ExtREAL_sequence such that

 A1: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = A and

 A2: for n be Nat holds (F . n) is nonnegative and

 A3: for n,m be Nat st n <= m holds for x be Element of X st x in A holds ((F . n) . x) <= ((F . m) . x) and

 A4: for n be Nat holds (G . n) is_simple_func_in S & ( dom (G . n)) = A and

 A5: for n be Nat holds (G . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in A holds ((G . n) . x) <= ((G . m) . x) and

 A7: for x be Element of X st x in A holds (F # x) is convergent & (G # x) is convergent & ( lim (F # x)) = ( lim (G # x)) and

 A8: for n be Nat holds (K . n) = ( integral' (M,(F . n))) & (L . n) = ( integral' (M,(G . n)));



 A9: for n0 be Nat holds L is convergent & ( sup ( rng L)) = ( lim L) & (K . n0) <= ( lim L)

 proof

 let n0 be Nat;

 reconsider f = (F . n0) as PartFunc of X, ExtREAL ;



 A10: f is_simple_func_in S by A1;



 A11: f is nonnegative by A2;



 A12: for x be Element of X st x in ( dom f) holds (G # x) is convergent & (f . x) <= ( lim (G # x))

 proof

 let x be Element of X;



 A13: ((F # x) . n0) <= ( sup ( rng (F # x))) by Th56;

 assume x in ( dom f);

 then

 A14: x in A by A1;

 now

 let n,m be Nat;

 assume

 A15: n <= m;



 A16: ((F # x) . m) = ((F . m) . x) by Def13;

 ((F # x) . n) = ((F . n) . x) by Def13;

 hence ((F # x) . n) <= ((F # x) . m) by A3, A14, A15, A16;

 end;

 then

 A17: ( lim (F # x)) = ( sup ( rng (F # x))) by Th54;

 (f . x) = ((F # x) . n0) by Def13;

 hence thesis by A7, A14, A17, A13;

 end;

 ( dom f) = A by A1;

 then

 consider FF be ExtREAL_sequence such that

 A18: for n be Nat holds (FF . n) = ( integral' (M,(G . n))) and

 A19: FF is convergent and

 A20: ( sup ( rng FF)) = ( lim FF) and

 A21: ( integral' (M,f)) <= ( lim FF) by A4, A5, A6, A12, A10, A11, Th75;

 now

 let n be Element of NAT ;

 (FF . n) = ( integral' (M,(G . n))) by A18;

 hence (FF . n) = (L . n) by A8;

 end;

 then FF = L by FUNCT_2: 63;

 hence thesis by A8, A19, A20, A21;

 end;



 A22: for n0 be Nat holds K is convergent & ( sup ( rng K)) = ( lim K) & (L . n0) <= ( lim K)

 proof

 let n0 be Nat;

 reconsider g = (G . n0) as PartFunc of X, ExtREAL ;



 A23: g is_simple_func_in S by A4;



 A24: g is nonnegative by A5;



 A25: for x be Element of X st x in ( dom g) holds (F # x) is convergent & (g . x) <= ( lim (F # x))

 proof

 let x be Element of X;



 A26: ((G # x) . n0) <= ( sup ( rng (G # x))) by Th56;

 assume x in ( dom g);

 then

 A27: x in A by A4;

 now

 let n,m be Nat;

 assume

 A28: n <= m;



 A29: ((G # x) . m) = ((G . m) . x) by Def13;

 ((G # x) . n) = ((G . n) . x) by Def13;

 hence ((G # x) . n) <= ((G # x) . m) by A6, A27, A28, A29;

 end;

 then

 A30: ( lim (G # x)) = ( sup ( rng (G # x))) by Th54;

 (g . x) = ((G # x) . n0) by Def13;

 hence thesis by A7, A27, A30, A26;

 end;

 ( dom g) = A by A4;

 then

 consider GG be ExtREAL_sequence such that

 A31: for n be Nat holds (GG . n) = ( integral' (M,(F . n))) and

 A32: GG is convergent and

 A33: ( sup ( rng GG)) = ( lim GG) and

 A34: ( integral' (M,g)) <= ( lim GG) by A1, A2, A3, A25, A23, A24, Th75;

 now

 let n be Element of NAT ;

 (GG . n) = ( integral' (M,(F . n))) by A31;

 hence (GG . n) = (K . n) by A8;

 end;

 then GG = K by FUNCT_2: 63;

 hence thesis by A8, A32, A33, A34;

 end;

 hence K is convergent & L is convergent by A9;



 A35: ( lim K) <= ( lim L) by A22, A9, Th57;

 ( lim L) <= ( lim K) by A22, A9, Th57;

 hence thesis by A35, XXREAL_0: 1;

 end;

 definition

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: f is nonnegative;

 :: MESFUNC5:def15

 func integral+ (M,f) -> Element of ExtREAL means

 : Def15: ex F be Functional_Sequence of X, ExtREAL , K be ExtREAL_sequence st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x)) & (for x be Element of X st x in ( dom f) holds (F # x) is convergent & ( lim (F # x)) = (f . x)) & (for n be Nat holds (K . n) = ( integral' (M,(F . n)))) & K is convergent & it = ( lim K);

 existence

 proof

 consider F be Functional_Sequence of X, ExtREAL such that

 A3: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

 A4: for n be Nat holds (F . n) is nonnegative and

 A5: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x) and

 A6: for x be Element of X st x in ( dom f) holds (F # x) is convergent & ( lim (F # x)) = (f . x) by A1, A2, Th64;

 reconsider g = (F . 0 ) as PartFunc of X, ExtREAL ;



 A7: g is_simple_func_in S by A3;



 A8: for x be Element of X st x in ( dom f) holds (F # x) is convergent & (g . x) <= ( lim (F # x))

 proof

 let x be Element of X such that

 A9: x in ( dom f);

 A10:

 now

 let n,m be Nat;

 assume

 A11: n <= m;



 A12: ((F # x) . m) = ((F . m) . x) by Def13;

 ((F # x) . n) = ((F . n) . x) by Def13;

 hence ((F # x) . n) <= ((F # x) . m) by A5, A9, A11, A12;

 end;



 A13: (g . x) = ((F # x) . 0 ) by Def13;

 ( lim (F # x)) = ( sup ( rng (F # x))) by A10, Th54;

 hence thesis by A10, A13, Th54, Th56;

 end;

 ( dom g) = ( dom f) by A3;

 then ex G be ExtREAL_sequence st (for n be Nat holds (G . n) = ( integral' (M,(F . n)))) & G is convergent & ( sup ( rng G)) = ( lim G) & ( integral' (M,g)) <= ( lim G) by A3, A4, A5, A8, A7, Th75;

 then

 consider G be ExtREAL_sequence such that

 A14: for n be Nat holds (G . n) = ( integral' (M,(F . n))) and

 A15: G is convergent and ( integral' (M,g)) <= ( lim G);

 take ( lim G);

 thus thesis by A3, A4, A5, A6, A14, A15;

 end;

 uniqueness

 proof

 let s1,s2 be Element of ExtREAL such that

 A16: ex F1 be Functional_Sequence of X, ExtREAL , K1 be ExtREAL_sequence st (for n be Nat holds (F1 . n) is_simple_func_in S & ( dom (F1 . n)) = ( dom f)) & (for n be Nat holds (F1 . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F1 . n) . x) <= ((F1 . m) . x)) & (for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & ( lim (F1 # x)) = (f . x)) & (for n be Nat holds (K1 . n) = ( integral' (M,(F1 . n)))) & K1 is convergent & s1 = ( lim K1) and

 A17: ex F2 be Functional_Sequence of X, ExtREAL , K2 be ExtREAL_sequence st (for n be Nat holds (F2 . n) is_simple_func_in S & ( dom (F2 . n)) = ( dom f)) & (for n be Nat holds (F2 . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F2 . n) . x) <= ((F2 . m) . x)) & (for x be Element of X st x in ( dom f) holds (F2 # x) is convergent & ( lim (F2 # x)) = (f . x)) & (for n be Nat holds (K2 . n) = ( integral' (M,(F2 . n)))) & K2 is convergent & s2 = ( lim K2);

 consider F1 be Functional_Sequence of X, ExtREAL , K1 be ExtREAL_sequence such that

 A18: for n be Nat holds (F1 . n) is_simple_func_in S & ( dom (F1 . n)) = ( dom f) and

 A19: for n be Nat holds (F1 . n) is nonnegative and

 A20: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F1 . n) . x) <= ((F1 . m) . x) and

 A21: for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & ( lim (F1 # x)) = (f . x) and

 A22: for n be Nat holds (K1 . n) = ( integral' (M,(F1 . n))) and K1 is convergent and

 A23: s1 = ( lim K1) by A16;

 consider F2 be Functional_Sequence of X, ExtREAL , K2 be ExtREAL_sequence such that

 A24: for n be Nat holds (F2 . n) is_simple_func_in S & ( dom (F2 . n)) = ( dom f) and

 A25: for n be Nat holds (F2 . n) is nonnegative and

 A26: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F2 . n) . x) <= ((F2 . m) . x) and

 A27: for x be Element of X st x in ( dom f) holds (F2 # x) is convergent & ( lim (F2 # x)) = (f . x) and

 A28: for n be Nat holds (K2 . n) = ( integral' (M,(F2 . n))) and K2 is convergent and

 A29: s2 = ( lim K2) by A17;

 for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & (F2 # x) is convergent & ( lim (F1 # x)) = ( lim (F2 # x))

 proof

 let x be Element of X;

 assume

 A30: x in ( dom f);



 then ( lim (F1 # x)) = (f . x) by A21

 .= ( lim (F2 # x)) by A27, A30;

 hence thesis by A21, A27, A30;

 end;

 hence thesis by A1, A18, A19, A20, A22, A23, A24, A25, A26, A28, A29, Th76;

 end;

 end

 theorem :: MESFUNC5:77



 Th77: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & f is nonnegative holds ( integral+ (M,f)) = ( integral' (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL such that

 A1: f is_simple_func_in S and

 A2: f is nonnegative;

 deffunc PF( Nat) = f;

 consider F be Functional_Sequence of X, ExtREAL such that

 A3: for n be Nat holds (F . n) = PF(n) from SEQFUNC:sch 1;



 A4: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let n,m be Nat;

 assume n <= m;

 let x be Element of X;

 assume x in ( dom f);

 ((F . n) . x) = (f . x) by A3;

 hence thesis by A3;

 end;

 deffunc PK( Nat) = ( integral' (M,(F . $1)));

 consider K be sequence of ExtREAL such that

 A5: for n be Element of NAT holds (K . n) = PK(n) from FUNCT_2:sch 4;

 A6:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (K . n) = PK(n) by A5;

 end;



 A7: for n be Nat holds (K . n) = ( integral' (M,f))

 proof

 let n be Nat;



 thus (K . n) = ( integral' (M,(F . n))) by A6

 .= ( integral' (M,f)) by A3;

 end;

 then

 A8: ( lim K) = ( integral' (M,f)) by Th60;

 ex GF be Finite_Sep_Sequence of S st ( dom f) = ( union ( rng GF)) & for n be Nat, x,y be Element of X st n in ( dom GF) & x in (GF . n) & y in (GF . n) holds (f . x) = (f . y) by A1, MESFUNC2:def 4;

 then

 reconsider A = ( dom f) as Element of S by MESFUNC2: 31;



 A9: f is A -measurable by A1, MESFUNC2: 34;



 A10: for x be Element of X st x in ( dom f) holds (F # x) is convergent & ( lim (F # x)) = (f . x)

 proof

 let x be Element of X;

 assume x in ( dom f);

 now

 let n be Nat;



 thus ((F # x) . n) = ((F . n) . x) by Def13

 .= (f . x) by A3;

 end;

 hence thesis by Th60;

 end;



 A11: for n be Nat holds (F . n) is nonnegative by A2, A3;



 A12: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) by A1, A3;

 K is convergent by A7, Th60;

 hence thesis by A2, A9, A6, A12, A11, A4, A10, A8, Def15;

 end;



 Lm10: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & A = ( dom g) & f is A -measurable & g is A -measurable) & f is nonnegative & g is nonnegative holds ( integral+ (M,(f + g))) = (( integral+ (M,f)) + ( integral+ (M,g)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: ex A be Element of S st A = ( dom f) & A = ( dom g) & f is A -measurable & g is A -measurable and

 A2: f is nonnegative and

 A3: g is nonnegative;

 consider F1 be Functional_Sequence of X, ExtREAL , K1 be ExtREAL_sequence such that

 A4: for n be Nat holds (F1 . n) is_simple_func_in S & ( dom (F1 . n)) = ( dom f) and

 A5: for n be Nat holds (F1 . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F1 . n) . x) <= ((F1 . m) . x) and

 A7: for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & ( lim (F1 # x)) = (f . x) and

 A8: for n be Nat holds (K1 . n) = ( integral' (M,(F1 . n))) and K1 is convergent and

 A9: ( integral+ (M,f)) = ( lim K1) by A1, A2, Def15;



 A10: (f + g) is nonnegative by A2, A3, Th19;

 consider A be Element of S such that

 A11: A = ( dom f) and

 A12: A = ( dom g) and

 A13: f is A -measurable and

 A14: g is A -measurable by A1;

 A = (( dom f) /\ ( dom g)) by A11, A12;

 then

 A15: A = ( dom (f + g)) by A2, A3, Th16;



 A16: for n,m be Nat st n <= m holds (K1 . n) <= (K1 . m)

 proof

 let n,m be Nat such that

 A17: n <= m;



 A18: ( dom (F1 . m)) = ( dom f) by A4;



 A19: ( dom (F1 . n)) = ( dom f) by A4;

 A20:

 now

 let x be object;

 assume x in ( dom ((F1 . m) - (F1 . n)));

 then x in ((( dom (F1 . m)) /\ ( dom (F1 . n))) \ ((((F1 . m) " { +infty }) /\ ((F1 . n) " { +infty })) \/ (((F1 . m) " { -infty }) /\ ((F1 . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F1 . m)) /\ ( dom (F1 . n))) by XBOOLE_0:def 5;

 hence ((F1 . n) . x) <= ((F1 . m) . x) by A6, A17, A19, A18;

 end;



 A21: (F1 . m) is nonnegative by A5;



 A22: (F1 . n) is nonnegative by A5;



 A23: (K1 . m) = ( integral' (M,(F1 . m))) by A8;



 A24: (K1 . n) = ( integral' (M,(F1 . n))) by A8;



 A25: (F1 . m) is_simple_func_in S by A4;



 A26: (F1 . n) is_simple_func_in S by A4;

 then

 A27: ( dom ((F1 . m) - (F1 . n))) = (( dom (F1 . m)) /\ ( dom (F1 . n))) by A25, A22, A21, A20, Th69;

 then

 A28: ((F1 . m) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . m) by A19, A18, GRFUNC_1: 23;

 ((F1 . n) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . n) by A19, A18, A27, GRFUNC_1: 23;

 hence thesis by A24, A23, A26, A25, A22, A21, A20, A28, Th70;

 end;

 consider F2 be Functional_Sequence of X, ExtREAL , K2 be ExtREAL_sequence such that

 A29: for n be Nat holds (F2 . n) is_simple_func_in S & ( dom (F2 . n)) = ( dom g) and

 A30: for n be Nat holds (F2 . n) is nonnegative and

 A31: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((F2 . n) . x) <= ((F2 . m) . x) and

 A32: for x be Element of X st x in ( dom g) holds (F2 # x) is convergent & ( lim (F2 # x)) = (g . x) and

 A33: for n be Nat holds (K2 . n) = ( integral' (M,(F2 . n))) and K2 is convergent and

 A34: ( integral+ (M,g)) = ( lim K2) by A1, A3, Def15;

 deffunc PF( Nat) = ((F1 . $1) + (F2 . $1));

 consider F be Functional_Sequence of X, ExtREAL such that

 A35: for n be Nat holds (F . n) = PF(n) from SEQFUNC:sch 1;



 A36: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom (f + g)) & (F . n) is nonnegative

 proof

 let n be Nat;



 A37: ( dom (F1 . n)) = ( dom f) by A4;



 A38: (F2 . n) is_simple_func_in S by A29;



 A39: (F2 . n) is nonnegative by A30;



 A40: (F . n) = ((F1 . n) + (F2 . n)) by A35;



 A41: (F1 . n) is_simple_func_in S by A4;

 hence (F . n) is_simple_func_in S by A38, A40, Th38;



 A42: ( dom (F2 . n)) = ( dom g) by A29;

 (F1 . n) is nonnegative by A5;

 then ( dom (F . n)) = (( dom (F1 . n)) /\ ( dom (F2 . n))) by A41, A38, A39, A40, Th65;

 hence ( dom (F . n)) = ( dom (f + g)) by A2, A3, A37, A42, Th16;



 A43: (F2 . n) is nonnegative by A30;



 A44: (F . n) = ((F1 . n) + (F2 . n)) by A35;

 (F1 . n) is nonnegative by A5;

 hence thesis by A43, A44, Th19;

 end;



 A45: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (f + g)) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let n,m be Nat;

 assume

 A46: n <= m;

 ( dom ((F1 . m) + (F2 . m))) = ( dom (F . m)) by A35;

 then

 A47: ( dom ((F1 . m) + (F2 . m))) = ( dom (f + g)) by A36;

 ( dom ((F1 . n) + (F2 . n))) = ( dom (F . n)) by A35;

 then

 A48: ( dom ((F1 . n) + (F2 . n))) = ( dom (f + g)) by A36;

 let x be Element of X;

 assume

 A49: x in ( dom (f + g));

 then

 A50: ((F2 . n) . x) <= ((F2 . m) . x) by A31, A12, A15, A46;

 ((F . m) . x) = (((F1 . m) + (F2 . m)) . x) by A35;

 then

 A51: ((F . m) . x) = (((F1 . m) . x) + ((F2 . m) . x)) by A49, A47, MESFUNC1:def 3;

 ((F . n) . x) = (((F1 . n) + (F2 . n)) . x) by A35;

 then

 A52: ((F . n) . x) = (((F1 . n) . x) + ((F2 . n) . x)) by A49, A48, MESFUNC1:def 3;

 ((F1 . n) . x) <= ((F1 . m) . x) by A6, A11, A15, A46, A49;

 hence thesis by A52, A51, A50, XXREAL_3: 36;

 end;

 now

 let n be set;

 assume n in ( dom K2);

 then

 reconsider n1 = n as Element of NAT ;



 A53: (F2 . n1) is_simple_func_in S by A29;

 (K2 . n1) = ( integral' (M,(F2 . n1))) by A33;

 hence -infty < (K2 . n) by A30, A53, Th68;

 end;

 then

 A54: K2 is without-infty by Th10;

 deffunc PK( Nat) = ( integral' (M,(F . $1)));

 consider K be ExtREAL_sequence such that

 A55: for n be Element of NAT holds (K . n) = PK(n) from FUNCT_2:sch 4;

 A56:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (K . n) = PK(n) by A55;

 end;



 A57: for n be Nat holds (K . n) = ((K1 . n) + (K2 . n))

 proof

 let n be Nat;



 A58: (F1 . n) is nonnegative by A5;



 A59: (F . n) = ((F1 . n) + (F2 . n)) by A35;



 A60: ( dom (F1 . n)) = ( dom f) by A4

 .= ( dom (F2 . n)) by A29, A11, A12;



 A61: (F2 . n) is_simple_func_in S by A29;



 A62: (K . n) = ( integral' (M,(F . n))) by A56;



 A63: (F2 . n) is nonnegative by A30;



 A64: (F1 . n) is_simple_func_in S by A4;

 then ( dom (F . n)) = (( dom (F1 . n)) /\ ( dom (F2 . n))) by A58, A61, A63, A59, Th65;

 then (K . n) = (( integral' (M,((F1 . n) | ( dom (F1 . n))))) + ( integral' (M,((F2 . n) | ( dom (F2 . n)))))) by A64, A58, A61, A63, A59, A60, A62, Th65;

 then (K . n) = (( integral' (M,(F1 . n))) + ( integral' (M,((F2 . n) | ( dom (F2 . n)))))) by GRFUNC_1: 23;

 then

 A65: (K . n) = (( integral' (M,(F1 . n))) + ( integral' (M,(F2 . n)))) by GRFUNC_1: 23;

 (K2 . n) = ( integral' (M,(F2 . n))) by A33;

 hence thesis by A8, A65;

 end;



 A66: for n,m be Nat st n <= m holds (K2 . n) <= (K2 . m)

 proof

 let n,m be Nat such that

 A67: n <= m;



 A68: ( dom (F2 . m)) = ( dom g) by A29;



 A69: ( dom (F2 . n)) = ( dom g) by A29;

 A70:

 now

 let x be object;

 assume x in ( dom ((F2 . m) - (F2 . n)));

 then x in ((( dom (F2 . m)) /\ ( dom (F2 . n))) \ ((((F2 . m) " { +infty }) /\ ((F2 . n) " { +infty })) \/ (((F2 . m) " { -infty }) /\ ((F2 . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F2 . m)) /\ ( dom (F2 . n))) by XBOOLE_0:def 5;

 hence ((F2 . n) . x) <= ((F2 . m) . x) by A31, A67, A69, A68;

 end;



 A71: (F2 . m) is nonnegative by A30;



 A72: (F2 . n) is nonnegative by A30;



 A73: (K2 . m) = ( integral' (M,(F2 . m))) by A33;



 A74: (K2 . n) = ( integral' (M,(F2 . n))) by A33;



 A75: (F2 . m) is_simple_func_in S by A29;



 A76: (F2 . n) is_simple_func_in S by A29;

 then

 A77: ( dom ((F2 . m) - (F2 . n))) = (( dom (F2 . m)) /\ ( dom (F2 . n))) by A75, A72, A71, A70, Th69;

 then

 A78: ((F2 . m) | ( dom ((F2 . m) - (F2 . n)))) = (F2 . m) by A69, A68, GRFUNC_1: 23;

 ((F2 . n) | ( dom ((F2 . m) - (F2 . n)))) = (F2 . n) by A69, A68, A77, GRFUNC_1: 23;

 hence thesis by A74, A73, A76, A75, A72, A71, A70, A78, Th70;

 end;

 now

 let n be set;

 assume n in ( dom K1);

 then

 reconsider n1 = n as Element of NAT ;



 A79: (F1 . n1) is_simple_func_in S by A4;

 (K1 . n1) = ( integral' (M,(F1 . n1))) by A8;

 hence -infty < (K1 . n) by A5, A79, Th68;

 end;

 then

 A80: K1 is without-infty by Th10;

 then

 A81: ( lim K) = (( lim K1) + ( lim K2)) by A16, A54, A66, A57, Th61;



 A82: for x be Element of X st x in ( dom (f + g)) holds (F # x) is convergent & ( lim (F # x)) = ((f + g) . x)

 proof

 let x be Element of X;

 A83:

 now

 let n be set;

 hereby

 assume n in ( dom (F1 # x));

 then

 reconsider n1 = n as Element of NAT ;



 A84: ((F1 # x) . n1) = ((F1 . n1) . x) by Def13;

 (F1 . n1) is nonnegative by A5;

 hence -infty < ((F1 # x) . n) by A84, Def5;

 end;

 assume n in ( dom (F2 # x));

 then

 reconsider n1 = n as Element of NAT ;



 A85: ((F2 # x) . n1) = ((F2 . n1) . x) by Def13;

 (F2 . n1) is nonnegative by A30;

 hence -infty < ((F2 # x) . n) by A85, Def5;

 end;

 then

 A86: (F2 # x) is without-infty by Th10;

 assume

 A87: x in ( dom (f + g));

 then (( lim (F1 # x)) + ( lim (F2 # x))) = ((f . x) + ( lim (F2 # x))) by A7, A11, A15;

 then (( lim (F1 # x)) + ( lim (F2 # x))) = ((f . x) + (g . x)) by A32, A12, A15, A87;

 then

 A88: (( lim (F1 # x)) + ( lim (F2 # x))) = ((f + g) . x) by A87, MESFUNC1:def 3;

 A89:

 now

 let n,m be Nat;

 assume

 A90: n <= m;



 A91: ((F2 # x) . m) = ((F2 . m) . x) by Def13;

 ((F2 # x) . n) = ((F2 . n) . x) by Def13;

 hence ((F2 # x) . n) <= ((F2 # x) . m) by A31, A12, A15, A87, A90, A91;

 end;

 A92:

 now

 let n,m be Nat;

 assume

 A93: n <= m;



 A94: ((F1 # x) . m) = ((F1 . m) . x) by Def13;

 ((F1 # x) . n) = ((F1 . n) . x) by Def13;

 hence ((F1 # x) . n) <= ((F1 # x) . m) by A6, A11, A15, A87, A93, A94;

 end;

 A95:

 now

 let n be Nat;

 ((F # x) . n) = ((F . n) . x) by Def13;

 then

 A96: ((F # x) . n) = (((F1 . n) + (F2 . n)) . x) by A35;

 ( dom ((F1 . n) + (F2 . n))) = ( dom (F . n)) by A35

 .= ( dom (f + g)) by A36;

 then ((F # x) . n) = (((F1 . n) . x) + ((F2 . n) . x)) by A87, A96, MESFUNC1:def 3;

 then ((F # x) . n) = (((F1 # x) . n) + ((F2 . n) . x)) by Def13;

 hence ((F # x) . n) = (((F1 # x) . n) + ((F2 # x) . n)) by Def13;

 end;

 (F1 # x) is without-infty by A83, Th10;

 hence thesis by A95, A86, A92, A89, A88, Th61;

 end;



 A97: (f + g) is A -measurable by A2, A3, A13, A14, Th31;

 K is convergent by A80, A16, A54, A66, A57, Th61;

 hence thesis by A9, A34, A97, A15, A10, A56, A36, A45, A82, A81, Def15;

 end;

 theorem :: MESFUNC5:78



 Th78: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & (ex B be Element of S st B = ( dom g) & g is B -measurable) & f is nonnegative & g is nonnegative holds ex C be Element of S st C = ( dom (f + g)) & ( integral+ (M,(f + g))) = (( integral+ (M,(f | C))) + ( integral+ (M,(g | C))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: ex B be Element of S st B = ( dom g) & g is B -measurable and

 A3: f is nonnegative and

 A4: g is nonnegative;

 set g1 = (g | (( dom f) /\ ( dom g)));



 A5: g1 is without-infty by A4, Th12, Th15;



 A6: g1 is nonnegative by A4, Th15;

 ( dom g1) = (( dom g) /\ (( dom f) /\ ( dom g))) by RELAT_1: 61;

 then

 A7: ( dom g1) = ((( dom g) /\ ( dom g)) /\ ( dom f)) by XBOOLE_1: 16;

 consider B be Element of S such that

 A8: B = ( dom g) and

 A9: g is B -measurable by A2;

 consider A be Element of S such that

 A10: A = ( dom f) and

 A11: f is A -measurable by A1;

 take C = (A /\ B);



 A12: C = ( dom (f + g)) by A3, A4, A10, A8, Th16;



 A13: C = (( dom g) /\ C) by A8, XBOOLE_1: 17, XBOOLE_1: 28;

 g is C -measurable by A9, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A14: (g | C) is C -measurable by A13, Th42;



 A15: C = (( dom f) /\ C) by A10, XBOOLE_1: 17, XBOOLE_1: 28;

 f is C -measurable by A11, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A16: (f | C) is C -measurable by A15, Th42;

 set f1 = (f | (( dom f) /\ ( dom g)));

 ( dom f1) = (( dom f) /\ (( dom f) /\ ( dom g))) by RELAT_1: 61;

 then

 A17: ( dom f1) = ((( dom f) /\ ( dom f)) /\ ( dom g)) by XBOOLE_1: 16;



 A18: f1 is without-infty by A3, Th12, Th15;

 then

 A19: ( dom (f1 + g1)) = (C /\ C) by A10, A8, A17, A7, A5, Th16;



 A20: ( dom (f1 + g1)) = (( dom f1) /\ ( dom g1)) by A18, A5, Th16;



 A21: for x be object st x in ( dom (f1 + g1)) holds ((f1 + g1) . x) = ((f + g) . x)

 proof

 let x be object;

 assume

 A22: x in ( dom (f1 + g1));

 then

 A23: x in ( dom f1) by A20, XBOOLE_0:def 4;



 A24: x in ( dom g1) by A20, A22, XBOOLE_0:def 4;

 ((f1 + g1) . x) = ((f1 . x) + (g1 . x)) by A22, MESFUNC1:def 3

 .= ((f . x) + (g1 . x)) by A23, FUNCT_1: 47

 .= ((f . x) + (g . x)) by A24, FUNCT_1: 47;

 hence thesis by A12, A19, A22, MESFUNC1:def 3;

 end;

 f1 is nonnegative by A3, Th15;

 then ( integral+ (M,(f1 + g1))) = (( integral+ (M,f1)) + ( integral+ (M,g1))) by A10, A8, A17, A7, A16, A14, A6, Lm10;

 hence thesis by A10, A8, A12, A19, A21, FUNCT_1: 2;

 end;

 theorem :: MESFUNC5:79



 Th79: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & f is nonnegative holds 0 <= ( integral+ (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: f is nonnegative;

 consider F1 be Functional_Sequence of X, ExtREAL , K1 be ExtREAL_sequence such that

 A3: for n be Nat holds (F1 . n) is_simple_func_in S & ( dom (F1 . n)) = ( dom f) and

 A4: for n be Nat holds (F1 . n) is nonnegative and

 A5: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F1 . n) . x) <= ((F1 . m) . x) and for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & ( lim (F1 # x)) = (f . x) and

 A6: for n be Nat holds (K1 . n) = ( integral' (M,(F1 . n))) and K1 is convergent and

 A7: ( integral+ (M,f)) = ( lim K1) by A1, A2, Def15;

 for n,m be Nat st n <= m holds (K1 . n) <= (K1 . m)

 proof

 let n,m be Nat;



 A8: (F1 . m) is_simple_func_in S by A3;



 A9: ( dom (F1 . m)) = ( dom f) by A3;



 A10: (K1 . m) = ( integral' (M,(F1 . m))) by A6;



 A11: ( dom (F1 . n)) = ( dom f) by A3;

 assume

 A12: n <= m;



 A13: for x be object st x in ( dom ((F1 . m) - (F1 . n))) holds ((F1 . n) . x) <= ((F1 . m) . x)

 proof

 let x be object;

 assume x in ( dom ((F1 . m) - (F1 . n)));

 then x in ((( dom (F1 . m)) /\ ( dom (F1 . n))) \ ((((F1 . m) " { +infty }) /\ ((F1 . n) " { +infty })) \/ (((F1 . m) " { -infty }) /\ ((F1 . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F1 . m)) /\ ( dom (F1 . n))) by XBOOLE_0:def 5;

 hence thesis by A5, A12, A11, A9;

 end;



 A14: (F1 . m) is nonnegative by A4;



 A15: (F1 . n) is nonnegative by A4;



 A16: (F1 . n) is_simple_func_in S by A3;

 then

 A17: ( dom ((F1 . m) - (F1 . n))) = (( dom (F1 . m)) /\ ( dom (F1 . n))) by A8, A15, A14, A13, Th69;

 then

 A18: ((F1 . n) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . n) by A11, A9, GRFUNC_1: 23;



 A19: ((F1 . m) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . m) by A11, A9, A17, GRFUNC_1: 23;

 ( integral' (M,((F1 . n) | ( dom ((F1 . m) - (F1 . n)))))) <= ( integral' (M,((F1 . m) | ( dom ((F1 . m) - (F1 . n)))))) by A16, A8, A15, A14, A13, Th70;

 hence thesis by A6, A10, A18, A19;

 end;

 then ( lim K1) = ( sup ( rng K1)) by Th54;

 then

 A20: (K1 . 0 ) <= ( lim K1) by Th56;

 for n be Nat holds 0 <= (K1 . n)

 proof

 let n be Nat;



 A21: (F1 . n) is_simple_func_in S by A3;

 (K1 . n) = ( integral' (M,(F1 . n))) by A6;

 hence thesis by A4, A21, Th68;

 end;

 hence thesis by A7, A20;

 end;

 theorem :: MESFUNC5:80



 Th80: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative holds 0 <= ( integral+ (M,(f | A)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative;

 consider E be Element of S such that

 A3: E = ( dom f) and

 A4: f is E -measurable by A1;

 set C = (E /\ A);



 A5: C = ( dom (f | A)) by A3, RELAT_1: 61;



 A6: ( dom (f | A)) = C by A3, RELAT_1: 61

 .= (( dom f) /\ C) by A3, XBOOLE_1: 17, XBOOLE_1: 28

 .= ( dom (f | C)) by RELAT_1: 61;



 A7: for x be object st x in ( dom (f | A)) holds ((f | A) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A8: x in ( dom (f | A));

 then ((f | A) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A6, A8, FUNCT_1: 47;

 end;



 A9: (( dom f) /\ C) = C by A3, XBOOLE_1: 17, XBOOLE_1: 28;

 f is C -measurable by A4, MESFUNC1: 30, XBOOLE_1: 17;

 then (f | C) is C -measurable by A9, Th42;

 then (f | A) is C -measurable by A6, A7, FUNCT_1: 2;

 hence thesis by A2, A5, Th15, Th79;

 end;

 theorem :: MESFUNC5:81



 Th81: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative & A misses B holds ( integral+ (M,(f | (A \/ B)))) = (( integral+ (M,(f | A))) + ( integral+ (M,(f | B))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative and

 A3: A misses B;

 consider F0 be Functional_Sequence of X, ExtREAL , K0 be ExtREAL_sequence such that

 A4: for n be Nat holds (F0 . n) is_simple_func_in S & ( dom (F0 . n)) = ( dom f) and

 A5: for n be Nat holds (F0 . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F0 . n) . x) <= ((F0 . m) . x) and

 A7: for x be Element of X st x in ( dom f) holds (F0 # x) is convergent & ( lim (F0 # x)) = (f . x) and for n be Nat holds (K0 . n) = ( integral' (M,(F0 . n))) and K0 is convergent and ( integral+ (M,f)) = ( lim K0) by A1, A2, Def15;

 deffunc PFB( Nat) = ((F0 . $1) | B);

 deffunc PFA( Nat) = ((F0 . $1) | A);

 consider FA be Functional_Sequence of X, ExtREAL such that

 A8: for n be Nat holds (FA . n) = PFA(n) from SEQFUNC:sch 1;

 consider E be Element of S such that

 A9: E = ( dom f) and

 A10: f is E -measurable by A1;

 consider FB be Functional_Sequence of X, ExtREAL such that

 A11: for n be Nat holds (FB . n) = PFB(n) from SEQFUNC:sch 1;

 set DB = (E /\ B);



 A12: DB = ( dom (f | B)) by A9, RELAT_1: 61;

 then

 A13: (( dom f) /\ DB) = DB by RELAT_1: 60, XBOOLE_1: 28;

 then

 A14: ( dom (f | DB)) = ( dom (f | B)) by A12, RELAT_1: 61;

 for x be object st x in ( dom (f | DB)) holds ((f | DB) . x) = ((f | B) . x)

 proof

 let x be object;

 assume

 A15: x in ( dom (f | DB));

 then ((f | B) . x) = (f . x) by A14, FUNCT_1: 47;

 hence thesis by A15, FUNCT_1: 47;

 end;

 then

 A16: (f | DB) = (f | B) by A12, A13, FUNCT_1: 2, RELAT_1: 61;

 set DA = (E /\ A);



 A17: DA = ( dom (f | A)) by A9, RELAT_1: 61;

 then

 A18: (( dom f) /\ DA) = DA by RELAT_1: 60, XBOOLE_1: 28;

 then

 A19: ( dom (f | DA)) = ( dom (f | A)) by A17, RELAT_1: 61;

 for x be object st x in ( dom (f | DA)) holds ((f | DA) . x) = ((f | A) . x)

 proof

 let x be object;

 assume

 A20: x in ( dom (f | DA));

 then ((f | A) . x) = (f . x) by A19, FUNCT_1: 47;

 hence thesis by A20, FUNCT_1: 47;

 end;

 then

 A21: (f | DA) = (f | A) by A17, A18, FUNCT_1: 2, RELAT_1: 61;



 A22: for n be Nat holds (FA . n) is_simple_func_in S & (FB . n) is_simple_func_in S & ( dom (FA . n)) = ( dom (f | A)) & ( dom (FB . n)) = ( dom (f | B))

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 A23: (FB . n1) = ((F0 . n1) | B) by A11;

 then

 A24: ( dom (FB . n)) = (( dom (F0 . n)) /\ B) by RELAT_1: 61;



 A25: (FA . n1) = ((F0 . n1) | A) by A8;

 hence (FA . n) is_simple_func_in S & (FB . n) is_simple_func_in S by A4, A23, Th34;

 ( dom (FA . n)) = (( dom (F0 . n)) /\ A) by A25, RELAT_1: 61;

 hence thesis by A9, A4, A17, A12, A24;

 end;



 A26: for x be Element of X st x in ( dom (f | A)) holds (FA # x) is convergent & ( lim (FA # x)) = ((f | A) . x)

 proof

 let x be Element of X;

 assume

 A27: x in ( dom (f | A));

 now

 let n be Element of NAT ;

 ((FA # x) . n) = ((FA . n) . x) by Def13;

 then

 A28: ((FA # x) . n) = (((F0 . n) | A) . x) by A8;

 ( dom ((F0 . n) | A)) = ( dom (FA . n)) by A8

 .= ( dom (f | A)) by A22;

 then ((FA # x) . n) = ((F0 . n) . x) by A27, A28, FUNCT_1: 47;

 hence ((FA # x) . n) = ((F0 # x) . n) by Def13;

 end;

 then

 A29: (FA # x) = (F0 # x) by FUNCT_2: 63;

 x in (( dom f) /\ A) by A27, RELAT_1: 61;

 then

 A30: x in ( dom f) by XBOOLE_0:def 4;

 then ( lim (F0 # x)) = (f . x) by A7;

 hence thesis by A7, A27, A30, A29, FUNCT_1: 47;

 end;



 A31: for x be Element of X st x in ( dom (f | B)) holds (FB # x) is convergent & ( lim (FB # x)) = ((f | B) . x)

 proof

 let x be Element of X;

 assume

 A32: x in ( dom (f | B));

 now

 let n be Element of NAT ;



 A33: ( dom ((F0 . n) | B)) = ( dom (FB . n)) by A11

 .= ( dom (f | B)) by A22;



 thus ((FB # x) . n) = ((FB . n) . x) by Def13

 .= (((F0 . n) | B) . x) by A11

 .= ((F0 . n) . x) by A32, A33, FUNCT_1: 47

 .= ((F0 # x) . n) by Def13;

 end;

 then

 A34: (FB # x) = (F0 # x) by FUNCT_2: 63;

 x in (( dom f) /\ B) by A32, RELAT_1: 61;

 then

 A35: x in ( dom f) by XBOOLE_0:def 4;

 then ( lim (F0 # x)) = (f . x) by A7;

 hence thesis by A7, A32, A35, A34, FUNCT_1: 47;

 end;

 set C = (E /\ (A \/ B));



 A36: C = (( dom f) /\ C) by A9, XBOOLE_1: 17, XBOOLE_1: 28;



 A37: ( dom (f | (A \/ B))) = C by A9, RELAT_1: 61;

 then

 A38: ( dom (f | (A \/ B))) = ( dom (f | C)) by A36, RELAT_1: 61;

 for x be object st x in ( dom (f | (A \/ B))) holds ((f | (A \/ B)) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A39: x in ( dom (f | (A \/ B)));

 then ((f | (A \/ B)) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A38, A39, FUNCT_1: 47;

 end;

 then

 A40: (f | (A \/ B)) = (f | C) by A38, FUNCT_1: 2;

 f is C -measurable by A10, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A41: (f | (A \/ B)) is C -measurable by A36, A40, Th42;

 f is DB -measurable by A10, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A42: (f | B) is DB -measurable by A13, A16, Th42;



 A43: (f | B) is nonnegative by A2, Th15;

 f is DA -measurable by A10, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A44: (f | A) is DA -measurable by A18, A21, Th42;



 A45: (f | A) is nonnegative by A2, Th15;

 deffunc PFAB( Nat) = ((F0 . $1) | (A \/ B));

 consider FAB be Functional_Sequence of X, ExtREAL such that

 A46: for n be Nat holds (FAB . n) = PFAB(n) from SEQFUNC:sch 1;



 A47: for n be Nat holds (FA . n) is nonnegative & (FB . n) is nonnegative

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;



 A48: ((F0 . n) | B) is nonnegative by A5, Th15;

 ((F0 . n) | A) is nonnegative by A5, Th15;

 hence thesis by A8, A11, A48;

 end;



 A49: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (f | B)) holds ((FB . n) . x) <= ((FB . m) . x)

 proof

 let n,m be Nat;

 assume

 A50: n <= m;

 reconsider n, m as Element of NAT by ORDINAL1:def 12;

 let x be Element of X;

 assume

 A51: x in ( dom (f | B));

 then x in (( dom f) /\ B) by RELAT_1: 61;

 then

 A52: x in ( dom f) by XBOOLE_0:def 4;

 ( dom ((F0 . m) | B)) = ( dom (FB . m)) by A11;

 then

 A53: ( dom ((F0 . m) | B)) = ( dom (f | B)) by A22;

 ((FB . m) . x) = (((F0 . m) | B) . x) by A11;

 then

 A54: ((FB . m) . x) = ((F0 . m) . x) by A51, A53, FUNCT_1: 47;

 ( dom ((F0 . n) | B)) = ( dom (FB . n)) by A11;

 then

 A55: ( dom ((F0 . n) | B)) = ( dom (f | B)) by A22;

 ((FB . n) . x) = (((F0 . n) | B) . x) by A11;

 then ((FB . n) . x) = ((F0 . n) . x) by A51, A55, FUNCT_1: 47;

 hence thesis by A6, A50, A52, A54;

 end;

 deffunc PKA( Nat) = ( integral' (M,(FA . $1)));

 consider KA be ExtREAL_sequence such that

 A56: for n be Element of NAT holds (KA . n) = PKA(n) from FUNCT_2:sch 4;

 deffunc PKB( Nat) = ( integral' (M,(FB . $1)));

 consider KB be ExtREAL_sequence such that

 A57: for n be Element of NAT holds (KB . n) = PKB(n) from FUNCT_2:sch 4;

 A58:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (KB . n) = PKB(n) by A57;

 end;

 A59:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (KA . n) = PKA(n) by A56;

 end;



 A60: for n be set holds (n in ( dom KA) implies -infty < (KA . n)) & (n in ( dom KB) implies -infty < (KB . n))

 proof

 let n be set;

 hereby

 assume n in ( dom KA);

 then

 reconsider n1 = n as Element of NAT ;



 A61: (FA . n1) is_simple_func_in S by A22;

 (KA . n1) = ( integral' (M,(FA . n1))) by A59;

 hence -infty < (KA . n) by A47, A61, Th68;

 end;

 assume n in ( dom KB);

 then

 reconsider n1 = n as Element of NAT ;



 A62: (FB . n1) is_simple_func_in S by A22;

 (KB . n1) = ( integral' (M,(FB . n1))) by A58;

 hence thesis by A47, A62, Th68;

 end;

 then

 A63: KB is without-infty by Th10;

 deffunc PK( Nat) = ( integral' (M,(FAB . $1)));

 consider KAB be ExtREAL_sequence such that

 A64: for n be Element of NAT holds (KAB . n) = PK(n) from FUNCT_2:sch 4;

 A65:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (KAB . n) = PK(n) by A64;

 end;



 A66: for n be Nat holds (KAB . n) = ((KA . n) + (KB . n))

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;



 A67: (FA . n) = ((F0 . n) | A) by A8;



 A68: (FB . n) = ((F0 . n) | B) by A11;



 A69: (KAB . n) = ( integral' (M,(FAB . n))) by A65

 .= ( integral' (M,((F0 . n) | (A \/ B)))) by A46;



 A70: (KA . n) = ( integral' (M,(FA . n))) by A59;

 (F0 . n) is_simple_func_in S by A4;

 then ( integral' (M,((F0 . n) | (A \/ B)))) = (( integral' (M,(FA . n))) + ( integral' (M,(FB . n)))) by A3, A5, A67, A68, Th67;

 hence thesis by A58, A69, A70;

 end;



 A71: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (f | A)) holds ((FA . n) . x) <= ((FA . m) . x)

 proof

 let n,m be Nat;

 assume

 A72: n <= m;

 reconsider n, m as Element of NAT by ORDINAL1:def 12;

 let x be Element of X;

 assume

 A73: x in ( dom (f | A));

 then x in (( dom f) /\ A) by RELAT_1: 61;

 then

 A74: x in ( dom f) by XBOOLE_0:def 4;

 ( dom ((F0 . m) | A)) = ( dom (FA . m)) by A8;

 then

 A75: ( dom ((F0 . m) | A)) = ( dom (f | A)) by A22;

 ((FA . m) . x) = (((F0 . m) | A) . x) by A8;

 then

 A76: ((FA . m) . x) = ((F0 . m) . x) by A73, A75, FUNCT_1: 47;

 ( dom ((F0 . n) | A)) = ( dom (FA . n)) by A8;

 then

 A77: ( dom ((F0 . n) | A)) = ( dom (f | A)) by A22;

 ((FA . n) . x) = (((F0 . n) | A) . x) by A8;

 then ((FA . n) . x) = ((F0 . n) . x) by A73, A77, FUNCT_1: 47;

 hence thesis by A6, A72, A74, A76;

 end;



 A78: for n,m be Nat st n <= m holds (KA . n) <= (KA . m) & (KB . n) <= (KB . m)

 proof

 let n,m be Nat;



 A79: (FA . m) is_simple_func_in S by A22;



 A80: ( dom (FA . m)) = ( dom (f | A)) by A22;



 A81: (KA . m) = ( integral' (M,(FA . m))) by A59;



 A82: ( dom (FA . n)) = ( dom (f | A)) by A22;

 assume

 A83: n <= m;



 A84: for x be object st x in ( dom ((FA . m) - (FA . n))) holds ((FA . n) . x) <= ((FA . m) . x)

 proof

 let x be object;

 assume x in ( dom ((FA . m) - (FA . n)));

 then x in ((( dom (FA . m)) /\ ( dom (FA . n))) \ ((((FA . m) " { +infty }) /\ ((FA . n) " { +infty })) \/ (((FA . m) " { -infty }) /\ ((FA . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (FA . m)) /\ ( dom (FA . n))) by XBOOLE_0:def 5;

 hence thesis by A71, A83, A82, A80;

 end;



 A85: (FA . m) is nonnegative by A47;



 A86: (FA . n) is nonnegative by A47;



 A87: (FA . n) is_simple_func_in S by A22;

 then

 A88: ( dom ((FA . m) - (FA . n))) = (( dom (FA . m)) /\ ( dom (FA . n))) by A79, A86, A85, A84, Th69;

 then

 A89: ((FA . m) | ( dom ((FA . m) - (FA . n)))) = (FA . m) by A82, A80, GRFUNC_1: 23;



 A90: ((FA . n) | ( dom ((FA . m) - (FA . n)))) = (FA . n) by A82, A80, A88, GRFUNC_1: 23;

 ( integral' (M,((FA . n) | ( dom ((FA . m) - (FA . n)))))) <= ( integral' (M,((FA . m) | ( dom ((FA . m) - (FA . n)))))) by A87, A79, A86, A85, A84, Th70;

 hence (KA . n) <= (KA . m) by A59, A81, A89, A90;



 A91: (FB . m) is_simple_func_in S by A22;



 A92: (FB . n) is nonnegative by A47;



 A93: (FB . m) is nonnegative by A47;



 A94: (KB . m) = ( integral' (M,(FB . m))) by A58;



 A95: ( dom (FB . m)) = ( dom (f | B)) by A22;



 A96: ( dom (FB . n)) = ( dom (f | B)) by A22;



 A97: for x be object st x in ( dom ((FB . m) - (FB . n))) holds ((FB . n) . x) <= ((FB . m) . x)

 proof

 let x be object;

 assume x in ( dom ((FB . m) - (FB . n)));

 then x in ((( dom (FB . m)) /\ ( dom (FB . n))) \ ((((FB . m) " { +infty }) /\ ((FB . n) " { +infty })) \/ (((FB . m) " { -infty }) /\ ((FB . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (FB . m)) /\ ( dom (FB . n))) by XBOOLE_0:def 5;

 hence thesis by A49, A83, A96, A95;

 end;



 A98: (FB . n) is_simple_func_in S by A22;

 then

 A99: ( dom ((FB . m) - (FB . n))) = (( dom (FB . m)) /\ ( dom (FB . n))) by A91, A92, A93, A97, Th69;

 then

 A100: ((FB . m) | ( dom ((FB . m) - (FB . n)))) = (FB . m) by A96, A95, GRFUNC_1: 23;



 A101: ((FB . n) | ( dom ((FB . m) - (FB . n)))) = (FB . n) by A96, A95, A99, GRFUNC_1: 23;

 ( integral' (M,((FB . n) | ( dom ((FB . m) - (FB . n)))))) <= ( integral' (M,((FB . m) | ( dom ((FB . m) - (FB . n)))))) by A98, A91, A92, A93, A97, Th70;

 hence thesis by A58, A94, A100, A101;

 end;

 then

 A102: for n,m be Nat st n <= m holds (KA . n) <= (KA . m);

 then KA is convergent by Th54;

 then

 A103: ( integral+ (M,(f | A))) = ( lim KA) by A17, A44, A45, A22, A47, A71, A26, A59, Def15;



 A104: (for n be Nat holds (FAB . n) is_simple_func_in S & ( dom (FAB . n)) = ( dom (f | (A \/ B)))) & (for n be Nat holds for x be Element of X st x in ( dom (f | (A \/ B))) holds ((FAB . n) . x) = ((F0 . n) . x)) & (for n be Nat holds (FAB . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (f | (A \/ B))) holds ((FAB . n) . x) <= ((FAB . m) . x)) & for x be Element of X st x in ( dom (f | (A \/ B))) holds (FAB # x) is convergent & ( lim (FAB # x)) = ((f | (A \/ B)) . x)

 proof

 now

 let n be Nat;

 (FAB . n) = ((F0 . n) | (A \/ B)) by A46;

 hence (FAB . n) is_simple_func_in S by A4, Th34;



 thus ( dom (FAB . n)) = ( dom ((F0 . n) | (A \/ B))) by A46

 .= (( dom (F0 . n)) /\ (A \/ B)) by RELAT_1: 61

 .= (( dom f) /\ (A \/ B)) by A4

 .= ( dom (f | (A \/ B))) by RELAT_1: 61;

 end;

 now

 let n be Nat, x be Element of X;

 assume x in ( dom (f | (A \/ B)));

 then

 A107: x in ( dom (FAB . n)) by A105;

 (FAB . n) = ((F0 . n) | (A \/ B)) by A46;

 hence ((FAB . n) . x) = ((F0 . n) . x) by A107, FUNCT_1: 47;

 end;

 hereby

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 ((F0 . n1) | (A \/ B)) is nonnegative by A5, Th15;

 hence (FAB . n) is nonnegative by A46;

 end;

 hereby

 let n,m be Nat such that

 A108: n <= m;

 now

 let x be Element of X;

 assume

 A109: x in ( dom (f | (A \/ B)));

 then

 A110: ((FAB . m) . x) = ((F0 . m) . x) by A106;

 x in (( dom f) /\ (A \/ B)) by A109, RELAT_1: 61;

 then

 A111: x in ( dom f) by XBOOLE_0:def 4;

 ((FAB . n) . x) = ((F0 . n) . x) by A106, A109;

 hence ((FAB . n) . x) <= ((FAB . m) . x) by A6, A108, A111, A110;

 end;

 hence for x be Element of X st x in ( dom (f | (A \/ B))) holds ((FAB . n) . x) <= ((FAB . m) . x);

 end;

 hereby

 let x be Element of X;

 assume

 A112: x in ( dom (f | (A \/ B)));

 then x in (( dom f) /\ (A \/ B)) by RELAT_1: 61;

 then

 A113: x in ( dom f) by XBOOLE_0:def 4;

 A114:

 now

 let n be Element of NAT ;



 thus ((FAB # x) . n) = ((FAB . n) . x) by Def13

 .= ((F0 . n) . x) by A106, A112

 .= ((F0 # x) . n) by Def13;

 end;

 then (FAB # x) = (F0 # x) by FUNCT_2: 63;

 hence (FAB # x) is convergent by A7, A113;



 thus ( lim (FAB # x)) = ( lim (F0 # x)) by A114, FUNCT_2: 63

 .= (f . x) by A7, A113

 .= ((f | (A \/ B)) . x) by A112, FUNCT_1: 47;

 end;

 end;

 for n,m be Nat st n <= m holds (KAB . n) <= (KAB . m)

 proof

 let n,m be Nat;

 assume

 A115: n <= m;

 reconsider n, m as Element of NAT by ORDINAL1:def 12;



 A116: ( dom (FAB . m)) = ( dom (f | (A \/ B))) by A104;



 A117: ( dom (FAB . n)) = ( dom (f | (A \/ B))) by A104;



 A118: for x be object st x in ( dom ((FAB . m) - (FAB . n))) holds ((FAB . n) . x) <= ((FAB . m) . x)

 proof

 let x be object;

 assume x in ( dom ((FAB . m) - (FAB . n)));

 then x in ((( dom (FAB . m)) /\ ( dom (FAB . n))) \ ((((FAB . m) " { +infty }) /\ ((FAB . n) " { +infty })) \/ (((FAB . m) " { -infty }) /\ ((FAB . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (FAB . m)) /\ ( dom (FAB . n))) by XBOOLE_0:def 5;

 hence thesis by A104, A115, A117, A116;

 end;



 A119: (KAB . m) = ( integral' (M,(FAB . m))) by A65;



 A120: (FAB . m) is_simple_func_in S by A104;



 A121: (FAB . m) is nonnegative by A104;



 A122: (FAB . n) is nonnegative by A104;



 A123: (FAB . n) is_simple_func_in S by A104;

 then

 A124: ( dom ((FAB . m) - (FAB . n))) = (( dom (FAB . m)) /\ ( dom (FAB . n))) by A120, A122, A121, A118, Th69;

 then

 A125: ((FAB . m) | ( dom ((FAB . m) - (FAB . n)))) = (FAB . m) by A117, A116, GRFUNC_1: 23;



 A126: ((FAB . n) | ( dom ((FAB . m) - (FAB . n)))) = (FAB . n) by A117, A116, A124, GRFUNC_1: 23;

 ( integral' (M,((FAB . n) | ( dom ((FAB . m) - (FAB . n)))))) <= ( integral' (M,((FAB . m) | ( dom ((FAB . m) - (FAB . n)))))) by A123, A120, A122, A121, A118, Th70;

 hence thesis by A65, A119, A125, A126;

 end;

 then

 A127: KAB is convergent by Th54;



 A128: for n,m be Nat st n <= m holds (KB . n) <= (KB . m) by A78;

 then KB is convergent by Th54;

 then

 A129: ( integral+ (M,(f | B))) = ( lim KB) by A12, A42, A43, A22, A47, A49, A31, A58, Def15;

 (f | (A \/ B)) is nonnegative by A2, Th15;

 then

 A130: ( integral+ (M,(f | (A \/ B)))) = ( lim KAB) by A37, A41, A65, A104, A127, Def15;

 KA is without-infty by A60, Th10;

 hence thesis by A130, A102, A128, A103, A129, A66, A63, Th61;

 end;

 theorem :: MESFUNC5:82



 Th82: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative & (M . A) = 0 holds ( integral+ (M,(f | A))) = 0

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative and

 A3: (M . A) = 0 ;

 consider F0 be Functional_Sequence of X, ExtREAL , K0 be ExtREAL_sequence such that

 A4: for n be Nat holds (F0 . n) is_simple_func_in S & ( dom (F0 . n)) = ( dom f) and

 A5: for n be Nat holds (F0 . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F0 . n) . x) <= ((F0 . m) . x) and

 A7: for x be Element of X st x in ( dom f) holds (F0 # x) is convergent & ( lim (F0 # x)) = (f . x) and for n be Nat holds (K0 . n) = ( integral' (M,(F0 . n))) and K0 is convergent and ( integral+ (M,f)) = ( lim K0) by A1, A2, Def15;

 deffunc PFA( Nat) = ((F0 . $1) | A);

 consider FA be Functional_Sequence of X, ExtREAL such that

 A8: for n be Nat holds (FA . n) = PFA(n) from SEQFUNC:sch 1;

 consider E be Element of S such that

 A9: E = ( dom f) and

 A10: f is E -measurable by A1;

 set C = (E /\ A);



 A11: (f | A) is nonnegative by A2, Th15;



 A12: (( dom f) /\ C) = C by A9, XBOOLE_1: 17, XBOOLE_1: 28;

 then

 A13: ( dom (f | C)) = C by RELAT_1: 61;

 then

 A14: ( dom (f | C)) = ( dom (f | A)) by A9, RELAT_1: 61;

 for x be object st x in ( dom (f | A)) holds ((f | A) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A15: x in ( dom (f | A));

 then ((f | A) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A14, A15, FUNCT_1: 47;

 end;

 then

 A16: (f | A) = (f | C) by A9, A13, FUNCT_1: 2, RELAT_1: 61;

 f is C -measurable by A10, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A17: (f | A) is C -measurable by A12, A16, Th42;



 A18: for n be Nat holds (FA . n) is nonnegative

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 ((F0 . n) | A) is nonnegative by A5, Th15;

 hence thesis by A8;

 end;

 deffunc PK( Nat) = ( integral' (M,(FA . $1)));

 consider KA be ExtREAL_sequence such that

 A19: for n be Element of NAT holds (KA . n) = PK(n) from FUNCT_2:sch 4;

 A20:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (KA . n) = PK(n) by A19;

 end;



 A21: for n be Nat holds (KA . n) = 0

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 (F0 . n) is_simple_func_in S by A4;

 then ( integral' (M,((F0 . n) | A))) = 0 by A3, A5, Th73;

 then ( integral' (M,(FA . n))) = 0 by A8;

 hence thesis by A20;

 end;

 then

 A22: ( lim KA) = 0 by Th60;



 A23: C = ( dom (f | A)) by A9, RELAT_1: 61;



 A24: for n be Nat holds (FA . n) is_simple_func_in S & ( dom (FA . n)) = ( dom (f | A))

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 (FA . n1) = ((F0 . n1) | A) by A8;

 hence (FA . n) is_simple_func_in S by A4, Th34;

 ( dom (FA . n1)) = ( dom ((F0 . n1) | A)) by A8;

 then ( dom (FA . n)) = (( dom (F0 . n)) /\ A) by RELAT_1: 61;

 hence thesis by A9, A4, A23;

 end;



 A25: for x be Element of X st x in ( dom (f | A)) holds (FA # x) is convergent & ( lim (FA # x)) = ((f | A) . x)

 proof

 let x be Element of X;

 assume

 A26: x in ( dom (f | A));

 now

 let n be Element of NAT ;



 A27: ( dom ((F0 . n) | A)) = ( dom (FA . n)) by A8

 .= ( dom (f | A)) by A24;



 thus ((FA # x) . n) = ((FA . n) . x) by Def13

 .= (((F0 . n) | A) . x) by A8

 .= ((F0 . n) . x) by A26, A27, FUNCT_1: 47

 .= ((F0 # x) . n) by Def13;

 end;

 then

 A28: (FA # x) = (F0 # x) by FUNCT_2: 63;

 x in (( dom f) /\ A) by A26, RELAT_1: 61;

 then

 A29: x in ( dom f) by XBOOLE_0:def 4;

 then ( lim (F0 # x)) = (f . x) by A7;

 hence thesis by A7, A26, A29, A28, FUNCT_1: 47;

 end;



 A30: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (f | A)) holds ((FA . n) . x) <= ((FA . m) . x)

 proof

 let n,m be Nat;

 assume

 A31: n <= m;

 let x be Element of X;

 reconsider n, m as Element of NAT by ORDINAL1:def 12;

 assume

 A32: x in ( dom (f | A));

 then x in (( dom f) /\ A) by RELAT_1: 61;

 then

 A33: x in ( dom f) by XBOOLE_0:def 4;

 ( dom ((F0 . m) | A)) = ( dom (FA . m)) by A8;

 then

 A34: ( dom ((F0 . m) | A)) = ( dom (f | A)) by A24;

 ((FA . m) . x) = (((F0 . m) | A) . x) by A8;

 then

 A35: ((FA . m) . x) = ((F0 . m) . x) by A32, A34, FUNCT_1: 47;

 ( dom ((F0 . n) | A)) = ( dom (FA . n)) by A8;

 then

 A36: ( dom ((F0 . n) | A)) = ( dom (f | A)) by A24;

 ((FA . n) . x) = (((F0 . n) | A) . x) by A8;

 then ((FA . n) . x) = ((F0 . n) . x) by A32, A36, FUNCT_1: 47;

 hence thesis by A6, A31, A33, A35;

 end;

 KA is convergent by A21, Th60;

 hence thesis by A17, A20, A23, A11, A24, A18, A30, A25, A22, Def15;

 end;

 theorem :: MESFUNC5:83



 Th83: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative & A c= B holds ( integral+ (M,(f | A))) <= ( integral+ (M,(f | B)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative and

 A3: A c= B;

 set A9 = (A /\ B);



 A4: A9 = A by A3, XBOOLE_1: 28;

 set B9 = (B \ A);



 A5: (A9 \/ B9) = B by XBOOLE_1: 51;

 ( integral+ (M,(f | (A9 \/ B9)))) = (( integral+ (M,(f | A9))) + ( integral+ (M,(f | B9)))) by A1, A2, Th81, XBOOLE_1: 89;

 hence thesis by A1, A2, A4, A5, Th80, XXREAL_3: 39;

 end;

 theorem :: MESFUNC5:84



 Th84: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , E,A be Element of S st f is nonnegative & E = ( dom f) & f is E -measurable & (M . A) = 0 holds ( integral+ (M,(f | (E \ A)))) = ( integral+ (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , E,A be Element of S such that

 A1: f is nonnegative and

 A2: E = ( dom f) and

 A3: f is E -measurable and

 A4: (M . A) = 0 ;

 set B = (E \ A);

 (A \/ B) = (A \/ E) by XBOOLE_1: 39;



 then

 A5: ( dom f) = (( dom f) /\ (A \/ B)) by A2, XBOOLE_1: 7, XBOOLE_1: 28

 .= ( dom (f | (A \/ B))) by RELAT_1: 61;

 for x be object st x in ( dom (f | (A \/ B))) holds ((f | (A \/ B)) . x) = (f . x) by FUNCT_1: 47;

 then

 A6: (f | (A \/ B)) = f by A5, FUNCT_1: 2;

 ( integral+ (M,(f | (A \/ B)))) = (( integral+ (M,(f | A))) + ( integral+ (M,(f | B)))) by A1, A2, A3, Th81, XBOOLE_1: 79;

 then ( integral+ (M,f)) = ( 0. + ( integral+ (M,(f | B)))) by A1, A2, A3, A4, A6, Th82;

 hence thesis by XXREAL_3: 4;

 end;

 theorem :: MESFUNC5:85



 Th85: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex E be Element of S st E = ( dom f) & E = ( dom g) & f is E -measurable & g is E -measurable) & f is nonnegative & g is nonnegative & (for x be Element of X st x in ( dom g) holds (g . x) <= (f . x)) holds ( integral+ (M,g)) <= ( integral+ (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: ex A be Element of S st A = ( dom f) & A = ( dom g) & f is A -measurable & g is A -measurable and

 A2: f is nonnegative and

 A3: g is nonnegative and

 A4: for x be Element of X st x in ( dom g) holds (g . x) <= (f . x);

 consider G be Functional_Sequence of X, ExtREAL , L be ExtREAL_sequence such that

 A5: for n be Nat holds (G . n) is_simple_func_in S & ( dom (G . n)) = ( dom g) and

 A6: for n be Nat holds (G . n) is nonnegative and

 A7: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom g) holds ((G . n) . x) <= ((G . m) . x) and

 A8: for x be Element of X st x in ( dom g) holds (G # x) is convergent & ( lim (G # x)) = (g . x) and

 A9: for n be Nat holds (L . n) = ( integral' (M,(G . n))) and L is convergent and

 A10: ( integral+ (M,g)) = ( lim L) by A1, A3, Def15;

 consider F be Functional_Sequence of X, ExtREAL , K be ExtREAL_sequence such that

 A11: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

 A12: for n be Nat holds (F . n) is nonnegative and

 A13: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F . n) . x) <= ((F . m) . x) and

 A14: for x be Element of X st x in ( dom f) holds (F # x) is convergent & ( lim (F # x)) = (f . x) and

 A15: for n be Nat holds (K . n) = ( integral' (M,(F . n))) and K is convergent and

 A16: ( integral+ (M,f)) = ( lim K) by A1, A2, Def15;

 consider A be Element of S such that

 A17: A = ( dom f) and

 A18: A = ( dom g) and f is A -measurable and g is A -measurable by A1;



 A19: for x be Element of X st x in A holds ( lim (G # x)) = ( sup ( rng (G # x)))

 proof

 let x be Element of X;

 assume

 A20: x in A;

 now

 let n,m be Nat;

 assume

 A21: n <= m;



 A22: ((G # x) . m) = ((G . m) . x) by Def13;

 ((G # x) . n) = ((G . n) . x) by Def13;

 hence ((G # x) . n) <= ((G # x) . m) by A18, A7, A20, A21, A22;

 end;

 hence thesis by Th54;

 end;



 A23: for n0 be Nat holds L is convergent & ( sup ( rng L)) = ( lim L)

 proof

 let n0 be Nat;

 set ff = (G . n0);



 A24: ( dom ff) = A by A18, A5;



 A25: for x be Element of X st x in ( dom ff) holds (G # x) is convergent & (ff . x) <= ( lim (G # x))

 proof

 let x be Element of X such that

 A26: x in ( dom ff);



 A27: ((G # x) . n0) <= ( sup ( rng (G # x))) by Th56;

 (ff . x) = ((G # x) . n0) by Def13;

 hence thesis by A18, A8, A19, A24, A26, A27;

 end;

 ff is_simple_func_in S by A5;

 then

 consider FF be ExtREAL_sequence such that

 A28: for n be Nat holds (FF . n) = ( integral' (M,(G . n))) and

 A29: FF is convergent and

 A30: ( sup ( rng FF)) = ( lim FF) and ( integral' (M,ff)) <= ( lim FF) by A18, A5, A6, A7, A24, A25, Th75;

 now

 let n be Element of NAT ;



 thus (FF . n) = ( integral' (M,(G . n))) by A28

 .= (L . n) by A9;

 end;

 then FF = L by FUNCT_2: 63;

 hence thesis by A29, A30;

 end;

 for n0 be Nat holds K is convergent & ( sup ( rng K)) = ( lim K) & (L . n0) <= ( lim K)

 proof

 let n0 be Nat;

 set gg = (G . n0);



 A31: gg is nonnegative by A6;



 A32: ( dom gg) = A by A18, A5;



 A33: for x be Element of X st x in ( dom gg) holds (F # x) is convergent & (gg . x) <= ( lim (F # x))

 proof

 let x be Element of X such that

 A34: x in ( dom gg);



 A35: ((G # x) . n0) <= ( sup ( rng (G # x))) by Th56;

 (gg . x) = ((G # x) . n0) by Def13;

 then (gg . x) <= ( lim (G # x)) by A19, A32, A34, A35;

 then

 A36: (gg . x) <= (g . x) by A18, A8, A32, A34;

 (g . x) <= (f . x) by A1, A4, A17, A32, A34;

 then (g . x) <= ( lim (F # x)) by A17, A14, A32, A34;

 hence thesis by A17, A14, A32, A34, A36, XXREAL_0: 2;

 end;

 gg is_simple_func_in S by A5;

 then

 consider GG be ExtREAL_sequence such that

 A37: for n be Nat holds (GG . n) = ( integral' (M,(F . n))) and

 A38: GG is convergent and

 A39: ( sup ( rng GG)) = ( lim GG) and

 A40: ( integral' (M,gg)) <= ( lim GG) by A17, A11, A12, A13, A32, A31, A33, Th75;

 now

 let n be Element of NAT ;

 (GG . n) = ( integral' (M,(F . n))) by A37;

 hence (GG . n) = (K . n) by A15;

 end;

 then GG = K by FUNCT_2: 63;

 hence thesis by A9, A38, A39, A40;

 end;

 hence thesis by A16, A10, A23, Th57;

 end;

 theorem :: MESFUNC5:86



 Th86: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real st 0 <= c & (ex A be Element of S st A = ( dom f) & f is A -measurable) & f is nonnegative holds ( integral+ (M,(c (#) f))) = (c * ( integral+ (M,f)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real such that

 A1: 0 <= c and

 A2: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A3: f is nonnegative;

 consider F1 be Functional_Sequence of X, ExtREAL , K1 be ExtREAL_sequence such that

 A4: for n be Nat holds (F1 . n) is_simple_func_in S & ( dom (F1 . n)) = ( dom f) and

 A5: for n be Nat holds (F1 . n) is nonnegative and

 A6: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom f) holds ((F1 . n) . x) <= ((F1 . m) . x) and

 A7: for x be Element of X st x in ( dom f) holds (F1 # x) is convergent & ( lim (F1 # x)) = (f . x) and

 A8: for n be Nat holds (K1 . n) = ( integral' (M,(F1 . n))) and K1 is convergent and

 A9: ( integral+ (M,f)) = ( lim K1) by A2, A3, Def15;

 deffunc PF( Nat) = (c (#) (F1 . $1));

 consider F be Functional_Sequence of X, ExtREAL such that

 A10: for n be Nat holds (F . n) = PF(n) from SEQFUNC:sch 1;



 A11: (c (#) f) is nonnegative by A1, A3, Th20;



 A12: for n be Nat holds (F . n) is nonnegative

 proof

 let n be Nat;

 reconsider n as Element of NAT by ORDINAL1:def 12;

 (F1 . n) is nonnegative by A5;

 then (c (#) (F1 . n)) is nonnegative by A1, Th20;

 hence thesis by A10;

 end;

 consider A be Element of S such that

 A13: A = ( dom f) and

 A14: f is A -measurable by A2;



 A15: (c (#) f) is A -measurable by A13, A14, MESFUNC1: 37;



 A16: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom (c (#) f))

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 A17: (F . n1) = (c (#) (F1 . n1)) by A10;

 hence (F . n) is_simple_func_in S by A4, Th39;



 thus ( dom (F . n)) = ( dom (F1 . n)) by A17, MESFUNC1:def 6

 .= A by A4, A13

 .= ( dom (c (#) f)) by A13, MESFUNC1:def 6;

 end;



 A18: for n,m be Nat st n <= m holds (K1 . n) <= (K1 . m)

 proof

 let n,m be Nat;



 A19: (K1 . n) = ( integral' (M,(F1 . n))) by A8;



 A20: (K1 . m) = ( integral' (M,(F1 . m))) by A8;



 A21: (F1 . m) is_simple_func_in S by A4;



 A22: (F1 . n) is nonnegative by A5;



 A23: ( dom (F1 . n)) = ( dom f) by A4;



 A24: (F1 . m) is nonnegative by A5;



 A25: ( dom (F1 . m)) = ( dom f) by A4;

 assume

 A26: n <= m;



 A27: for x be object st x in ( dom ((F1 . m) - (F1 . n))) holds ((F1 . n) . x) <= ((F1 . m) . x)

 proof

 let x be object;

 assume x in ( dom ((F1 . m) - (F1 . n)));

 then x in ((( dom (F1 . m)) /\ ( dom (F1 . n))) \ ((((F1 . m) " { +infty }) /\ ((F1 . n) " { +infty })) \/ (((F1 . m) " { -infty }) /\ ((F1 . n) " { -infty })))) by MESFUNC1:def 4;

 then x in (( dom (F1 . m)) /\ ( dom (F1 . n))) by XBOOLE_0:def 5;

 hence thesis by A6, A26, A23, A25;

 end;



 A28: (F1 . n) is_simple_func_in S by A4;

 then

 A29: ( dom ((F1 . m) - (F1 . n))) = (( dom (F1 . m)) /\ ( dom (F1 . n))) by A21, A22, A24, A27, Th69;

 then

 A30: ((F1 . m) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . m) by A23, A25, GRFUNC_1: 23;

 ((F1 . n) | ( dom ((F1 . m) - (F1 . n)))) = (F1 . n) by A23, A25, A29, GRFUNC_1: 23;

 hence thesis by A19, A20, A28, A21, A22, A24, A27, A30, Th70;

 end;

 deffunc PK( Nat) = ( integral' (M,(F . $1)));

 consider K be ExtREAL_sequence such that

 A31: for n be Element of NAT holds (K . n) = PK(n) from FUNCT_2:sch 4;

 A32:

 now

 let n be Nat;

 n in NAT by ORDINAL1:def 12;

 hence (K . n) = PK(n) by A31;

 end;



 A33: for n be Nat holds (K . n) = (c * (K1 . n))

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 A34: (F1 . n) is_simple_func_in S by A4;



 A35: (F . n1) = (c (#) (F1 . n1)) by A10;



 thus (K . n) = ( integral' (M,(F . n1))) by A32

 .= (c * ( integral' (M,(F1 . n)))) by A1, A5, A34, A35, Th66

 .= (c * (K1 . n)) by A8;

 end;



 A36: A = ( dom (c (#) f)) by A13, MESFUNC1:def 6;



 A37: for x be Element of X st x in ( dom (c (#) f)) holds (F # x) is convergent & ( lim (F # x)) = ((c (#) f) . x)

 proof

 let x be Element of X;

 now

 let n1 be set;

 assume n1 in ( dom (F1 # x));

 then

 reconsider n = n1 as Element of NAT ;



 A38: ((F1 # x) . n) = ((F1 . n) . x) by Def13;

 (F1 . n) is nonnegative by A5;

 hence -infty < ((F1 # x) . n1) by A38, Def5;

 end;

 then

 A39: (F1 # x) is without-infty by Th10;

 assume

 A40: x in ( dom (c (#) f));

 A41:

 now

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 A42: ( dom (c (#) (F1 . n1))) = ( dom (F . n1)) by A10

 .= ( dom (c (#) f)) by A16;



 thus ((F # x) . n) = ((F . n) . x) by Def13

 .= ((c (#) (F1 . n1)) . x) by A10

 .= (c * ((F1 . n) . x)) by A40, A42, MESFUNC1:def 6

 .= (c * ((F1 # x) . n)) by Def13;

 end;

 A43:

 now

 let n,m be Nat;

 assume

 A44: n <= m;



 A45: ((F1 # x) . m) = ((F1 . m) . x) by Def13;

 ((F1 # x) . n) = ((F1 . n) . x) by Def13;

 hence ((F1 # x) . n) <= ((F1 # x) . m) by A6, A13, A36, A40, A44, A45;

 end;

 (c * ( lim (F1 # x))) = (c * (f . x)) by A7, A13, A36, A40

 .= ((c (#) f) . x) by A40, MESFUNC1:def 6;

 hence thesis by A1, A41, A39, A43, Th63;

 end;

 now

 let n1 be set;

 assume n1 in ( dom K1);

 then

 reconsider n = n1 as Element of NAT ;



 A46: (F1 . n) is_simple_func_in S by A4;

 (K1 . n) = ( integral' (M,(F1 . n))) by A8;

 hence -infty < (K1 . n1) by A5, A46, Th68;

 end;

 then

 A47: K1 is without-infty by Th10;

 then

 A48: ( lim K) = (c * ( lim K1)) by A1, A18, A33, Th63;



 A49: for n,m be Nat st n <= m holds for x be Element of X st x in ( dom (c (#) f)) holds ((F . n) . x) <= ((F . m) . x)

 proof

 let n,m be Nat;

 assume

 A50: n <= m;

 reconsider n, m as Element of NAT by ORDINAL1:def 12;

 let x be Element of X;

 assume

 A51: x in ( dom (c (#) f));

 ( dom (c (#) (F1 . m))) = ( dom (F . m)) by A10;

 then

 A52: ( dom (c (#) (F1 . m))) = ( dom (c (#) f)) by A16;

 ((F . m) . x) = ((c (#) (F1 . m)) . x) by A10;

 then

 A53: ((F . m) . x) = (c * ((F1 . m) . x)) by A51, A52, MESFUNC1:def 6;

 ( dom (c (#) (F1 . n))) = ( dom (F . n)) by A10;

 then

 A54: ( dom (c (#) (F1 . n))) = ( dom (c (#) f)) by A16;

 ((F . n) . x) = ((c (#) (F1 . n)) . x) by A10;

 then ((F . n) . x) = (c * ((F1 . n) . x)) by A51, A54, MESFUNC1:def 6;

 hence thesis by A1, A6, A13, A36, A50, A51, A53, XXREAL_3: 71;

 end;

 K is convergent by A1, A47, A18, A33, Th63;

 hence thesis by A9, A36, A15, A11, A32, A16, A12, A49, A37, A48, Def15;

 end;

 theorem :: MESFUNC5:87



 Th87: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & (for x be Element of X st x in ( dom f) holds 0 = (f . x)) holds ( integral+ (M,f)) = 0

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL such that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: for x be Element of X st x in ( dom f) holds 0 = (f . x);



 A3: for x be object st x in ( dom f) holds 0 <= (f . x) by A2;



 A4: ( dom ( 0 (#) f)) = ( dom f) by MESFUNC1:def 6;

 now

 let x be object;

 assume

 A5: x in ( dom ( 0 (#) f));



 hence (( 0 (#) f) . x) = ( 0 * (f . x)) by MESFUNC1:def 6

 .= 0

 .= (f . x) by A2, A4, A5;

 end;



 hence ( integral+ (M,f)) = ( integral+ (M,( 0 (#) f))) by A4, FUNCT_1: 2

 .= ( 0 * ( integral+ (M,f))) by A1, A3, Th86, SUPINF_2: 52

 .= 0 ;

 end;

 begin

 definition

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 ::$Notion-Name

 ::$Notion-Name

 :: MESFUNC5:def16

 func Integral (M,f) -> Element of ExtREAL equals (( integral+ (M,( max+ f))) - ( integral+ (M,( max- f))));

 coherence ;

 end

 theorem :: MESFUNC5:88



 Th88: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & f is nonnegative holds ( Integral (M,f)) = ( integral+ (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: f is nonnegative;



 A3: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 A4:

 now

 let x be object;



 A5: 0 <= (f . x) by A2, SUPINF_2: 51;

 assume x in ( dom f);



 hence (( max+ f) . x) = ( max ((f . x), 0 )) by A3, MESFUNC2:def 2

 .= (f . x) by A5, XXREAL_0:def 10;

 end;



 A6: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 A7:

 now

 let x be Element of X;

 assume x in ( dom ( max- f));

 then (( max+ f) . x) = (f . x) by A4, A6;

 hence (( max- f) . x) = 0 by MESFUNC2: 19;

 end;



 A8: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 f = ( max+ f) by A3, A4, FUNCT_1: 2;



 hence ( Integral (M,f)) = (( integral+ (M,f)) - 0 ) by A1, A7, A8, Th87, MESFUNC2: 26

 .= ( integral+ (M,f)) by XXREAL_3: 15;

 end;

 theorem :: MESFUNC5:89

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_simple_func_in S & f is nonnegative holds ( Integral (M,f)) = ( integral+ (M,f)) & ( Integral (M,f)) = ( integral' (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_simple_func_in S and

 A2: f is nonnegative;

 reconsider A = ( dom f) as Element of S by A1, Th37;

 f is A -measurable by A1, MESFUNC2: 34;

 hence ( Integral (M,f)) = ( integral+ (M,f)) by A2, Th88;

 hence thesis by A1, A2, Th77;

 end;

 theorem :: MESFUNC5:90

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & f is nonnegative holds 0 <= ( Integral (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: f is nonnegative;

 0 <= ( integral+ (M,f)) by A1, A2, Th79;

 hence thesis by A1, A2, Th88;

 end;

 theorem :: MESFUNC5:91

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative & A misses B holds ( Integral (M,(f | (A \/ B)))) = (( Integral (M,(f | A))) + ( Integral (M,(f | B))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative and

 A3: A misses B;

 consider E be Element of S such that

 A4: E = ( dom f) and

 A5: f is E -measurable by A1;

 ex C be Element of S st C = ( dom (f | A)) & (f | A) is C -measurable

 proof

 take C = (E /\ A);

 thus ( dom (f | A)) = C by A4, RELAT_1: 61;



 A6: C = (( dom f) /\ C) by A4, XBOOLE_1: 17, XBOOLE_1: 28;



 A7: ( dom (f | A)) = C by A4, RELAT_1: 61

 .= ( dom (f | C)) by A6, RELAT_1: 61;

 for x be object st x in ( dom (f | A)) holds ((f | A) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A8: x in ( dom (f | A));

 then ((f | A) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A7, A8, FUNCT_1: 47;

 end;

 then

 A9: (f | C) = (f | A) by A7, FUNCT_1: 2;

 f is C -measurable by A5, MESFUNC1: 30, XBOOLE_1: 17;

 hence thesis by A6, A9, Th42;

 end;

 then

 A10: ( Integral (M,(f | A))) = ( integral+ (M,(f | A))) by A2, Th15, Th88;

 ex C be Element of S st C = ( dom (f | (A \/ B))) & (f | (A \/ B)) is C -measurable

 proof

 reconsider C = (E /\ (A \/ B)) as Element of S;

 take C;

 thus ( dom (f | (A \/ B))) = C by A4, RELAT_1: 61;



 A11: C = (( dom f) /\ C) by A4, XBOOLE_1: 17, XBOOLE_1: 28;



 A12: ( dom (f | (A \/ B))) = C by A4, RELAT_1: 61

 .= ( dom (f | C)) by A11, RELAT_1: 61;



 A13: for x be object st x in ( dom (f | (A \/ B))) holds ((f | (A \/ B)) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A14: x in ( dom (f | (A \/ B)));

 then ((f | (A \/ B)) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A12, A14, FUNCT_1: 47;

 end;

 f is C -measurable by A5, MESFUNC1: 30, XBOOLE_1: 17;

 then (f | C) is C -measurable by A11, Th42;

 hence thesis by A12, A13, FUNCT_1: 2;

 end;

 then

 A15: ( Integral (M,(f | (A \/ B)))) = ( integral+ (M,(f | (A \/ B)))) by A2, Th15, Th88;



 A16: ex C be Element of S st C = ( dom (f | B)) & (f | B) is C -measurable

 proof

 take C = (E /\ B);

 thus ( dom (f | B)) = C by A4, RELAT_1: 61;



 A17: C = (( dom f) /\ C) by A4, XBOOLE_1: 17, XBOOLE_1: 28;



 A18: ( dom (f | B)) = C by A4, RELAT_1: 61

 .= ( dom (f | C)) by A17, RELAT_1: 61;

 for x be object st x in ( dom (f | B)) holds ((f | B) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A19: x in ( dom (f | B));

 then ((f | B) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A18, A19, FUNCT_1: 47;

 end;

 then

 A20: (f | C) = (f | B) by A18, FUNCT_1: 2;

 f is C -measurable by A5, MESFUNC1: 30, XBOOLE_1: 17;

 hence thesis by A17, A20, Th42;

 end;

 ( integral+ (M,(f | (A \/ B)))) = (( integral+ (M,(f | A))) + ( integral+ (M,(f | B)))) by A1, A2, A3, Th81;

 hence thesis by A2, A15, A10, A16, Th15, Th88;

 end;

 theorem :: MESFUNC5:92

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative holds 0 <= ( Integral (M,(f | A)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative;

 consider E be Element of S such that

 A3: E = ( dom f) and

 A4: f is E -measurable by A1;



 A5: ex C be Element of S st C = ( dom (f | A)) & (f | A) is C -measurable

 proof

 take C = (E /\ A);

 thus ( dom (f | A)) = C by A3, RELAT_1: 61;



 A6: C = (( dom f) /\ C) by A3, XBOOLE_1: 17, XBOOLE_1: 28;



 A7: ( dom (f | A)) = C by A3, RELAT_1: 61

 .= ( dom (f | C)) by A6, RELAT_1: 61;



 A8: for x be object st x in ( dom (f | A)) holds ((f | A) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A9: x in ( dom (f | A));

 then ((f | A) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A7, A9, FUNCT_1: 47;

 end;

 f is C -measurable by A4, MESFUNC1: 30, XBOOLE_1: 17;

 then (f | C) is C -measurable by A6, Th42;

 hence thesis by A7, A8, FUNCT_1: 2;

 end;

 then 0 <= ( integral+ (M,(f | A))) by A2, Th15, Th79;

 hence thesis by A2, A5, Th15, Th88;

 end;

 theorem :: MESFUNC5:93

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & f is nonnegative & A c= B holds ( Integral (M,(f | A))) <= ( Integral (M,(f | B)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S;

 assume that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: f is nonnegative and

 A3: A c= B;

 consider E be Element of S such that

 A4: E = ( dom f) and

 A5: f is E -measurable by A1;



 A6: ex C be Element of S st C = ( dom (f | A)) & (f | A) is C -measurable

 proof

 take C = (E /\ A);

 thus ( dom (f | A)) = C by A4, RELAT_1: 61;



 A7: C = (( dom f) /\ C) by A4, XBOOLE_1: 17, XBOOLE_1: 28;



 A8: ( dom (f | A)) = C by A4, RELAT_1: 61

 .= ( dom (f | C)) by A7, RELAT_1: 61;



 A9: for x be object st x in ( dom (f | A)) holds ((f | A) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A10: x in ( dom (f | A));

 then ((f | A) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A8, A10, FUNCT_1: 47;

 end;

 f is C -measurable by A5, MESFUNC1: 30, XBOOLE_1: 17;

 then (f | C) is C -measurable by A7, Th42;

 hence thesis by A8, A9, FUNCT_1: 2;

 end;



 A11: ex C be Element of S st C = ( dom (f | B)) & (f | B) is C -measurable

 proof

 take C = (E /\ B);

 thus ( dom (f | B)) = C by A4, RELAT_1: 61;



 A12: C = (( dom f) /\ C) by A4, XBOOLE_1: 17, XBOOLE_1: 28;



 A13: ( dom (f | B)) = C by A4, RELAT_1: 61

 .= ( dom (f | C)) by A12, RELAT_1: 61;



 A14: for x be object st x in ( dom (f | B)) holds ((f | B) . x) = ((f | C) . x)

 proof

 let x be object;

 assume

 A15: x in ( dom (f | B));

 then ((f | B) . x) = (f . x) by FUNCT_1: 47;

 hence thesis by A13, A15, FUNCT_1: 47;

 end;

 f is C -measurable by A5, MESFUNC1: 30, XBOOLE_1: 17;

 then (f | C) is C -measurable by A12, Th42;

 hence thesis by A13, A14, FUNCT_1: 2;

 end;

 ( integral+ (M,(f | A))) <= ( integral+ (M,(f | B))) by A1, A2, A3, Th83;

 then ( Integral (M,(f | A))) <= ( integral+ (M,(f | B))) by A2, A6, Th15, Th88;

 hence thesis by A2, A11, Th15, Th88;

 end;

 theorem :: MESFUNC5:94

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st (ex E be Element of S st E = ( dom f) & f is E -measurable) & (M . A) = 0 holds ( Integral (M,(f | A))) = 0

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S such that

 A1: ex E be Element of S st E = ( dom f) & f is E -measurable and

 A2: (M . A) = 0 ;



 A3: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 ( max+ f) is nonnegative by Lm1;

 then

 A4: ( integral+ (M,(( max+ f) | A))) = 0 by A1, A2, A3, Th82, MESFUNC2: 25;



 A5: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;



 A6: ( max- f) is nonnegative by Lm1;

 ( Integral (M,(f | A))) = (( integral+ (M,(( max+ f) | A))) - ( integral+ (M,( max- (f | A))))) by Th28

 .= (( integral+ (M,(( max+ f) | A))) - ( integral+ (M,(( max- f) | A)))) by Th28

 .= ( 0. - 0. ) by A1, A2, A5, A6, A4, Th82, MESFUNC2: 26;

 hence thesis;

 end;

 theorem :: MESFUNC5:95



 Th95: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , E,A be Element of S st E = ( dom f) & f is E -measurable & (M . A) = 0 holds ( Integral (M,(f | (E \ A)))) = ( Integral (M,f))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , E,A be Element of S such that

 A1: E = ( dom f) and

 A2: f is E -measurable and

 A3: (M . A) = 0 ;

 set B = (E \ A);



 A4: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;



 A5: ( max- f) is nonnegative by Lm1;



 A6: ( max+ f) is nonnegative by Lm1;



 A7: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 ( Integral (M,(f | B))) = (( integral+ (M,(( max+ f) | B))) - ( integral+ (M,( max- (f | B))))) by Th28

 .= (( integral+ (M,(( max+ f) | B))) - ( integral+ (M,(( max- f) | B)))) by Th28

 .= (( integral+ (M,( max+ f))) - ( integral+ (M,(( max- f) | B)))) by A1, A2, A3, A4, A6, Th84, MESFUNC2: 25;

 hence thesis by A1, A2, A3, A7, A5, Th84, MESFUNC2: 26;

 end;

 definition

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 :: MESFUNC5:def17

 pred f is_integrable_on M means (ex A be Element of S st A = ( dom f) & f is A -measurable) & ( integral+ (M,( max+ f))) < +infty & ( integral+ (M,( max- f))) < +infty ;

 end

 theorem :: MESFUNC5:96



 Th96: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_integrable_on M holds 0 <= ( integral+ (M,( max+ f))) & 0 <= ( integral+ (M,( max- f))) & -infty < ( Integral (M,f)) & ( Integral (M,f)) < +infty

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL such that

 A1: f is_integrable_on M;

 consider A be Element of S such that

 A2: A = ( dom f) and

 A3: f is A -measurable by A1;



 A4: ( integral+ (M,( max+ f))) <> +infty by A1;



 A5: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;



 A6: ( max+ f) is nonnegative by Lm1;

 then -infty <> ( integral+ (M,( max+ f))) by A2, A3, A5, Th79, MESFUNC2: 25;

 then

 reconsider maxf1 = ( integral+ (M,( max+ f))) as Element of REAL by A4, XXREAL_0: 14;



 A7: ( max+ f) is A -measurable by A3, MESFUNC2: 25;



 A8: ( integral+ (M,( max- f))) <> +infty by A1;



 A9: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;



 A10: ( max- f) is nonnegative by Lm1;

 then -infty <> ( integral+ (M,( max- f))) by A2, A3, A9, Th79, MESFUNC2: 26;

 then

 reconsider maxf2 = ( integral+ (M,( max- f))) as Element of REAL by A8, XXREAL_0: 14;

 (( integral+ (M,( max+ f))) - ( integral+ (M,( max- f)))) = (maxf1 - maxf2) by SUPINF_2: 3;

 hence thesis by A2, A3, A5, A9, A6, A10, A7, Th79, MESFUNC2: 26, XXREAL_0: 9, XXREAL_0: 12;

 end;

 theorem :: MESFUNC5:97



 Th97: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S st f is_integrable_on M holds ( integral+ (M,( max+ (f | A)))) <= ( integral+ (M,( max+ f))) & ( integral+ (M,( max- (f | A)))) <= ( integral+ (M,( max- f))) & (f | A) is_integrable_on M

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A be Element of S;



 A1: ( max+ f) is nonnegative by Lm1;

 assume

 A2: f is_integrable_on M;

 then

 consider E be Element of S such that

 A3: E = ( dom f) and

 A4: f is E -measurable;



 A5: ( max+ f) is E -measurable by A4, MESFUNC2: 25;



 A6: f is (E /\ A) -measurable by A4, MESFUNC1: 30, XBOOLE_1: 17;

 (( dom f) /\ (E /\ A)) = (E /\ A) by A3, XBOOLE_1: 17, XBOOLE_1: 28;

 then (f | (E /\ A)) is (E /\ A) -measurable by A6, Th42;

 then ((f | E) | A) is (E /\ A) -measurable by RELAT_1: 71;

 then

 A7: (f | A) is (E /\ A) -measurable by A3, GRFUNC_1: 23;



 A8: ( integral+ (M,( max- f))) < +infty by A2;



 A9: ( max- f) is nonnegative by Lm1;



 A10: ( integral+ (M,( max+ f))) < +infty by A2;



 A11: (( max+ f) | (E /\ A)) = ((( max+ f) | E) | A) by RELAT_1: 71;



 A12: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 then (( max+ f) | E) = ( max+ f) by A3, GRFUNC_1: 23;

 then

 A13: ( integral+ (M,(( max+ f) | A))) <= ( integral+ (M,( max+ f))) by A3, A5, A12, A1, A11, Th83, XBOOLE_1: 17;

 then ( integral+ (M,( max+ (f | A)))) <= ( integral+ (M,( max+ f))) by Th28;

 then

 A14: ( integral+ (M,( max+ (f | A)))) < +infty by A10, XXREAL_0: 2;



 A15: ( max- f) is E -measurable by A3, A4, MESFUNC2: 26;



 A16: (( max- f) | (E /\ A)) = ((( max- f) | E) | A) by RELAT_1: 71;



 A17: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 then (( max- f) | E) = ( max- f) by A3, GRFUNC_1: 23;

 then

 A18: ( integral+ (M,(( max- f) | A))) <= ( integral+ (M,( max- f))) by A3, A15, A17, A9, A16, Th83, XBOOLE_1: 17;

 then ( integral+ (M,( max- (f | A)))) <= ( integral+ (M,( max- f))) by Th28;

 then

 A19: ( integral+ (M,( max- (f | A)))) < +infty by A8, XXREAL_0: 2;

 (E /\ A) = ( dom (f | A)) by A3, RELAT_1: 61;

 hence thesis by A13, A18, A7, A14, A19, Th28;

 end;

 theorem :: MESFUNC5:98



 Th98: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st f is_integrable_on M & A misses B holds ( Integral (M,(f | (A \/ B)))) = (( Integral (M,(f | A))) + ( Integral (M,(f | B))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S;

 assume that

 A1: f is_integrable_on M and

 A2: A misses B;

 consider E be Element of S such that

 A3: E = ( dom f) and

 A4: f is E -measurable by A1;

 set AB = (E /\ (A \/ B));



 A5: ( max+ (f | A)) = (( max+ f) | A) by Th28;



 A6: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 then (( max- f) | (A \/ B)) = ((( max- f) | E) | (A \/ B)) by A3, GRFUNC_1: 23;

 then

 A7: (( max- f) | (A \/ B)) = (( max- f) | AB) by RELAT_1: 71;

 ( max- f) is nonnegative by Lm1;

 then

 A8: ( integral+ (M,(( max- f) | (A \/ B)))) = (( integral+ (M,(( max- f) | A))) + ( integral+ (M,(( max- f) | B)))) by A2, A3, A4, A6, Th81, MESFUNC2: 26;



 A9: (f | A) is_integrable_on M by A1, Th97;

 then

 A10: 0 <= ( integral+ (M,( max+ (f | A)))) by Th96;



 A11: (f | B) is_integrable_on M by A1, Th97;

 then

 A12: 0 <= ( integral+ (M,( max+ (f | B)))) by Th96;



 A13: 0 <= ( integral+ (M,( max- (f | B)))) by A11, Th96;

 ( integral+ (M,( max- (f | B)))) < +infty by A11;

 then

 reconsider g2 = ( integral+ (M,( max- (f | B)))) as Element of REAL by A13, XXREAL_0: 14;

 ( integral+ (M,( max+ (f | B)))) < +infty by A11;

 then

 reconsider g1 = ( integral+ (M,( max+ (f | B)))) as Element of REAL by A12, XXREAL_0: 14;



 A14: (( integral+ (M,( max+ (f | B)))) - ( integral+ (M,( max- (f | B))))) = (g1 - g2) by SUPINF_2: 3;



 A15: ( max- (f | A)) = (( max- f) | A) by Th28;



 A16: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 then (( max+ f) | (A \/ B)) = ((( max+ f) | E) | (A \/ B)) by A3, GRFUNC_1: 23;

 then

 A17: (( max+ f) | (A \/ B)) = (( max+ f) | AB) by RELAT_1: 71;

 ( max+ f) is nonnegative by Lm1;

 then

 A18: ( integral+ (M,(( max+ f) | (A \/ B)))) = (( integral+ (M,(( max+ f) | A))) + ( integral+ (M,(( max+ f) | B)))) by A2, A3, A4, A16, Th81, MESFUNC2: 25;



 A19: ( max- (f | B)) = (( max- f) | B) by Th28;



 A20: ( max+ (f | B)) = (( max+ f) | B) by Th28;

 ( integral+ (M,( max+ (f | A)))) < +infty by A9;

 then

 reconsider f1 = ( integral+ (M,( max+ (f | A)))) as Element of REAL by A10, XXREAL_0: 14;



 A21: (( integral+ (M,( max+ (f | A)))) + ( integral+ (M,( max+ (f | B))))) = (f1 + g1) by SUPINF_2: 1;



 A22: 0 <= ( integral+ (M,( max- (f | A)))) by A9, Th96;

 ( integral+ (M,( max- (f | A)))) < +infty by A9;

 then

 reconsider f2 = ( integral+ (M,( max- (f | A)))) as Element of REAL by A22, XXREAL_0: 14;



 A23: (( integral+ (M,( max- (f | A)))) + ( integral+ (M,( max- (f | B))))) = (f2 + g2) by SUPINF_2: 1;

 ( Integral (M,(f | (A \/ B)))) = ( Integral (M,((f | E) | (A \/ B)))) by A3, GRFUNC_1: 23

 .= ( Integral (M,(f | AB))) by RELAT_1: 71

 .= (( integral+ (M,(( max+ f) | AB))) - ( integral+ (M,( max- (f | AB))))) by Th28

 .= (( integral+ (M,(( max+ f) | AB))) - ( integral+ (M,(( max- f) | AB)))) by Th28;

 then ( Integral (M,(f | (A \/ B)))) = ((f1 + g1) - (f2 + g2)) by A18, A8, A17, A7, A5, A15, A20, A19, A21, A23, SUPINF_2: 3;

 then

 A24: ( Integral (M,(f | (A \/ B)))) = ((f1 - f2) + (g1 - g2));

 (( integral+ (M,( max+ (f | A)))) - ( integral+ (M,( max- (f | A))))) = (f1 - f2) by SUPINF_2: 3;

 hence thesis by A24, A14, SUPINF_2: 1;

 end;

 theorem :: MESFUNC5:99

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S st f is_integrable_on M & B = (( dom f) \ A) holds (f | A) is_integrable_on M & ( Integral (M,f)) = (( Integral (M,(f | A))) + ( Integral (M,(f | B))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , A,B be Element of S such that

 A1: f is_integrable_on M and

 A2: B = (( dom f) \ A);

 (A \/ B) = (A \/ ( dom f)) by A2, XBOOLE_1: 39;

 then

 A3: (( dom f) /\ (A \/ B)) = ( dom f) by XBOOLE_1: 7, XBOOLE_1: 28;



 A4: (f | (A \/ B)) = ((f | ( dom f)) | (A \/ B)) by GRFUNC_1: 23

 .= (f | (( dom f) /\ (A \/ B))) by RELAT_1: 71

 .= f by A3, GRFUNC_1: 23;

 A misses B by A2, XBOOLE_1: 79;

 hence thesis by A1, A4, Th97, Th98;

 end;

 theorem :: MESFUNC5:100



 Th100: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) holds f is_integrable_on M iff |.f.| is_integrable_on M

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL ;



 A1: ( dom |.f.|) = ( dom ( max- |.f.|)) by MESFUNC2:def 3;



 A2: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;

 A3:

 now

 let x be object;

 assume x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 hence 0 <= ( |.f.| . x) by EXTREAL1: 14;

 end;



 A4: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;



 A5: |.f.| = (( max+ f) + ( max- f)) by MESFUNC2: 24;



 A6: ( max+ f) is nonnegative by Lm1;

 assume

 A7: ex A be Element of S st A = ( dom f) & f is A -measurable;

 then

 consider A be Element of S such that

 A8: A = ( dom f) and

 A9: f is A -measurable;



 A10: ( max- f) is A -measurable by A8, A9, MESFUNC2: 26;



 A11: |.f.| is A -measurable by A8, A9, MESFUNC2: 27;



 A12: A = ( dom |.f.|) by A8, MESFUNC1:def 10;



 A13: ( max+ f) is A -measurable by A9, MESFUNC2: 25;



 A14: ( dom |.f.|) = ( dom ( max+ |.f.|)) by MESFUNC2:def 2;

 hereby

 A15:

 now

 let x be object;

 assume

 A16: x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 then

 A17: 0 <= ( |.f.| . x) by EXTREAL1: 14;

 (( max+ |.f.|) . x) = ( max (( |.f.| . x), 0 )) by A14, A16, MESFUNC2:def 2;

 hence (( max+ |.f.|) . x) = ( |.f.| . x) by A17, XXREAL_0:def 10;

 end;

 now

 let x be Element of X;

 assume x in ( dom ( max- |.f.|));

 then (( max+ |.f.|) . x) = ( |.f.| . x) by A1, A15;

 hence (( max- |.f.|) . x) = 0 by MESFUNC2: 19;

 end;

 then

 A18: ( integral+ (M,( max- |.f.|))) = 0 by A1, A12, A11, Th87, MESFUNC2: 26;

 ( max- f) is nonnegative by Lm1;

 then

 A19: ( integral+ (M,(( max+ f) + ( max- f)))) = (( integral+ (M,( max+ f))) + ( integral+ (M,( max- f)))) by A8, A4, A2, A13, A10, A6, Lm10;

 assume

 A20: f is_integrable_on M;

 then

 A21: ( integral+ (M,( max+ f))) < +infty ;



 A22: ( integral+ (M,( max- f))) < +infty by A20;

 |.f.| = ( max+ |.f.|) by A14, A15, FUNCT_1: 2;

 then ( integral+ (M,( max+ |.f.|))) < +infty by A5, A21, A22, A19, XXREAL_0: 4, XXREAL_3: 16;

 hence |.f.| is_integrable_on M by A12, A11, A18;

 end;

 assume |.f.| is_integrable_on M;

 then ( Integral (M, |.f.|)) < +infty by Th96;

 then

 A23: ( integral+ (M,(( max+ f) + ( max- f)))) < +infty by A12, A11, A5, A3, Th88, SUPINF_2: 52;

 ( max- f) is nonnegative by Lm1;

 then

 A24: ( integral+ (M,(( max+ f) + ( max- f)))) = (( integral+ (M,( max+ f))) + ( integral+ (M,( max- f)))) by A8, A4, A2, A13, A10, A6, Lm10;

 -infty <> ( integral+ (M,( max- f))) by A8, A2, A10, Lm1, Th79;

 then ( integral+ (M,( max+ f))) <> +infty by A24, A23, XXREAL_3:def 2;

 then

 A25: ( integral+ (M,( max+ f))) < +infty by XXREAL_0: 4;

 -infty <> ( integral+ (M,( max+ f))) by A8, A4, A13, Lm1, Th79;

 then ( integral+ (M,( max- f))) <> +infty by A24, A23, XXREAL_3:def 2;

 then ( integral+ (M,( max- f))) < +infty by XXREAL_0: 4;

 hence thesis by A7, A25;

 end;

 theorem :: MESFUNC5:101

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_integrable_on M holds |.( Integral (M,f)).| <= ( Integral (M, |.f.|))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL such that

 A1: f is_integrable_on M;



 A2: |.(( integral+ (M,( max+ f))) - ( integral+ (M,( max- f)))).| <= ( |.( integral+ (M,( max+ f))).| + |.( integral+ (M,( max- f))).|) by EXTREAL1: 32;



 A3: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 A4:

 now

 let x be object;

 assume x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 hence 0 <= ( |.f.| . x) by EXTREAL1: 14;

 end;



 A5: ( dom f) = ( dom ( max- f)) by MESFUNC2:def 3;



 A6: |.f.| = (( max+ f) + ( max- f)) by MESFUNC2: 24;

 consider A be Element of S such that

 A7: A = ( dom f) and

 A8: f is A -measurable by A1;



 A9: ( max- f) is A -measurable by A7, A8, MESFUNC2: 26;



 A10: ( max+ f) is nonnegative by Lm1;

 then 0 <= ( integral+ (M,( max+ f))) by A7, A8, A3, Th79, MESFUNC2: 25;

 then

 A11: |.( Integral (M,f)).| <= (( integral+ (M,( max+ f))) + |.( integral+ (M,( max- f))).|) by A2, EXTREAL1:def 1;



 A12: ( max+ f) is A -measurable by A8, MESFUNC2: 25;



 A13: A = ( dom |.f.|) by A7, MESFUNC1:def 10;



 A14: ( max- f) is nonnegative by Lm1;

 then

 A15: 0 <= ( integral+ (M,( max- f))) by A7, A8, A5, Th79, MESFUNC2: 26;

 |.f.| is A -measurable by A7, A8, MESFUNC2: 27;



 then ( Integral (M, |.f.|)) = ( integral+ (M,(( max+ f) + ( max- f)))) by A13, A4, A6, Th88, SUPINF_2: 52

 .= (( integral+ (M,( max+ f))) + ( integral+ (M,( max- f)))) by A7, A3, A5, A10, A14, A12, A9, Lm10;

 hence thesis by A15, A11, EXTREAL1:def 1;

 end;

 theorem :: MESFUNC5:102



 Th102: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex A be Element of S st A = ( dom f) & f is A -measurable) & ( dom f) = ( dom g) & g is_integrable_on M & (for x be Element of X st x in ( dom f) holds |.(f . x).| <= (g . x)) holds f is_integrable_on M & ( Integral (M, |.f.|)) <= ( Integral (M,g))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: ex A be Element of S st A = ( dom f) & f is A -measurable and

 A2: ( dom f) = ( dom g) and

 A3: g is_integrable_on M and

 A4: for x be Element of X st x in ( dom f) holds |.(f . x).| <= (g . x);



 A5: ex AA be Element of S st AA = ( dom g) & g is AA -measurable by A3;

 A6:

 now

 let x be object;

 assume x in ( dom g);

 then |.(f . x).| <= (g . x) by A2, A4;

 hence 0 <= (g . x) by EXTREAL1: 14;

 end;

 then

 A7: g is nonnegative by SUPINF_2: 52;



 A8: ( dom g) = ( dom ( max+ g)) by MESFUNC2:def 2;

 now

 let x be object;



 A9: 0 <= (g . x) by A7, SUPINF_2: 51;

 assume x in ( dom g);



 hence (( max+ g) . x) = ( max ((g . x), 0 )) by A8, MESFUNC2:def 2

 .= (g . x) by A9, XXREAL_0:def 10;

 end;

 then

 A10: g = ( max+ g) by A8, FUNCT_1: 2;



 A11: ( dom |.f.|) = ( dom ( max+ |.f.|)) by MESFUNC2:def 2;

 A12:

 now

 let x be object;

 assume

 A13: x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 then

 A14: 0 <= ( |.f.| . x) by EXTREAL1: 14;



 thus (( max+ |.f.|) . x) = ( max (( |.f.| . x), 0 )) by A11, A13, MESFUNC2:def 2

 .= ( |.f.| . x) by A14, XXREAL_0:def 10;

 end;

 then

 A15: |.f.| = ( max+ |.f.|) by A11, FUNCT_1: 2;

 consider A be Element of S such that

 A16: A = ( dom f) and

 A17: f is A -measurable by A1;



 A18: |.f.| is A -measurable by A16, A17, MESFUNC2: 27;



 A19: A = ( dom |.f.|) by A16, MESFUNC1:def 10;



 A20: for x be Element of X st x in ( dom |.f.|) holds ( |.f.| . x) <= (g . x)

 proof

 let x be Element of X;

 assume

 A21: x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 hence thesis by A4, A16, A19, A21;

 end;

 A22:

 now

 let x be object;

 assume x in ( dom |.f.|);

 then ( |.f.| . x) = |.(f . x).| by MESFUNC1:def 10;

 hence 0 <= ( |.f.| . x) by EXTREAL1: 14;

 end;

 then |.f.| is nonnegative by SUPINF_2: 52;

 then

 A23: ( integral+ (M, |.f.|)) <= ( integral+ (M,g)) by A2, A16, A5, A19, A18, A7, A20, Th85;



 A24: ( dom |.f.|) = ( dom ( max- |.f.|)) by MESFUNC2:def 3;

 now

 let x be Element of X;

 assume x in ( dom ( max- |.f.|));

 then (( max+ |.f.|) . x) = ( |.f.| . x) by A24, A12;

 hence (( max- |.f.|) . x) = 0 by MESFUNC2: 19;

 end;

 then

 A25: ( integral+ (M,( max- |.f.|))) = 0 by A19, A18, A24, Th87, MESFUNC2: 26;

 ( integral+ (M,( max+ g))) < +infty by A3;

 then ( integral+ (M,( max+ |.f.|))) < +infty by A15, A10, A23, XXREAL_0: 2;

 then |.f.| is_integrable_on M by A19, A18, A25;

 hence f is_integrable_on M by A1, Th100;

 ( Integral (M,g)) = ( integral+ (M,g)) by A5, A6, Th88, SUPINF_2: 52;

 hence thesis by A19, A18, A22, A23, Th88, SUPINF_2: 52;

 end;

 theorem :: MESFUNC5:103



 Th103: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , r be Real st ( dom f) in S & 0 <= r & ( dom f) <> {} & (for x be object st x in ( dom f) holds (f . x) = r) holds ( integral (M,f)) = (r * (M . ( dom f)))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let r be Real;

 assume that

 A1: ( dom f) in S and

 A2: 0 <= r and

 A3: ( dom f) <> {} and

 A4: for x be object st x in ( dom f) holds (f . x) = r;

 for x be object st x in ( dom f) holds 0 <= (f . x) by A2, A4;

 then

 a5: f is nonnegative by SUPINF_2: 52;

 f is_simple_func_in S by A1, A4, Lm4;

 then

 consider F be Finite_Sep_Sequence of S, a,v be FinSequence of ExtREAL such that

 A6: (F,a) are_Re-presentation_of f and

 A7: ( dom v) = ( dom F) and

 A8: for n be Nat st n in ( dom v) holds (v . n) = ((a . n) * ((M * F) . n)) and

 A9: ( integral (M,f)) = ( Sum v) by A3, a5, MESFUNC4: 4;



 A10: ( dom f) = ( union ( rng F)) by A6, MESFUNC3:def 1;



 A11: for n be Nat st n in ( dom v) holds (v . n) = (r * ((M * F) . n))

 proof

 let n be Nat;

 assume

 A12: n in ( dom v);

 then

 A13: (F . n) c= ( union ( rng F)) by A7, FUNCT_1: 3, ZFMISC_1: 74;



 A14: (v . n) = ((a . n) * ((M * F) . n)) by A8, A12;

 per cases ;

 suppose (F . n) = {} ;

 then (M . (F . n)) = 0 by VALUED_0:def 19;

 then

 A15: ((M * F) . n) = 0 by A7, A12, FUNCT_1: 13;

 then (v . n) = 0 by A14;

 hence thesis by A15;

 end;

 suppose (F . n) <> {} ;

 then

 consider x be object such that

 A16: x in (F . n) by XBOOLE_0:def 1;

 (a . n) = (f . x) by A6, A7, A12, A16, MESFUNC3:def 1;

 hence thesis by A4, A10, A13, A14, A16;

 end;

 end;

 reconsider rr = r as R_eal by XXREAL_0:def 1;

 ( dom v) = ( dom (M * F)) by A7, MESFUNC3: 8;



 then ( integral (M,f)) = (rr * ( Sum (M * F))) by A9, A11, MESFUNC3: 10

 .= (rr * (M . ( union ( rng F)))) by MESFUNC3: 9;

 hence thesis by A6, MESFUNC3:def 1;

 end;

 theorem :: MESFUNC5:104



 Th104: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , r be Real st ( dom f) in S & 0 <= r & (for x be object st x in ( dom f) holds (f . x) = r) holds ( integral' (M,f)) = (r * (M . ( dom f)))

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let r be Real;

 assume that

 A1: ( dom f) in S and

 A2: 0 <= r and

 A3: for x be object st x in ( dom f) holds (f . x) = r;

 per cases ;

 suppose

 A4: ( dom f) = {} ;

 then

 A5: (M . ( dom f)) = 0 by VALUED_0:def 19;

 ( integral' (M,f)) = 0 by A4, Def14;

 hence thesis by A5;

 end;

 suppose

 A6: ( dom f) <> {} ;

 then ( integral' (M,f)) = ( integral (M,f)) by Def14;

 hence thesis by A1, A2, A3, A6, Th103;

 end;

 end;

 theorem :: MESFUNC5:105



 Th105: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st f is_integrable_on M holds (f " { +infty }) in S & (f " { -infty }) in S & (M . (f " { +infty })) = 0 & (M . (f " { -infty })) = 0 & ((f " { +infty }) \/ (f " { -infty })) in S & (M . ((f " { +infty }) \/ (f " { -infty }))) = 0

 proof

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;



 A1: ( max+ f) is nonnegative by Lm1;

 assume

 A2: f is_integrable_on M;

 then

 A3: ( integral+ (M,( max+ f))) < +infty ;

 consider A be Element of S such that

 A4: A = ( dom f) and

 A5: f is A -measurable by A2;



 A6: for x be object holds (x in ( eq_dom (f, +infty )) implies x in A) & (x in ( eq_dom (f, -infty )) implies x in A) by A4, MESFUNC1:def 15;

 then

 A7: ( eq_dom (f, +infty )) c= A;

 then

 A8: (A /\ ( eq_dom (f, +infty ))) = ( eq_dom (f, +infty )) by XBOOLE_1: 28;



 A9: ( eq_dom (f, -infty )) c= A by A6;

 then

 A10: (A /\ ( eq_dom (f, -infty ))) = ( eq_dom (f, -infty )) by XBOOLE_1: 28;



 A11: (A /\ ( eq_dom (f, +infty ))) in S by A4, A5, MESFUNC1: 33;

 then

 A12: (f " { +infty }) in S by A8, Th30;



 A13: (A /\ ( eq_dom (f, -infty ))) in S by A5, MESFUNC1: 34;

 then

 reconsider B2 = (f " { -infty }) as Element of S by A10, Th30;



 A14: (f " { -infty }) in S by A13, A10, Th30;

 thus (f " { +infty }) in S & (f " { -infty }) in S by A11, A13, A8, A10, Th30;

 set C2 = (A \ B2);



 A15: ( integral+ (M,( max- f))) < +infty by A2;

 reconsider B1 = (f " { +infty }) as Element of S by A11, A8, Th30;



 A16: A = ( dom ( max+ f)) by A4, MESFUNC2:def 2;

 then

 A17: B1 c= ( dom ( max+ f)) by A7, Th30;

 then

 A18: B1 = (( dom ( max+ f)) /\ B1) by XBOOLE_1: 28;



 A19: ( max+ f) is A -measurable by A5, MESFUNC2: 25;

 then ( max+ f) is B1 -measurable by A16, A17, MESFUNC1: 30;

 then

 A20: (( max+ f) | B1) is B1 -measurable by A18, Th42;

 set C1 = (A \ B1);



 A21: for x be Element of X holds (x in ( dom (( max+ f) | (B1 \/ C1))) implies ((( max+ f) | (B1 \/ C1)) . x) = (( max+ f) . x)) & (x in ( dom (( max- f) | (B2 \/ C2))) implies ((( max- f) | (B2 \/ C2)) . x) = (( max- f) . x)) by FUNCT_1: 47;

 (B1 \/ C1) = A by A16, A17, XBOOLE_1: 45;

 then ( dom (( max+ f) | (B1 \/ C1))) = (( dom ( max+ f)) /\ ( dom ( max+ f))) by A16, RELAT_1: 61;

 then (( max+ f) | (B1 \/ C1)) = ( max+ f) by A21, PARTFUN1: 5;

 then ( integral+ (M,( max+ f))) = (( integral+ (M,(( max+ f) | B1))) + ( integral+ (M,(( max+ f) | C1)))) by A1, A16, A19, Th81, XBOOLE_1: 106;

 then

 A22: ( integral+ (M,(( max+ f) | B1))) <= ( integral+ (M,( max+ f))) by A1, A16, A19, Th80, XXREAL_3: 65;

 now



 A23: for r be Real st 0 < r holds (r * (M . B1)) <= ( integral+ (M,( max+ f)))

 proof

 defpred P[ object] means $1 in ( dom (( max+ f) | B1));

 let r be Real;

 deffunc F( object) = ( In (r, ExtREAL ));



 A24: for x be object st P[x] holds F(x) in ExtREAL ;

 consider g be PartFunc of X, ExtREAL such that

 A25: (for x be object holds x in ( dom g) iff x in X & P[x]) & for x be object st x in ( dom g) holds (g . x) = F(x) from PARTFUN1:sch 3( A24);

 assume

 A26: 0 < r;

 then for x be object st x in ( dom g) holds 0 <= (g . x) by A25;

 then

 A27: g is nonnegative by SUPINF_2: 52;

 ( dom (( max+ f) | B1)) = (( dom ( max+ f)) /\ B1) by RELAT_1: 61;

 then

 A28: ( dom (( max+ f) | B1)) = B1 by A17, XBOOLE_1: 28;

 for x be object holds x in ( dom g) iff x in X & x in ( dom (( max+ f) | B1)) by A25;

 then ( dom g) = (X /\ ( dom (( max+ f) | B1))) by XBOOLE_0:def 4;

 then

 A29: ( dom g) = ( dom (( max+ f) | B1)) by XBOOLE_1: 28;

 then

 A30: ( integral' (M,g)) = (r * (M . ( dom g))) by A26, A25, A28, Th104;



 A31: for x be Element of X st x in ( dom g) holds (g . x) <= ((( max+ f) | B1) . x)

 proof

 let x be Element of X;

 assume

 A32: x in ( dom g);

 then x in ( dom f) by A29, A28, FUNCT_1:def 7;

 then

 A33: x in ( dom ( max+ f)) by MESFUNC2:def 2;

 (f . x) in { +infty } by A29, A28, A32, FUNCT_1:def 7;

 then

 A34: (f . x) = +infty by TARSKI:def 1;

 then ( max ((f . x), 0 )) = (f . x) by XXREAL_0:def 10;

 then (( max+ f) . x) = +infty by A34, A33, MESFUNC2:def 2;

 then ((( max+ f) | B1) . x) = +infty by A29, A28, A32, FUNCT_1: 49;

 hence thesis by XXREAL_0: 4;

 end;

 ( dom ( chi (B1,X))) = X by FUNCT_3:def 3;

 then

 A35: B1 = (( dom ( chi (B1,X))) /\ B1) by XBOOLE_1: 28;

 then

 A36: (( chi (B1,X)) | B1) is B1 -measurable by Th42, MESFUNC2: 29;



 A37: B1 = ( dom (( chi (B1,X)) | B1)) by A35, RELAT_1: 61;



 A38: for x be Element of X st x in ( dom g) holds (g . x) = ((r (#) (( chi (B1,X)) | B1)) . x)

 proof

 let x be Element of X;

 assume

 A39: x in ( dom g);

 then x in ( dom (( chi (B1,X)) | B1)) by A29, A28, A35, RELAT_1: 61;

 then x in ( dom (r (#) (( chi (B1,X)) | B1))) by MESFUNC1:def 6;



 then

 A40: ((r (#) (( chi (B1,X)) | B1)) . x) = (r * ((( chi (B1,X)) | B1) . x)) by MESFUNC1:def 6

 .= (r * (( chi (B1,X)) . x)) by A29, A28, A37, A39, FUNCT_1: 47;

 (( chi (B1,X)) . x) = 1 by A29, A28, A39, FUNCT_3:def 3;

 then ((r (#) (( chi (B1,X)) | B1)) . x) = r by A40, XXREAL_3: 81;

 hence thesis by A25, A39;

 end;

 ( dom g) = ( dom (r (#) (( chi (B1,X)) | B1))) by A29, A28, A37, MESFUNC1:def 6;

 then g = (r (#) (( chi (B1,X)) | B1)) by A38, PARTFUN1: 5;

 then

 A41: g is B1 -measurable by A37, A36, MESFUNC1: 37;

 (( max+ f) | B1) is nonnegative by Lm1, Th15;

 then ( integral+ (M,g)) <= ( integral+ (M,(( max+ f) | B1))) by A20, A29, A28, A41, A27, A31, Th85;

 then ( integral+ (M,g)) <= ( integral+ (M,( max+ f))) by A22, XXREAL_0: 2;

 hence thesis by A25, A29, A28, A27, A30, Lm4, Th77;

 end;

 assume

 A42: (M . (f " { +infty })) <> 0 ;

 then

 A43: 0 < (M . (f " { +infty })) by A12, Th45;

 per cases ;

 suppose

 A44: (M . B1) = +infty ;

 (jj * (M . B1)) <= ( integral+ (M,( max+ f))) by A23;

 hence contradiction by A3, A44, XXREAL_3: 81;

 end;

 suppose (M . B1) <> +infty ;

 then

 reconsider MB = (M . B1) as Element of REAL by A43, XXREAL_0: 14;

 (jj * (M . B1)) <= ( integral+ (M,( max+ f))) by A23;

 then

 A45: 0 < ( integral+ (M,( max+ f))) by A43;

 then

 reconsider I = ( integral+ (M,( max+ f))) as Element of REAL by A3, XXREAL_0: 14;



 A46: (((2 * I) / MB) * (M . B1)) = (((2 * I) / MB) * MB);

 (((2 * I) / MB) * (M . B1)) <= ( integral+ (M,( max+ f))) by A43, A23, A45;

 then (2 * I) <= I by A42, A46, XCMPLX_1: 87;

 hence contradiction by A45, XREAL_1: 155;

 end;

 end;

 then

 reconsider B1 as measure_zero of M by MEASURE1:def 7;



 A47: ( max- f) is nonnegative by Lm1;



 A48: A = ( dom ( max- f)) by A4, MESFUNC2:def 3;

 then

 A49: B2 c= ( dom ( max- f)) by A9, Th30;

 then

 A50: B2 = (( dom ( max- f)) /\ B2) by XBOOLE_1: 28;



 A51: ( max- f) is A -measurable by A4, A5, MESFUNC2: 26;

 then ( max- f) is B2 -measurable by A48, A49, MESFUNC1: 30;

 then

 A52: (( max- f) | B2) is B2 -measurable by A50, Th42;

 (B2 \/ C2) = A by A48, A49, XBOOLE_1: 45;

 then ( dom (( max- f) | (B2 \/ C2))) = (( dom ( max- f)) /\ ( dom ( max- f))) by A48, RELAT_1: 61;

 then (( max- f) | (B2 \/ C2)) = ( max- f) by A21, PARTFUN1: 5;

 then ( integral+ (M,( max- f))) = (( integral+ (M,(( max- f) | B2))) + ( integral+ (M,(( max- f) | C2)))) by A47, A48, A51, Th81, XBOOLE_1: 106;

 then

 A53: ( integral+ (M,(( max- f) | B2))) <= ( integral+ (M,( max- f))) by A47, A48, A51, Th80, XXREAL_3: 65;

 now



 A55: for r be Real st 0 < r holds (r * (M . B2)) <= ( integral+ (M,( max- f)))

 proof

 defpred P[ object] means $1 in ( dom (( max- f) | B2));

 let r be Real;

 deffunc F( object) = ( In (r, ExtREAL ));



 A56: for x be object st P[x] holds F(x) in ExtREAL ;

 consider g be PartFunc of X, ExtREAL such that

 A57: (for x be object holds x in ( dom g) iff x in X & P[x]) & for x be object st x in ( dom g) holds (g . x) = F(x) from PARTFUN1:sch 3( A56);

 assume

 A58: 0 < r;

 then for x be object st x in ( dom g) holds 0 <= (g . x) by A57;

 then

 A59: g is nonnegative by SUPINF_2: 52;

 ( dom (( max- f) | B2)) = (( dom ( max- f)) /\ B2) by RELAT_1: 61;

 then

 A60: ( dom (( max- f) | B2)) = B2 by A49, XBOOLE_1: 28;

 for x be object holds x in ( dom g) iff x in X & x in ( dom (( max- f) | B2)) by A57;

 then ( dom g) = (X /\ ( dom (( max- f) | B2))) by XBOOLE_0:def 4;

 then

 A61: ( dom g) = ( dom (( max- f) | B2)) by XBOOLE_1: 28;

 then

 A62: ( integral' (M,g)) = (r * (M . ( dom g))) by A58, A57, A60, Th104;

 ( dom ( chi (B2,X))) = X by FUNCT_3:def 3;

 then

 A63: B2 = (( dom ( chi (B2,X))) /\ B2) by XBOOLE_1: 28;

 then

 A64: B2 = ( dom (( chi (B2,X)) | B2)) by RELAT_1: 61;



 A65: for x be Element of X st x in ( dom g) holds (g . x) = ((r (#) (( chi (B2,X)) | B2)) . x)

 proof

 let x be Element of X;

 assume

 A66: x in ( dom g);

 then x in ( dom (r (#) (( chi (B2,X)) | B2))) by A61, A60, A64, MESFUNC1:def 6;



 then

 A67: ((r (#) (( chi (B2,X)) | B2)) . x) = (r * ((( chi (B2,X)) | B2) . x)) by MESFUNC1:def 6

 .= (r * (( chi (B2,X)) . x)) by A61, A60, A64, A66, FUNCT_1: 47;

 (( chi (B2,X)) . x) = 1 by A61, A60, A66, FUNCT_3:def 3;

 then ((r (#) (( chi (B2,X)) | B2)) . x) = r by A67, XXREAL_3: 81;

 hence thesis by A57, A66;

 end;



 A68: for x be Element of X st x in ( dom g) holds (g . x) <= ((( max- f) | B2) . x)

 proof

 let x be Element of X;

 assume

 A69: x in ( dom g);

 then x in ( dom f) by A61, A60, FUNCT_1:def 7;

 then

 A70: x in ( dom ( max- f)) by MESFUNC2:def 3;

 (f . x) in { -infty } by A61, A60, A69, FUNCT_1:def 7;

 then

 A71: ( - (f . x)) = +infty by TARSKI:def 1, XXREAL_3: 5;

 then ( max (( - (f . x)), 0 )) = ( - (f . x)) by XXREAL_0:def 10;

 then (( max- f) . x) = +infty by A71, A70, MESFUNC2:def 3;

 then ((( max- f) | B2) . x) = +infty by A61, A60, A69, FUNCT_1: 49;

 hence thesis by XXREAL_0: 4;

 end;



 A72: (( chi (B2,X)) | B2) is B2 -measurable by A63, Th42, MESFUNC2: 29;

 ( dom g) = ( dom (r (#) (( chi (B2,X)) | B2))) by A61, A60, A64, MESFUNC1:def 6;

 then g = (r (#) (( chi (B2,X)) | B2)) by A65, PARTFUN1: 5;

 then

 A73: g is B2 -measurable by A64, A72, MESFUNC1: 37;

 (( max- f) | B2) is nonnegative by Lm1, Th15;

 then ( integral+ (M,g)) <= ( integral+ (M,(( max- f) | B2))) by A52, A61, A60, A73, A59, A68, Th85;

 then ( integral+ (M,g)) <= ( integral+ (M,( max- f))) by A53, XXREAL_0: 2;

 hence thesis by A57, A61, A60, A59, A62, Lm4, Th77;

 end;

 assume

 A74: (M . (f " { -infty })) <> 0 ;



 A75: 0 <= (M . (f " { -infty })) by A14, Th45;

 per cases ;

 suppose

 A76: (M . B2) = +infty ;

 (jj * (M . B2)) <= ( integral+ (M,( max- f))) by A55;

 hence contradiction by A15, A76, XXREAL_3: 81;

 end;

 suppose (M . B2) <> +infty ;

 then

 reconsider MB = (M . B2) as Element of REAL by A75, XXREAL_0: 14;

 (jj * (M . B2)) <= ( integral+ (M,( max- f))) by A55;

 then

 A77: 0 < ( integral+ (M,( max- f))) by A74, A75;

 then

 reconsider I = ( integral+ (M,( max- f))) as Element of REAL by A15, XXREAL_0: 14;



 A78: (((2 * I) / MB) * (M . B2)) = (((2 * I) / MB) * MB);

 (((2 * I) / MB) * (M . B2)) <= ( integral+ (M,( max- f))) by A74, A75, A55, A77;

 then (2 * I) <= I by A74, A78, XCMPLX_1: 87;

 hence contradiction by A77, XREAL_1: 155;

 end;

 end;

 thus ((f " { +infty }) \/ (f " { -infty })) in S by A12, A14, PROB_1: 3;



 thus (M . ((f " { +infty }) \/ (f " { -infty }))) = (M . (B1 \/ B2))

 .= 0 by A54, MEASURE1: 38;

 end;

 theorem :: MESFUNC5:106



 Th106: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is nonnegative holds (f + g) is_integrable_on M

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M and

 A3: f is nonnegative and

 A4: g is nonnegative;



 A5: ( integral+ (M,( max+ g))) < +infty by A2;



 A6: ( dom g) = ( dom ( max+ g)) by MESFUNC2:def 2;

 now

 let x be object;

 assume x in ( dom g);

 then

 A7: (( max+ g) . x) = ( max ((g . x), 0 )) by A6, MESFUNC2:def 2;

 0 <= (g . x) by A4, SUPINF_2: 51;

 hence (( max+ g) . x) = (g . x) by A7, XXREAL_0:def 10;

 end;

 then

 A8: g = ( max+ g) by A6, FUNCT_1: 2;

 consider B be Element of S such that

 A9: B = ( dom g) and

 A10: g is B -measurable by A2;

 consider A be Element of S such that

 A11: A = ( dom f) and

 A12: f is A -measurable by A1;



 A13: g is (A /\ B) -measurable by A10, MESFUNC1: 30, XBOOLE_1: 17;

 f is (A /\ B) -measurable by A12, MESFUNC1: 30, XBOOLE_1: 17;

 then

 A14: (f + g) is (A /\ B) -measurable by A3, A4, A13, Th31;

 consider C be Element of S such that

 A15: C = ( dom (f + g)) and

 A16: ( integral+ (M,(f + g))) = (( integral+ (M,(f | C))) + ( integral+ (M,(g | C)))) by A3, A4, A11, A12, A9, A10, Th78;



 A17: (A /\ B) = ( dom (f + g)) by A3, A4, A11, A9, Th16;

 then ( integral+ (M,(g | C))) <= ( integral+ (M,(g | B))) by A4, A9, A10, A15, Th83, XBOOLE_1: 17;

 then

 A18: ( integral+ (M,(g | C))) <= ( integral+ (M,( max+ g))) by A9, A8, GRFUNC_1: 23;

 ( integral+ (M,( max+ f))) < +infty by A1;

 then

 A19: (( integral+ (M,( max+ f))) + ( integral+ (M,( max+ g)))) < +infty by A5, XXREAL_0: 4, XXREAL_3: 16;



 A20: ( dom f) = ( dom ( max+ f)) by MESFUNC2:def 2;

 now

 let x be object;

 assume x in ( dom f);

 then

 A21: (( max+ f) . x) = ( max ((f . x), 0 )) by A20, MESFUNC2:def 2;

 0 <= (f . x) by A3, SUPINF_2: 51;

 hence (( max+ f) . x) = (f . x) by A21, XXREAL_0:def 10;

 end;

 then

 A22: f = ( max+ f) by A20, FUNCT_1: 2;



 A23: ( dom (f + g)) = ( dom ( max+ (f + g))) by MESFUNC2:def 2;

 A24:

 now

 let x be object;

 assume

 A25: x in ( dom (f + g));

 then

 A26: ((f + g) . x) = ((f . x) + (g . x)) by MESFUNC1:def 3;



 A27: 0 <= (g . x) by A4, SUPINF_2: 51;



 A28: 0 <= (f . x) by A3, SUPINF_2: 51;

 (( max+ (f + g)) . x) = ( max (((f + g) . x), 0 )) by A23, A25, MESFUNC2:def 2;

 hence (( max+ (f + g)) . x) = ((f + g) . x) by A26, A28, A27, XXREAL_0:def 10;

 end;

 then

 A29: (f + g) = ( max+ (f + g)) by A23, FUNCT_1: 2;

 A30:

 now

 let x be Element of X;

 assume x in ( dom ( max- (f + g)));

 then x in ( dom (f + g)) by MESFUNC2:def 3;

 then (( max+ (f + g)) . x) = ((f + g) . x) by A24;

 hence (( max- (f + g)) . x) = 0 by MESFUNC2: 19;

 end;

 ( integral+ (M,(f | C))) <= ( integral+ (M,(f | A))) by A3, A11, A12, A17, A15, Th83, XBOOLE_1: 17;

 then ( integral+ (M,(f | C))) <= ( integral+ (M,( max+ f))) by A11, A22, GRFUNC_1: 23;

 then ( integral+ (M,( max+ (f + g)))) <= (( integral+ (M,( max+ f))) + ( integral+ (M,( max+ g)))) by A29, A16, A18, XXREAL_3: 36;

 then

 A31: ( integral+ (M,( max+ (f + g)))) < +infty by A19, XXREAL_0: 4;

 ( dom (f + g)) = ( dom ( max- (f + g))) by MESFUNC2:def 3;

 then ( integral+ (M,( max- (f + g)))) = 0 by A17, A14, A30, Th87, MESFUNC2: 26;

 hence thesis by A17, A14, A31;

 end;

 theorem :: MESFUNC5:107



 Th107: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M holds ( dom (f + g)) in S

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M;



 A3: (f " { -infty }) in S by A1, Th105;



 A4: ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable by A2;



 A5: ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable by A1;



 A6: (g " { -infty }) in S by A2, Th105;



 A7: (g " { +infty }) in S by A2, Th105;

 (f " { +infty }) in S by A1, Th105;

 hence thesis by A3, A7, A6, A5, A4, Th46;

 end;



 Lm11: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st (ex E0 be Element of S st ( dom (f + g)) = E0 & (f + g) is E0 -measurable) & f is_integrable_on M & g is_integrable_on M holds (f + g) is_integrable_on M

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: ex E0 be Element of S st ( dom (f + g)) = E0 & (f + g) is E0 -measurable and

 A2: f is_integrable_on M and

 A3: g is_integrable_on M;

 consider E be Element of S such that

 A4: ( dom (f + g)) = E and

 A5: (f + g) is E -measurable by A1;



 A6: ( |.f.| | E) is nonnegative by Lm1, Th15;

 |.g.| is_integrable_on M by A3, Th100;

 then

 A7: ( |.g.| | E) is_integrable_on M by Th97;



 A8: ( |.g.| | E) is nonnegative by Lm1, Th15;



 A9: ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then ( dom (f + g)) c= ( dom g) by XBOOLE_1: 18, XBOOLE_1: 36;

 then

 A10: E c= ( dom |.g.|) by A4, MESFUNC1:def 10;

 then

 A11: (( dom |.g.|) /\ E) = E by XBOOLE_1: 28;

 ( dom (f + g)) c= ( dom f) by A9, XBOOLE_1: 18, XBOOLE_1: 36;

 then

 A12: E c= ( dom |.f.|) by A4, MESFUNC1:def 10;

 then (( dom |.f.|) /\ E) = E by XBOOLE_1: 28;

 then

 A13: E = ( dom ( |.f.| | E)) by RELAT_1: 61;

 then

 A14: (( dom ( |.f.| | E)) /\ ( dom ( |.g.| | E))) = (E /\ E) by A11, RELAT_1: 61;

 then

 A15: ( dom (( |.f.| | E) + ( |.g.| | E))) = E by A6, A8, Th22;



 A16: E = ( dom ( |.g.| | E)) by A11, RELAT_1: 61;

 A17:

 now

 let x be Element of X;



 A18: |.((f . x) + (g . x)).| <= ( |.(f . x).| + |.(g . x).|) by EXTREAL1: 24;

 assume

 A19: x in ( dom (f + g));

 then

 A20: x in ( dom (( |.f.| | E) + ( |.g.| | E))) by A4, A6, A8, A14, Th22;

 ( |.(f . x).| + |.(g . x).|) = (( |.f.| . x) + |.(g . x).|) by A4, A12, A19, MESFUNC1:def 10

 .= (( |.f.| . x) + ( |.g.| . x)) by A4, A10, A19, MESFUNC1:def 10

 .= ((( |.f.| | E) . x) + ( |.g.| . x)) by A4, A13, A19, FUNCT_1: 47

 .= ((( |.f.| | E) . x) + (( |.g.| | E) . x)) by A4, A16, A19, FUNCT_1: 47

 .= ((( |.f.| | E) + ( |.g.| | E)) . x) by A20, MESFUNC1:def 3;

 hence |.((f + g) . x).| <= ((( |.f.| | E) + ( |.g.| | E)) . x) by A19, A18, MESFUNC1:def 3;

 end;

 |.f.| is_integrable_on M by A2, Th100;

 then ( |.f.| | E) is_integrable_on M by Th97;

 then (( |.f.| | E) + ( |.g.| | E)) is_integrable_on M by A7, A6, A8, Th106;

 hence thesis by A4, A5, A17, A15, Th102;

 end;

 theorem :: MESFUNC5:108



 Th108: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M holds (f + g) is_integrable_on M

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL such that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M;



 A3: ex E2 be Element of S st E2 = ( dom g) & g is E2 -measurable by A2;

 ex E1 be Element of S st E1 = ( dom f) & f is E1 -measurable by A1;

 then ex K0 be Element of S st K0 = ( dom (f + g)) & (f + g) is K0 -measurable by A3, Th47;

 hence thesis by A1, A2, Lm11;

 end;



 Lm12: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M & ( dom f) = ( dom g) holds ex E,NFG,NFPG be Element of S st E c= ( dom f) & NFG c= ( dom f) & E = (( dom f) \ NFG) & (f | E) is real-valued & E = ( dom (f | E)) & (f | E) is E -measurable & (f | E) is_integrable_on M & ( Integral (M,f)) = ( Integral (M,(f | E))) & E c= ( dom g) & NFG c= ( dom g) & E = (( dom g) \ NFG) & (g | E) is real-valued & E = ( dom (g | E)) & (g | E) is E -measurable & (g | E) is_integrable_on M & ( Integral (M,g)) = ( Integral (M,(g | E))) & E c= ( dom (f + g)) & NFPG c= ( dom (f + g)) & E = (( dom (f + g)) \ NFPG) & (M . NFG) = 0 & (M . NFPG) = 0 & E = ( dom ((f + g) | E)) & ((f + g) | E) is E -measurable & ((f + g) | E) is_integrable_on M & ((f + g) | E) = ((f | E) + (g | E)) & ( Integral (M,((f + g) | E))) = (( Integral (M,(f | E))) + ( Integral (M,(g | E))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M and

 A3: ( dom f) = ( dom g);



 A4: ((f " { +infty }) /\ (g " { -infty })) c= (g " { -infty }) by XBOOLE_1: 17;

 reconsider NG = ((g " { +infty }) \/ (g " { -infty })) as Element of S by A2, Th105;

 reconsider NF = ((f " { +infty }) \/ (f " { -infty })) as Element of S by A1, Th105;

 set NFG = (NF \/ NG);

 consider E0 be Element of S such that

 A5: E0 = ( dom f) and

 A6: f is E0 -measurable by A1;

 set E = (E0 \ NFG);

 set f1 = (f | E);

 set g1 = (g | E);



 A7: E c= ( dom f) by A5, XBOOLE_1: 36;

 reconsider DFPG = ( dom (f + g)) as Element of S by A1, A2, Th107;



 A8: ((f " { -infty }) /\ (g " { +infty })) c= (f " { -infty }) by XBOOLE_1: 17;



 A9: for x be object holds (x in (f " { +infty }) implies x in ( dom f)) & (x in (f " { -infty }) implies x in ( dom f)) & (x in (g " { +infty }) implies x in ( dom g)) & (x in (g " { -infty }) implies x in ( dom g)) by FUNCT_1:def 7;

 then

 A10: (g " { -infty }) c= ( dom g);

 set NFPG = (DFPG \ E);



 A11: ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then (DFPG \ (E0 \ NFG)) c= (E0 \ (E0 \ NFG)) by A3, A5, XBOOLE_1: 33, XBOOLE_1: 36;

 then

 A12: NFPG c= (E0 /\ NFG) by XBOOLE_1: 48;

 (g " { -infty }) c= NG by XBOOLE_1: 7;

 then

 A13: ((f " { +infty }) /\ (g " { -infty })) c= NG by A4;

 (f " { -infty }) c= NF by XBOOLE_1: 7;

 then ((f " { -infty }) /\ (g " { +infty })) c= NF by A8;

 then

 A14: (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty }))) c= (NF \/ NG) by A13, XBOOLE_1: 13;

 then

 A15: E c= ( dom (f + g)) by A3, A5, A11, XBOOLE_1: 34;

 then

 A16: ((f + g) | E) = (f1 + g1) by Th29;

 (DFPG \ NFPG) = (DFPG /\ E) by XBOOLE_1: 48;

 then

 A17: E = (DFPG \ NFPG) by A3, A5, A11, A14, XBOOLE_1: 28, XBOOLE_1: 34;



 A18: ( dom (f1 + g1)) = E by A15, Th29;



 A19: for x be set st x in ( dom g1) holds -infty < (g1 . x) & (g1 . x) < +infty

 proof

 let x be set;

 for x be object st x in ( dom g) holds (g . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider gg = g as Function of ( dom g), ExtREAL by FUNCT_2: 3;

 assume

 A20: x in ( dom g1);

 then

 A21: x in (( dom g) /\ E) by RELAT_1: 61;

 then

 A22: x in ( dom g) by XBOOLE_0:def 4;

 x in E by A21, XBOOLE_0:def 4;

 then

 A23: not x in NFG by XBOOLE_0:def 5;

 A24:

 now

 assume (g1 . x) = -infty ;

 then (g . x) = -infty by A20, FUNCT_1: 47;

 then (gg . x) in { -infty } by TARSKI:def 1;

 then

 A25: x in (gg " { -infty }) by A22, FUNCT_2: 38;

 (g " { -infty }) c= NG by XBOOLE_1: 7;

 hence contradiction by A23, A25, XBOOLE_0:def 3;

 end;

 now

 assume (g1 . x) = +infty ;

 then (g . x) = +infty by A20, FUNCT_1: 47;

 then (gg . x) in { +infty } by TARSKI:def 1;

 then

 A26: x in (gg " { +infty }) by A22, FUNCT_2: 38;

 (g " { +infty }) c= NG by XBOOLE_1: 7;

 hence contradiction by A23, A26, XBOOLE_0:def 3;

 end;

 hence thesis by A24, XXREAL_0: 4, XXREAL_0: 6;

 end;

 then for x be set st x in ( dom g1) holds -infty < (g1 . x);

 then

 A27: g1 is without-infty by Th10;

 now

 let x be Element of X;



 A28: ( - +infty ) = -infty by XXREAL_3:def 3;

 assume

 A29: x in ( dom g1);

 then

 A30: (g1 . x) < +infty by A19;

 -infty < (g1 . x) by A19, A29;

 hence |.(g1 . x).| < +infty by A30, A28, EXTREAL1: 22;

 end;

 then

 A31: g1 is real-valued by MESFUNC2:def 1;



 A32: for x be set st x in ( dom f1) holds (f1 . x) < +infty & -infty < (f1 . x)

 proof

 let x be set;

 for x be object st x in ( dom f) holds (f . x) in ExtREAL by XXREAL_0:def 1;

 then

 reconsider ff = f as Function of ( dom f), ExtREAL by FUNCT_2: 3;

 assume

 A33: x in ( dom f1);

 then

 A34: x in (( dom f) /\ E) by RELAT_1: 61;

 then

 A35: x in ( dom f) by XBOOLE_0:def 4;

 x in E by A34, XBOOLE_0:def 4;

 then

 A36: not x in NFG by XBOOLE_0:def 5;

 A37:

 now

 assume (f1 . x) = -infty ;

 then (f . x) = -infty by A33, FUNCT_1: 47;

 then (ff . x) in { -infty } by TARSKI:def 1;

 then

 A38: x in (ff " { -infty }) by A35, FUNCT_2: 38;

 (f " { -infty }) c= NF by XBOOLE_1: 7;

 hence contradiction by A36, A38, XBOOLE_0:def 3;

 end;

 now

 assume (f1 . x) = +infty ;

 then (f . x) = +infty by A33, FUNCT_1: 47;

 then (ff . x) in { +infty } by TARSKI:def 1;

 then

 A39: x in (ff " { +infty }) by A35, FUNCT_2: 38;

 (f " { +infty }) c= NF by XBOOLE_1: 7;

 hence contradiction by A36, A39, XBOOLE_0:def 3;

 end;

 hence thesis by A37, XXREAL_0: 4, XXREAL_0: 6;

 end;

 then for x be set st x in ( dom f1) holds -infty < (f1 . x);

 then

 A40: f1 is without-infty by Th10;

 then

 A41: ( dom (( max- (f1 + g1)) + ( max+ f1))) = (( dom f1) /\ ( dom g1)) by A27, Th24;



 A42: (( max+ (f1 + g1)) + ( max- f1)) is nonnegative by A40, A27, Th24;



 A43: ( dom (( max+ (f1 + g1)) + ( max- f1))) = (( dom f1) /\ ( dom g1)) by A40, A27, Th24;



 A44: (( max- (f1 + g1)) + ( max+ f1)) is nonnegative by A40, A27, Th24;



 A45: ( max+ f1) is nonnegative by Lm1;



 A46: ( dom ( max+ (f1 + g1))) = ( dom (f1 + g1)) by MESFUNC2:def 2;



 A47: ( dom g1) = (( dom g) /\ E) by RELAT_1: 61;

 then

 A48: E = ( dom g1) by A3, A5, XBOOLE_1: 28, XBOOLE_1: 36;

 then

 A49: ( dom ( max- g1)) = E by MESFUNC2:def 3;



 A50: ex Gf be Element of S st Gf = ( dom g) & g is Gf -measurable by A2;

 then g is E -measurable by A3, A5, MESFUNC1: 30, XBOOLE_1: 36;

 then

 A51: g1 is E -measurable by A47, A48, Th42;

 then

 A52: ( max- g1) is E -measurable by A48, MESFUNC2: 26;



 A53: ( dom ( max+ g1)) = E by A48, MESFUNC2:def 2;



 A54: ( max+ g1) is nonnegative by Lm1;



 A55: ( max- g1) is nonnegative by Lm1;



 A56: ( dom f1) = (( dom f) /\ E) by RELAT_1: 61;

 then

 A57: E = ( dom f1) by A5, XBOOLE_1: 28, XBOOLE_1: 36;

 (M . NG) = 0 by A2, Th105;

 then

 A58: NG is measure_zero of M by MEASURE1:def 7;

 (M . NF) = 0 by A1, Th105;

 then NF is measure_zero of M by MEASURE1:def 7;

 then

 A59: NFG is measure_zero of M by A58, MEASURE1: 37;

 then

 A60: (M . NFG) = 0 by MEASURE1:def 7;

 then

 A61: ( Integral (M,f)) = ( Integral (M,f1)) by A5, A6, Th95;

 (E0 /\ NFG) c= NFG by XBOOLE_1: 17;

 then NFPG is measure_zero of M by A59, A12, MEASURE1: 36, XBOOLE_1: 1;

 then

 A62: (M . NFPG) = 0 by MEASURE1:def 7;



 A63: ( max- (f1 + g1)) is nonnegative by Lm1;



 A64: ( max+ (f1 + g1)) is nonnegative by Lm1;

 for x be set st x in ( dom g1) holds (g1 . x) < +infty by A19;

 then

 A65: g1 is without+infty by Th11;



 A66: ( dom ( max+ f1)) = ( dom f1) by MESFUNC2:def 2;

 for x be set st x in ( dom g1) holds -infty < (g1 . x) by A19;

 then

 A67: g1 is without-infty by Th10;



 A68: ( dom ( max- f1)) = ( dom f1) by MESFUNC2:def 3;



 A69: (f " { -infty }) c= ( dom f) by A9;

 (g " { +infty }) c= ( dom g) by A9;

 then

 A70: NG c= ( dom g) by A10, XBOOLE_1: 8;

 (f " { +infty }) c= ( dom f) by A9;

 then NF c= ( dom g) by A3, A69, XBOOLE_1: 8;

 then

 A71: (NF \/ NG) c= ( dom g) by A70, XBOOLE_1: 8;



 A72: NFPG c= ( dom (f + g)) by XBOOLE_1: 36;



 A73: g1 is_integrable_on M by A2, Th97;

 then

 A74: 0 <= ( integral+ (M,( max+ g1))) by Th96;

 for x be set st x in ( dom f1) holds (f1 . x) < +infty by A32;

 then

 A75: f1 is without+infty by Th11;

 for x be set st x in ( dom f1) holds -infty < (f1 . x) by A32;

 then f1 is without-infty by Th10;

 then

 A76: ((( max+ (f1 + g1)) + ( max- f1)) + ( max- g1)) = ((( max- (f1 + g1)) + ( max+ f1)) + ( max+ g1)) by A75, A67, A65, Th25;



 A77: ( max- f1) is nonnegative by Lm1;



 A78: ( dom ( max- (f1 + g1))) = ( dom (f1 + g1)) by MESFUNC2:def 3;



 A79: ( integral+ (M,( max+ g1))) <> +infty by A73;



 A80: 0 <= ( integral+ (M,( max- g1))) by A73, Th96;

 f is E -measurable by A6, MESFUNC1: 30, XBOOLE_1: 36;

 then

 A81: f1 is E -measurable by A56, A57, Th42;

 then

 A82: ( max- f1) is E -measurable by A57, MESFUNC2: 26;

 now

 let x be Element of X;



 A83: ( - +infty ) = -infty by XXREAL_3:def 3;

 assume

 A84: x in ( dom f1);

 then

 A85: (f1 . x) < +infty by A32;

 -infty < (f1 . x) by A32, A84;

 hence |.(f1 . x).| < +infty by A85, A83, EXTREAL1: 22;

 end;

 then

 A86: f1 is real-valued by MESFUNC2:def 1;

 then

 A87: (f1 + g1) is E -measurable by A81, A51, A31, MESFUNC2: 7;

 then

 A88: ( max+ (f1 + g1)) is E -measurable by MESFUNC2: 25;

 (( dom f1) /\ ( dom g1)) = E by A3, A5, A56, A47, XBOOLE_1: 28, XBOOLE_1: 36;

 then

 A89: (( max- (f1 + g1)) + ( max+ f1)) is E -measurable by A81, A51, A40, A27, Th44;

 E = ( dom (f1 + g1)) by A15, Th29;

 then

 A90: ( max- (f1 + g1)) is E -measurable by A87, MESFUNC2: 26;



 A91: ( max+ f1) is E -measurable by A81, MESFUNC2: 25;



 A92: ( integral+ (M,( max- g1))) <> +infty by A73;

 (( max+ (f1 + g1)) + ( max- f1)) is E -measurable by A57, A81, A51, A40, A27, Th43;



 then

 A93: ( integral+ (M,((( max+ (f1 + g1)) + ( max- f1)) + ( max- g1)))) = (( integral+ (M,(( max+ (f1 + g1)) + ( max- f1)))) + ( integral+ (M,( max- g1)))) by A57, A48, A43, A49, A42, A55, A52, Lm10

 .= ((( integral+ (M,( max+ (f1 + g1)))) + ( integral+ (M,( max- f1)))) + ( integral+ (M,( max- g1)))) by A18, A57, A68, A46, A77, A64, A88, A82, Lm10;

 ( max+ g1) is E -measurable by A51, MESFUNC2: 25;



 then ( integral+ (M,((( max- (f1 + g1)) + ( max+ f1)) + ( max+ g1)))) = (( integral+ (M,(( max- (f1 + g1)) + ( max+ f1)))) + ( integral+ (M,( max+ g1)))) by A57, A48, A41, A53, A44, A54, A89, Lm10

 .= ((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + ( integral+ (M,( max+ g1)))) by A18, A57, A66, A78, A45, A63, A90, A91, Lm10;

 then ((( integral+ (M,( max+ (f1 + g1)))) + ( integral+ (M,( max- f1)))) + (( integral+ (M,( max- g1))) - ( integral+ (M,( max- g1))))) = (((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + ( integral+ (M,( max+ g1)))) - ( integral+ (M,( max- g1)))) by A76, A80, A92, A93, XXREAL_3: 30;

 then ((( integral+ (M,( max+ (f1 + g1)))) + ( integral+ (M,( max- f1)))) + (( integral+ (M,( max- g1))) - ( integral+ (M,( max- g1))))) = ((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by A74, A79, A80, A92, XXREAL_3: 30;

 then ((( integral+ (M,( max+ (f1 + g1)))) + ( integral+ (M,( max- f1)))) + 0. ) = ((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by XXREAL_3: 7;

 then

 A94: (( integral+ (M,( max+ (f1 + g1)))) + ( integral+ (M,( max- f1)))) = ((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by XXREAL_3: 4;



 A95: f1 is_integrable_on M by A1, Th97;

 then

 A96: 0 <= ( integral+ (M,( max+ f1))) by Th96;



 A97: (f1 + g1) is_integrable_on M by A95, A73, Th108;

 then

 A98: ( integral+ (M,( max+ (f1 + g1)))) <> +infty ;



 A99: ( integral+ (M,( max- (f1 + g1)))) <> +infty by A97;

 then

 A100: ( - ( integral+ (M,( max- (f1 + g1))))) <> -infty by XXREAL_3: 23;



 A101: 0 <= ( integral+ (M,( max- (f1 + g1)))) by A97, Th96;



 A102: ( integral+ (M,( max- f1))) <> +infty by A95;

 then

 A103: ( - ( integral+ (M,( max- f1)))) <> -infty by XXREAL_3: 23;



 A104: ( integral+ (M,( max+ f1))) <> +infty by A95;



 A105: 0 <= ( integral+ (M,( max- f1))) by A95, Th96;

 0 <= ( integral+ (M,( max+ (f1 + g1)))) by A97, Th96;

 then ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1))))) + ( integral+ (M,( max- f1)))) = (( - ( integral+ (M,( max- (f1 + g1))))) + ((( integral+ (M,( max- (f1 + g1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1)))))) by A105, A102, A98, A94, XXREAL_3: 29;

 then ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1))))) + ( integral+ (M,( max- f1)))) = (( - ( integral+ (M,( max- (f1 + g1))))) + (( integral+ (M,( max- (f1 + g1)))) + (( integral+ (M,( max+ f1))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))))) by A96, A104, A101, A99, XXREAL_3: 29;

 then ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1))))) + ( integral+ (M,( max- f1)))) = ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max- (f1 + g1))))) + (( integral+ (M,( max+ f1))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1)))))) by A101, A99, A100, XXREAL_3: 29;

 then ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1))))) + ( integral+ (M,( max- f1)))) = ( 0 + (( integral+ (M,( max+ f1))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1)))))) by XXREAL_3: 7;

 then ((( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1))))) + ( integral+ (M,( max- f1)))) = (( integral+ (M,( max+ f1))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by XXREAL_3: 4;

 then ((( - ( integral+ (M,( max- f1)))) + ( integral+ (M,( max- f1)))) + (( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1)))))) = (( - ( integral+ (M,( max- f1)))) + (( integral+ (M,( max+ f1))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1)))))) by A105, A102, A103, XXREAL_3: 29;

 then ((( - ( integral+ (M,( max- f1)))) + ( integral+ (M,( max- f1)))) + (( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1)))))) = ((( - ( integral+ (M,( max- f1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by A96, A104, A105, A103, XXREAL_3: 29;

 then ( 0 + (( - ( integral+ (M,( max- (f1 + g1))))) + ( integral+ (M,( max+ (f1 + g1)))))) = ((( - ( integral+ (M,( max- f1)))) + ( integral+ (M,( max+ f1)))) + (( integral+ (M,( max+ g1))) - ( integral+ (M,( max- g1))))) by XXREAL_3: 7;

 then

 A106: ( Integral (M,(f1 + g1))) = (( Integral (M,f1)) + ( Integral (M,g1))) by XXREAL_3: 4;

 ( Integral (M,g)) = ( Integral (M,g1)) by A3, A5, A50, A60, Th95;

 hence thesis by A3, A5, A60, A71, A15, A16, A18, A62, A17, A72, A7, A57, A48, A86, A31, A87, A95, A73, A106, A61, Th108;

 end;



 Lm13: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M & ( dom f) = ( dom g) holds (f + g) is_integrable_on M & ( Integral (M,(f + g))) = (( Integral (M,f)) + ( Integral (M,g)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M and

 A3: ( dom f) = ( dom g);

 thus (f + g) is_integrable_on M by A1, A2, Th108;

 then

 A4: ex K0 be Element of S st K0 = ( dom (f + g)) & (f + g) is K0 -measurable;

 ex E,NFG,NFPG be Element of S st E c= ( dom f) & NFG c= ( dom f) & E = (( dom f) \ NFG) & (f | E) is real-valued & E = ( dom (f | E)) & (f | E) is E -measurable & (f | E) is_integrable_on M & ( Integral (M,f)) = ( Integral (M,(f | E))) & E c= ( dom g) & NFG c= ( dom g) & E = (( dom g) \ NFG) & (g | E) is real-valued & E = ( dom (g | E)) & (g | E) is E -measurable & (g | E) is_integrable_on M & ( Integral (M,g)) = ( Integral (M,(g | E))) & E c= ( dom (f + g)) & NFPG c= ( dom (f + g)) & E = (( dom (f + g)) \ NFPG) & (M . NFG) = 0 & (M . NFPG) = 0 & E = ( dom ((f + g) | E)) & ((f + g) | E) is E -measurable & ((f + g) | E) is_integrable_on M & ((f + g) | E) = ((f | E) + (g | E)) & ( Integral (M,((f + g) | E))) = (( Integral (M,(f | E))) + ( Integral (M,(g | E)))) by A1, A2, A3, Lm12;

 hence thesis by A4, Th95;

 end;

 theorem :: MESFUNC5:109

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL st f is_integrable_on M & g is_integrable_on M holds ex E be Element of S st E = (( dom f) /\ ( dom g)) & ( Integral (M,(f + g))) = (( Integral (M,(f | E))) + ( Integral (M,(g | E))))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL ;

 assume that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M;

 consider B be Element of S such that

 A3: B = ( dom g) and g is B -measurable by A2;

 consider A be Element of S such that

 A4: A = ( dom f) and f is A -measurable by A1;

 set E = (A /\ B);

 set g1 = (g | E);

 set f1 = (f | E);

 take E = (A /\ B);



 A5: ( dom f1) = (( dom f) /\ (A /\ B)) by RELAT_1: 61

 .= ((A /\ A) /\ B) by A4, XBOOLE_1: 16;



 A6: (f1 " { +infty }) = (E /\ (f " { +infty })) by FUNCT_1: 70;

 (g1 " { -infty }) = (E /\ (g " { -infty })) by FUNCT_1: 70;



 then

 A7: ((f1 " { +infty }) /\ (g1 " { -infty })) = ((((f " { +infty }) /\ E) /\ E) /\ (g " { -infty })) by A6, XBOOLE_1: 16

 .= (((f " { +infty }) /\ (E /\ E)) /\ (g " { -infty })) by XBOOLE_1: 16

 .= (E /\ ((f " { +infty }) /\ (g " { -infty }))) by XBOOLE_1: 16;



 A8: (g1 " { +infty }) = (E /\ (g " { +infty })) by FUNCT_1: 70;

 (f1 " { -infty }) = (E /\ (f " { -infty })) by FUNCT_1: 70;



 then ((f1 " { -infty }) /\ (g1 " { +infty })) = ((((f " { -infty }) /\ E) /\ E) /\ (g " { +infty })) by A8, XBOOLE_1: 16

 .= (((f " { -infty }) /\ (E /\ E)) /\ (g " { +infty })) by XBOOLE_1: 16

 .= (E /\ ((f " { -infty }) /\ (g " { +infty }))) by XBOOLE_1: 16;

 then

 A9: (((f1 " { -infty }) /\ (g1 " { +infty })) \/ ((f1 " { +infty }) /\ (g1 " { -infty }))) = (E /\ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by A7, XBOOLE_1: 23;



 A10: ( dom g1) = (( dom g) /\ (A /\ B)) by RELAT_1: 61

 .= ((B /\ B) /\ A) by A3, XBOOLE_1: 16;



 A11: ( dom (f1 + g1)) = ((( dom f1) /\ ( dom g1)) \ (((f1 " { -infty }) /\ (g1 " { +infty })) \/ ((f1 " { +infty }) /\ (g1 " { -infty })))) by MESFUNC1:def 3

 .= (E \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by A5, A10, A9, XBOOLE_1: 47

 .= ( dom (f + g)) by A4, A3, MESFUNC1:def 3;



 A12: for x be object st x in ( dom (f1 + g1)) holds ((f1 + g1) . x) = ((f + g) . x)

 proof

 let x be object;

 assume

 A13: x in ( dom (f1 + g1));

 then x in ((( dom f1) /\ ( dom g1)) \ (((f1 " { -infty }) /\ (g1 " { +infty })) \/ ((f1 " { +infty }) /\ (g1 " { -infty })))) by MESFUNC1:def 3;

 then

 A14: x in (( dom f1) /\ ( dom g1)) by XBOOLE_0:def 5;

 then

 A15: x in ( dom f1) by XBOOLE_0:def 4;



 A16: x in ( dom g1) by A14, XBOOLE_0:def 4;

 ((f1 + g1) . x) = ((f1 . x) + (g1 . x)) by A13, MESFUNC1:def 3

 .= ((f . x) + (g1 . x)) by A15, FUNCT_1: 47

 .= ((f . x) + (g . x)) by A16, FUNCT_1: 47;

 hence thesis by A11, A13, MESFUNC1:def 3;

 end;

 thus E = (( dom f) /\ ( dom g)) by A4, A3;



 A17: g1 is_integrable_on M by A2, Th97;

 f1 is_integrable_on M by A1, Th97;

 then ( Integral (M,(f1 + g1))) = (( Integral (M,f1)) + ( Integral (M,g1))) by A17, A5, A10, Lm13;

 hence thesis by A11, A12, FUNCT_1: 2;

 end;

 theorem :: MESFUNC5:110



 Th110: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real st f is_integrable_on M holds (c (#) f) is_integrable_on M & ( Integral (M,(c (#) f))) = (c * ( Integral (M,f)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real such that

 A1: f is_integrable_on M;



 A2: ( integral+ (M,( max+ f))) <> +infty by A1;

 consider A be Element of S such that

 A3: A = ( dom f) and

 A4: f is A -measurable by A1;



 A5: (c (#) f) is A -measurable by A3, A4, Th49;



 A6: ( dom ( max- f)) = A by A3, MESFUNC2:def 3;



 A7: ( integral+ (M,( max- f))) <> +infty by A1;

 0 <= ( integral+ (M,( max- f))) by A1, Th96;

 then

 reconsider I2 = ( integral+ (M,( max- f))) as Element of REAL by A7, XXREAL_0: 14;



 A8: ( max- f) is nonnegative by Lm1;

 0 <= ( integral+ (M,( max+ f))) by A1, Th96;

 then

 reconsider I1 = ( integral+ (M,( max+ f))) as Element of REAL by A2, XXREAL_0: 14;



 A9: ( max+ f) is nonnegative by Lm1;



 A10: ( dom (c (#) f)) = A by A3, MESFUNC1:def 6;



 A11: ( dom ( max+ f)) = A by A3, MESFUNC2:def 2;

 per cases ;

 suppose

 A12: 0 <= c;

 (c * I1) in REAL by XREAL_0:def 1;

 then

 A13: (c * ( integral+ (M,( max+ f)))) in REAL ;



 A14: ( max+ (c (#) f)) = (c (#) ( max+ f)) by A12, Th26;

 ( integral+ (M,(c (#) ( max+ f)))) = (c * ( integral+ (M,( max+ f)))) by A4, A9, A11, A12, Th86, MESFUNC2: 25;

 then

 A15: ( integral+ (M,( max+ (c (#) f)))) < +infty by A14, A13, XXREAL_0: 9;

 (c * I2) in REAL by XREAL_0:def 1;

 then (c * ( integral+ (M,( max- f)))) is Element of REAL ;

 then

 A16: (c * ( integral+ (M,( max- f)))) < +infty by XXREAL_0: 9;



 A17: ( max- (c (#) f)) = (c (#) ( max- f)) by A12, Th26;

 ( integral+ (M,(c (#) ( max- f)))) = (c * ( integral+ (M,( max- f)))) by A3, A4, A8, A6, A12, Th86, MESFUNC2: 26;

 hence (c (#) f) is_integrable_on M by A5, A10, A17, A15, A16;



 thus ( Integral (M,(c (#) f))) = (( integral+ (M,(c (#) ( max+ f)))) - ( integral+ (M,( max- (c (#) f))))) by A12, Th26

 .= (( integral+ (M,(c (#) ( max+ f)))) - ( integral+ (M,(c (#) ( max- f))))) by A12, Th26

 .= ((c * ( integral+ (M,( max+ f)))) - ( integral+ (M,(c (#) ( max- f))))) by A4, A9, A11, A12, Th86, MESFUNC2: 25

 .= ((c * ( integral+ (M,( max+ f)))) - (c * ( integral+ (M,( max- f))))) by A3, A4, A8, A6, A12, Th86, MESFUNC2: 26

 .= (c * ( Integral (M,f))) by XXREAL_3: 100;

 end;

 suppose

 A18: c < 0 ;

 ( - ( - c)) = c;

 then

 consider a be Real such that

 A19: c = ( - a) and

 A20: a > 0 by A18;



 A21: ( max+ (c (#) f)) = (a (#) ( max- f)) by A19, A20, Th27;



 A22: ( max- (c (#) f)) = (a (#) ( max+ f)) by A19, A20, Th27;

 (a * I2) in REAL by XREAL_0:def 1;

 then

 A23: (a * ( integral+ (M,( max- f)))) in REAL ;

 ( integral+ (M,(a (#) ( max- f)))) = (a * ( integral+ (M,( max- f)))) by A3, A4, A8, A6, A20, Th86, MESFUNC2: 26;

 then

 A24: ( integral+ (M,( max+ (c (#) f)))) < +infty by A21, A23, XXREAL_0: 9;

 (a * I1) in REAL by XREAL_0:def 1;

 then (a * ( integral+ (M,( max+ f)))) is Element of REAL ;

 then

 A25: (a * ( integral+ (M,( max+ f)))) < +infty by XXREAL_0: 9;

 ( integral+ (M,(a (#) ( max+ f)))) = (a * ( integral+ (M,( max+ f)))) by A4, A9, A11, A20, Th86, MESFUNC2: 25;

 hence (c (#) f) is_integrable_on M by A5, A10, A22, A24, A25;



 thus ( Integral (M,(c (#) f))) = ((a * ( integral+ (M,( max- f)))) - ( integral+ (M,(a (#) ( max+ f))))) by A3, A4, A8, A6, A20, A21, A22, Th86, MESFUNC2: 26

 .= ((a * ( integral+ (M,( max- f)))) - (a * ( integral+ (M,( max+ f))))) by A4, A9, A11, A20, Th86, MESFUNC2: 25

 .= (a * (( integral+ (M,( max- f))) - ( integral+ (M,( max+ f))))) by XXREAL_3: 100

 .= (a * ( - (( integral+ (M,( max+ f))) - ( integral+ (M,( max- f)))))) by XXREAL_3: 26

 .= ( - (a * (( integral+ (M,( max+ f))) - ( integral+ (M,( max- f)))))) by XXREAL_3: 92

 .= (c * ( Integral (M,f))) by A19, XXREAL_3: 92;

 end;

 end;

 definition

 let X be non empty set;

 let S be SigmaField of X;

 let M be sigma_Measure of S;

 let f be PartFunc of X, ExtREAL ;

 let B be Element of S;

 :: MESFUNC5:def18

 func Integral_on (M,B,f) -> Element of ExtREAL equals ( Integral (M,(f | B)));

 coherence ;

 end

 theorem :: MESFUNC5:111

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL , B be Element of S st f is_integrable_on M & g is_integrable_on M & B c= ( dom (f + g)) holds (f + g) is_integrable_on M & ( Integral_on (M,B,(f + g))) = (( Integral_on (M,B,f)) + ( Integral_on (M,B,g)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL , B be Element of S such that

 A1: f is_integrable_on M and

 A2: g is_integrable_on M and

 A3: B c= ( dom (f + g));



 A4: ( dom (f | B)) = (( dom f) /\ B) by RELAT_1: 61;

 ( dom (f + g)) = ((( dom f) /\ ( dom g)) \ (((f " { -infty }) /\ (g " { +infty })) \/ ((f " { +infty }) /\ (g " { -infty })))) by MESFUNC1:def 3;

 then

 A5: ( dom (f + g)) c= (( dom f) /\ ( dom g)) by XBOOLE_1: 36;

 (( dom f) /\ ( dom g)) c= ( dom f) by XBOOLE_1: 17;

 then ( dom (f + g)) c= ( dom f) by A5;

 then

 A6: ( dom (f | B)) = B by A3, A4, XBOOLE_1: 1, XBOOLE_1: 28;



 A7: ( Integral_on (M,B,(f + g))) = ( Integral (M,((f | B) + (g | B)))) by A3, Th29;



 A8: (g | B) is_integrable_on M by A2, Th97;



 A9: ( dom (g | B)) = (( dom g) /\ B) by RELAT_1: 61;

 (( dom f) /\ ( dom g)) c= ( dom g) by XBOOLE_1: 17;

 then ( dom (f + g)) c= ( dom g) by A5;

 then

 A10: ( dom (g | B)) = B by A3, A9, XBOOLE_1: 1, XBOOLE_1: 28;

 (f | B) is_integrable_on M by A1, Th97;

 hence thesis by A1, A2, A6, A8, A10, A7, Lm13, Th108;

 end;

 theorem :: MESFUNC5:112

 for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real, B be Element of S st f is_integrable_on M holds (f | B) is_integrable_on M & ( Integral_on (M,B,(c (#) f))) = (c * ( Integral_on (M,B,f)))

 proof

 let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL , c be Real, B be Element of S;

 assume f is_integrable_on M;

 then

 A1: (f | B) is_integrable_on M by Th97;



 A2: for x be object st x in ( dom ((c (#) f) | B)) holds (((c (#) f) | B) . x) = ((c (#) (f | B)) . x)

 proof

 let x be object;

 assume

 A3: x in ( dom ((c (#) f) | B));

 then

 A4: (((c (#) f) | B) . x) = ((c (#) f) . x) by FUNCT_1: 47;



 A5: x in (( dom (c (#) f)) /\ B) by A3, RELAT_1: 61;

 then x in (( dom f) /\ B) by MESFUNC1:def 6;

 then

 A6: x in ( dom (f | B)) by RELAT_1: 61;

 x in ( dom (c (#) f)) by A5, XBOOLE_0:def 4;

 then (((c (#) f) | B) . x) = (c * (f . x)) by A4, MESFUNC1:def 6;

 then

 A7: (((c (#) f) | B) . x) = (c * ((f | B) . x)) by A6, FUNCT_1: 47;

 x in ( dom (c (#) (f | B))) by A6, MESFUNC1:def 6;

 hence thesis by A7, MESFUNC1:def 6;

 end;

 ( dom ((c (#) f) | B)) = (( dom (c (#) f)) /\ B) by RELAT_1: 61;

 then ( dom ((c (#) f) | B)) = (( dom f) /\ B) by MESFUNC1:def 6;

 then ( dom ((c (#) f) | B)) = ( dom (f | B)) by RELAT_1: 61;

 then ( dom ((c (#) f) | B)) = ( dom (c (#) (f | B))) by MESFUNC1:def 6;

 then ((c (#) f) | B) = (c (#) (f | B)) by A2, FUNCT_1: 2;

 hence thesis by A1, Th110;

 end;