dblseq_3.miz

begin



 Lm1: for X be set, x be object holds ((X --> 0. ) . x) = 0

 proof

 let X be set, x be object;

 set M = (X --> 0. );

 per cases ;

 suppose x in ( dom M);

 then x in X;

 hence ((X --> 0. ) . x) = 0 by FUNCOP_1: 7;

 end;

 suppose not x in ( dom M);

 then (M . x) = 0 by FUNCT_1:def 2;

 hence ((X --> 0. ) . x) = 0 ;

 end;

 end;

 registration

 let X be non empty set;

 cluster nonnegative nonpositive for Function of X, REAL ;

 existence

 proof

 reconsider M = (X --> 0 ) as Function of X, REAL by FUNCOP_1: 45, XREAL_0:def 1;

 take M;



 A1: M is Function of X, ExtREAL by FUNCOP_1: 45, XXREAL_0:def 1;

 for x be object holds 0 <= (M . x);

 hence M is nonnegative by A1, SUPINF_2: 51;

 for x be object holds (M . x) <= 0 by Lm1;

 hence M is nonpositive by A1, MESFUNC5: 8;

 end;

 end

 registration

 let X be non empty set;

 cluster without-infty without+infty nonnegative nonpositive for Function of X, ExtREAL ;

 existence

 proof

 reconsider M = (X --> 0. ) as Function of X, ExtREAL ;

 take M;

 for x be object holds -infty < (M . x);

 hence M is without-infty by MESFUNC5:def 5;

 for x be object holds (M . x) < +infty by Lm1;

 hence M is without+infty by MESFUNC5:def 6;

 for x be object holds 0 <= (M . x);

 hence M is nonnegative by SUPINF_2: 51;

 for x be object holds (M . x) <= 0 by Lm1;

 hence M is nonpositive by MESFUNC5: 8;

 end;

 end



 Lm2: for X be non empty set holds (X --> 0. ) is without-infty Function of X, ExtREAL & (X --> 0. ) is without+infty Function of X, ExtREAL

 proof

 let X be non empty set;

 reconsider P = (X --> 0. ) as Function of X, ExtREAL ;

 for x be object holds -infty < (P . x);

 hence (X --> 0. ) is without-infty Function of X, ExtREAL by MESFUNC5:def 5;

 reconsider M = (X --> 0. ) as Function of X, ExtREAL ;

 for x be object holds (M . x) < +infty by Lm1;

 hence (X --> 0. ) is without+infty Function of X, ExtREAL by MESFUNC5:def 6;

 end;

 registration

 let X be non empty set;

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

 correctness

 proof

 now

 let f be nonnegative Function of X, ExtREAL ;

 for x be object holds (f . x) > -infty by SUPINF_2: 51;

 hence f is without-infty by MESFUNC5:def 5;

 end;

 hence thesis;

 end;

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

 correctness

 proof

 now

 let f be nonpositive Function of X, ExtREAL ;

 for x be object holds (f . x) < +infty by MESFUNC5: 8;

 hence f is without+infty by MESFUNC5:def 6;

 end;

 hence thesis;

 end;

 cluster without-infty for without+infty Function of X, ExtREAL ;

 existence

 proof

 reconsider f = (X --> 0. ) as without+infty Function of X, ExtREAL by Lm2;

 take f;

 thus f is without-infty by Lm2;

 end;

 end

 definition

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

 :: original: -

 redefine

 func - f -> Function of X, ExtREAL ;

 correctness

 proof

 ( dom ( - f)) = ( dom f) by MESFUNC1:def 7;

 then

 A1: ( dom ( - f)) = X by FUNCT_2:def 1;

 now

 let x be object;

 assume x in X;

 then

 reconsider x1 = x as Element of X;

 (( - f) . x) = ( - (f . x1)) by A1, MESFUNC1:def 7;

 hence (( - f) . x) in ExtREAL ;

 end;

 hence ( - f) is Function of X, ExtREAL by A1, FUNCT_2: 3;

 end;

 end

 registration

 let X be non empty set, f be without-infty Function of X, ExtREAL ;

 cluster ( - f) -> without+infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose

 A1: x in ( dom ( - f));

 then

 reconsider x1 = x as Element of X;

 (( - f) . x) = ( - (f . x1)) by A1, MESFUNC1:def 7;

 hence (( - f) . x) < +infty by XXREAL_3: 5, XXREAL_3: 38, MESFUNC5:def 5;

 end;

 suppose not x in ( dom ( - f));

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

 end;

 end;

 hence ( - f) is without+infty by MESFUNC5:def 6;

 end;

 end

 registration

 let X be non empty set, f be without+infty Function of X, ExtREAL ;

 cluster ( - f) -> without-infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose

 A1: x in ( dom ( - f));

 then

 reconsider x1 = x as Element of X;

 (( - f) . x) = ( - (f . x1)) by A1, MESFUNC1:def 7;

 hence (( - f) . x) > -infty by XXREAL_3: 6, XXREAL_3: 38, MESFUNC5:def 6;

 end;

 suppose not x in ( dom ( - f));

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

 end;

 end;

 hence ( - f) is without-infty by MESFUNC5:def 5;

 end;

 end

 registration

 let X be non empty set, f be nonnegative Function of X, ExtREAL ;

 cluster ( - f) -> nonpositive;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose

 A1: x in ( dom ( - f));

 then

 reconsider x1 = x as Element of X;



 A2: (( - f) . x) = ( - (f . x1)) by A1, MESFUNC1:def 7;

 (f . x1) >= 0 by SUPINF_2: 51;

 hence (( - f) . x) <= 0 by A2;

 end;

 suppose not x in ( dom ( - f));

 hence (( - f) . x) <= 0 by FUNCT_1:def 2;

 end;

 end;

 hence ( - f) is nonpositive by MESFUNC5: 8;

 end;

 end

 registration

 let X be non empty set, f be nonpositive Function of X, ExtREAL ;

 cluster ( - f) -> nonnegative;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose

 A1: x in ( dom ( - f));

 then

 reconsider x1 = x as Element of X;



 A2: (( - f) . x) = ( - (f . x1)) by A1, MESFUNC1:def 7;

 (f . x1) <= 0 by MESFUNC5: 8;

 hence (( - f) . x) >= 0 by A2;

 end;

 suppose not x in ( dom ( - f));

 hence (( - f) . x) >= 0 by FUNCT_1:def 2;

 end;

 end;

 hence ( - f) is nonnegative by SUPINF_2: 51;

 end;

 end

 registration

 let A,B be non empty set;

 let f be without-infty Function of [:A, B:], ExtREAL ;

 cluster ( ~ f) -> without-infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose x in ( dom ( ~ f));

 then

 consider b,a be object such that

 A1: b in B & a in A & x = [b, a] by ZFMISC_1:def 2;

 reconsider a as Element of A by A1;

 reconsider b as Element of B by A1;

 (f . (a,b)) > -infty by MESFUNC5:def 5;

 then (( ~ f) . (b,a)) > -infty by FUNCT_4:def 8;

 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 thesis by MESFUNC5:def 5;

 end;

 end

 registration

 let A,B be non empty set;

 let f be without+infty Function of [:A, B:], ExtREAL ;

 cluster ( ~ f) -> without+infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose x in ( dom ( ~ f));

 then

 consider b,a be object such that

 A1: b in B & a in A & x = [b, a] by ZFMISC_1:def 2;

 reconsider a as Element of A by A1;

 reconsider b as Element of B by A1;

 (f . (a,b)) < +infty by MESFUNC5:def 6;

 then (( ~ f) . (b,a)) < +infty by FUNCT_4:def 8;

 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 thesis by MESFUNC5:def 6;

 end;

 end

 registration

 let A,B be non empty set;

 let f be nonnegative Function of [:A, B:], ExtREAL ;

 cluster ( ~ f) -> nonnegative;

 correctness

 proof

 now

 let x be object;

 assume x in ( dom ( ~ f));

 then

 consider b,a be object such that

 A1: b in B & a in A & x = [b, a] by ZFMISC_1:def 2;

 reconsider a as Element of A by A1;

 reconsider b as Element of B by A1;

 (f . (a,b)) >= 0 by SUPINF_2: 51;

 then (( ~ f) . (b,a)) >= 0 by FUNCT_4:def 8;

 hence (( ~ f) . x) >= 0 by A1;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 end

 registration

 let A,B be non empty set;

 let f be nonpositive Function of [:A, B:], ExtREAL ;

 cluster ( ~ f) -> nonpositive;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose not x in ( dom ( ~ f));

 hence (( ~ f) . x) <= 0. by FUNCT_1:def 2;

 end;

 suppose x in ( dom ( ~ f));

 then

 consider b,a be object such that

 A1: b in B & a in A & x = [b, a] by ZFMISC_1:def 2;

 reconsider a as Element of A by A1;

 reconsider b as Element of B by A1;

 (f . (a,b)) <= 0 by MESFUNC5: 8;

 then (( ~ f) . (b,a)) <= 0 by FUNCT_4:def 8;

 hence (( ~ f) . x) <= 0. by A1;

 end;

 end;

 hence ( ~ f) is nonpositive by MESFUNC5: 8;

 end;

 end

 theorem :: DBLSEQ_3:1



 Th1: for seq be ExtREAL_sequence holds ( Partial_Sums ( - seq)) = ( - ( Partial_Sums seq))

 proof

 let seq be ExtREAL_sequence;



 A1: ( dom ( - seq)) = NAT & ( dom ( - ( Partial_Sums seq))) = NAT by FUNCT_2:def 1;

 defpred Q[ Nat] means (( - ( Partial_Sums seq)) . $1) = ( - (( Partial_Sums seq) . $1));



 A3: Q[ 0 ] by A1, MESFUNC1:def 7;



 A4: for n be Nat st Q[n] holds Q[(n + 1)] by A1, MESFUNC1:def 7;



 A5: for n be Nat holds Q[n] from NAT_1:sch 2( A3, A4);

 defpred P[ Nat] means (( Partial_Sums ( - seq)) . $1) = (( - ( Partial_Sums seq)) . $1);

 (( Partial_Sums ( - seq)) . 0 ) = (( - seq) . 0 ) by MESFUNC9:def 1;

 then

 A6: (( Partial_Sums ( - seq)) . 0 ) = ( - (seq . 0 )) by A1, MESFUNC1:def 7;

 (( Partial_Sums seq) . 0 ) = (seq . 0 ) by MESFUNC9:def 1;

 then

 A7: P[ 0 ] by A1, A6, MESFUNC1:def 7;



 A8: for n be Nat st P[n] holds P[(n + 1)]

 proof

 let n be Nat;

 assume

 A9: P[n];

 (( Partial_Sums ( - seq)) . (n + 1)) = ((( - ( Partial_Sums seq)) . n) + (( - seq) . (n + 1))) by A9, MESFUNC9:def 1

 .= ((( - ( Partial_Sums seq)) . n) + ( - (seq . (n + 1)))) by A1, MESFUNC1:def 7

 .= (( - (( Partial_Sums seq) . n)) - (seq . (n + 1))) by A5

 .= ( - ((( Partial_Sums seq) . n) + (seq . (n + 1)))) by XXREAL_3: 25

 .= ( - (( Partial_Sums seq) . (n + 1))) by MESFUNC9:def 1;

 hence P[(n + 1)] by A1, MESFUNC1:def 7;

 end;

 for n be Nat holds P[n] from NAT_1:sch 2( A7, A8);

 then for n be Element of NAT holds (( Partial_Sums ( - seq)) . n) = (( - ( Partial_Sums seq)) . n);

 hence thesis by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:2



 Th2: for X be non empty set, f be PartFunc of X, ExtREAL holds ( - ( - f)) = f

 proof

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



 A1: ( dom f) = ( dom ( - f)) by MESFUNC1:def 7;

 then

 A2: ( dom f) = ( dom ( - ( - f))) by MESFUNC1:def 7;

 now

 let x be object;

 assume

 A3: x in ( dom f);

 then (( - f) . x) = ( - (f . x)) by A1, MESFUNC1:def 7;

 then (( - ( - f)) . x) = ( - ( - (f . x))) by A2, A3, MESFUNC1:def 7;

 hence (( - ( - f)) . x) = (f . x);

 end;

 hence thesis by A2, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:3

 for X,Y be non empty set, f be Function of [:X, Y:], ExtREAL holds ( ~ ( - f)) = ( - ( ~ f))

 proof

 let X,Y be non empty set, f be Function of [:X, Y:], ExtREAL ;

 now

 let z be Element of [:Y, X:];

 consider y,x be object such that

 A1: y in Y & x in X & z = [y, x] by ZFMISC_1:def 2;



 A2: ( dom ( - f)) = [:X, Y:] & ( dom ( - ( ~ f))) = [:Y, X:] by FUNCT_2:def 1;

 reconsider y as Element of Y by A1;

 reconsider x as Element of X by A1;

 reconsider z1 = [x, y] as Element of [:X, Y:] by ZFMISC_1: 87;

 (( ~ ( - f)) . z) = (( ~ ( - f)) . (y,x)) by A1;

 then (( ~ ( - f)) . z) = (( - f) . (x,y)) by FUNCT_4:def 8;

 then (( ~ ( - f)) . z) = ( - (f . z1)) by A2, MESFUNC1:def 7;

 then (( ~ ( - f)) . z) = ( - (f . (x,y)));

 then (( ~ ( - f)) . z) = ( - (( ~ f) . (y,x))) by FUNCT_4:def 8;

 hence (( ~ ( - f)) . z) = (( - ( ~ f)) . z) by A1, A2, MESFUNC1:def 7;

 end;

 hence thesis by FUNCT_2:def 8;

 end;

 registration

 let seq be nonnegative ExtREAL_sequence;

 cluster ( Partial_Sums seq) -> nonnegative;

 correctness by MESFUNC9: 16;

 end

 registration

 let seq be nonpositive ExtREAL_sequence;

 cluster ( Partial_Sums seq) -> nonpositive;

 correctness

 proof

 set f = ( - seq);

 ( Partial_Sums f) is nonnegative;

 then ( - ( Partial_Sums seq)) is nonnegative by Th1;

 then ( - ( - ( Partial_Sums seq))) is nonpositive;

 hence ( Partial_Sums seq) is nonpositive by Th2;

 end;

 end

 theorem :: DBLSEQ_3:4



 Th4: for seq be nonnegative ExtREAL_sequence, m be Nat holds (seq . m) <= (( Partial_Sums seq) . m)

 proof

 let seq be nonnegative ExtREAL_sequence, m be Nat;

 defpred P[ Nat] means (seq . $1) <= (( Partial_Sums seq) . $1);



 A1: P[ 0 ] by MESFUNC9:def 1;



 A2: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume P[k];

 (( Partial_Sums seq) . (k + 1)) = ((( Partial_Sums seq) . k) + (seq . (k + 1))) by MESFUNC9:def 1;

 hence P[(k + 1)] by XXREAL_3: 39, SUPINF_2: 51;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A1, A2);

 hence (seq . m) <= (( Partial_Sums seq) . m);

 end;

 theorem :: DBLSEQ_3:5

 for seq be nonpositive ExtREAL_sequence, m be Nat holds (seq . m) >= (( Partial_Sums seq) . m)

 proof

 let seq be nonpositive ExtREAL_sequence, m be Nat;

 reconsider f = ( - seq) as nonnegative ExtREAL_sequence;



 A1: ( dom f) = NAT & ( dom seq) = NAT & ( dom ( - f)) = NAT & ( dom ( Partial_Sums seq)) = NAT & ( dom ( - ( Partial_Sums seq))) = NAT & ( dom ( - ( - ( Partial_Sums seq)))) = NAT by FUNCT_2:def 1;



 A2: m in NAT by ORDINAL1:def 12;

 (f . m) <= (( Partial_Sums f) . m) by Th4;

 then (f . m) <= (( - ( Partial_Sums seq)) . m) by Th1;

 then ( - (( - seq) . m)) >= ( - (( - ( Partial_Sums seq)) . m)) by XXREAL_3: 38;

 then (( - ( - seq)) . m) >= ( - (( - ( Partial_Sums seq)) . m)) by A1, A2, MESFUNC1:def 7;

 then (seq . m) >= ( - (( - ( Partial_Sums seq)) . m)) by Th2;

 then (seq . m) >= ( - ( - (( Partial_Sums seq) . m))) by A1, A2, MESFUNC1:def 7;

 hence thesis;

 end;

 theorem :: DBLSEQ_3:6

 for X be non empty set, f be without-infty without+infty Function of X, ExtREAL holds f is Function of X, REAL

 proof

 let X be non empty set, f be without-infty without+infty Function of X, ExtREAL ;



 A1: ( dom f) = X by FUNCT_2:def 1;

 now

 let x be object;

 assume x in X;

 then (f . x) in ExtREAL & (f . x) > -infty & (f . x) < +infty by FUNCT_2: 5, MESFUNC5:def 5, MESFUNC5:def 6;

 hence (f . x) in REAL by XXREAL_0: 14;

 end;

 hence thesis by A1, FUNCT_2: 3;

 end;

 definition

 let X be non empty set;

 let f1,f2 be without-infty Function of X, ExtREAL ;

 :: original: +

 redefine

 func f1 + f2 -> without-infty Function of X, ExtREAL ;

 correctness

 proof



 A1: ( dom f1) = X & ( dom f2) = X by FUNCT_2:def 1;

 { -infty } misses ( rng f1) & { -infty } misses ( rng f2) by ZFMISC_1: 50, MESFUNC5:def 3;

 then (f1 " { -infty }) = {} & (f2 " { -infty }) = {} by RELAT_1: 138;

 then ((( dom f1) /\ ( dom f2)) \ (((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty })))) = X by A1;

 then

 A2: ( dom (f1 + f2)) = X by MESFUNC1:def 3;

 now

 assume -infty in ( rng (f1 + f2));

 then

 consider x be object such that

 A3: x in ( dom (f1 + f2)) & ((f1 + f2) . x) = -infty by FUNCT_1:def 3;

 reconsider x as Element of X by A3;

 ((f1 + f2) . x) = ((f1 . x) + (f2 . x)) by A3, MESFUNC1:def 3;

 then (f1 . x) = -infty or (f2 . x) = -infty by A3, XXREAL_3: 17;

 hence contradiction by MESFUNC5:def 5;

 end;

 hence thesis by A2, FUNCT_2:def 1, MESFUNC5:def 3;

 end;

 end

 definition

 let X be non empty set;

 let f1,f2 be without+infty Function of X, ExtREAL ;

 :: original: +

 redefine

 func f1 + f2 -> without+infty Function of X, ExtREAL ;

 correctness

 proof



 A1: ( dom f1) = X & ( dom f2) = X by FUNCT_2:def 1;

 { +infty } misses ( rng f1) & { +infty } misses ( rng f2) by ZFMISC_1: 50, MESFUNC5:def 4;

 then (f1 " { +infty }) = {} & (f2 " { +infty }) = {} by RELAT_1: 138;

 then ((( dom f1) /\ ( dom f2)) \ (((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty })))) = X by A1;

 then

 A2: ( dom (f1 + f2)) = X by MESFUNC1:def 3;

 now

 assume +infty in ( rng (f1 + f2));

 then

 consider x be object such that

 A3: x in ( dom (f1 + f2)) & ((f1 + f2) . x) = +infty by FUNCT_1:def 3;

 reconsider x as Element of X by A3;

 ((f1 + f2) . x) = ((f1 . x) + (f2 . x)) by A3, MESFUNC1:def 3;

 then (f1 . x) = +infty or (f2 . x) = +infty by A3, XXREAL_3: 16;

 hence contradiction by MESFUNC5:def 6;

 end;

 hence thesis by A2, FUNCT_2:def 1, MESFUNC5:def 4;

 end;

 end

 definition

 let X be non empty set;

 let f1 be without-infty Function of X, ExtREAL ;

 let f2 be without+infty Function of X, ExtREAL ;

 :: original: -

 redefine

 func f1 - f2 -> without-infty Function of X, ExtREAL ;

 correctness

 proof



 A1: ( dom f1) = X & ( dom f2) = X by FUNCT_2:def 1;

 { -infty } misses ( rng f1) & { +infty } misses ( rng f2) by ZFMISC_1: 50, MESFUNC5:def 3, MESFUNC5:def 4;

 then (f1 " { -infty }) = {} & (f2 " { +infty }) = {} by RELAT_1: 138;

 then ((( dom f1) /\ ( dom f2)) \ (((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty })))) = X by A1;

 then

 A2: ( dom (f1 - f2)) = X by MESFUNC1:def 4;

 now

 assume -infty in ( rng (f1 - f2));

 then

 consider x be object such that

 A3: x in ( dom (f1 - f2)) & ((f1 - f2) . x) = -infty by FUNCT_1:def 3;

 reconsider x as Element of X by A3;

 ((f1 - f2) . x) = ((f1 . x) - (f2 . x)) by A3, MESFUNC1:def 4;

 then (f1 . x) = -infty or (f2 . x) = +infty by A3, XXREAL_3: 19;

 hence contradiction by MESFUNC5:def 5, MESFUNC5:def 6;

 end;

 hence thesis by A2, FUNCT_2:def 1, MESFUNC5:def 3;

 end;

 end

 definition

 let X be non empty set;

 let f1 be without+infty Function of X, ExtREAL ;

 let f2 be without-infty Function of X, ExtREAL ;

 :: original: -

 redefine

 func f1 - f2 -> without+infty Function of X, ExtREAL ;

 correctness

 proof



 A1: ( dom f1) = X & ( dom f2) = X by FUNCT_2:def 1;

 { -infty } misses ( rng f2) & { +infty } misses ( rng f1) by ZFMISC_1: 50, MESFUNC5:def 3, MESFUNC5:def 4;

 then (f2 " { -infty }) = {} & (f1 " { +infty }) = {} by RELAT_1: 138;

 then ((( dom f1) /\ ( dom f2)) \ (((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty })))) = X by A1;

 then

 A2: ( dom (f1 - f2)) = X by MESFUNC1:def 4;

 now

 assume +infty in ( rng (f1 - f2));

 then

 consider x be object such that

 A3: x in ( dom (f1 - f2)) & ((f1 - f2) . x) = +infty by FUNCT_1:def 3;

 reconsider x as Element of X by A3;

 ((f1 - f2) . x) = ((f1 . x) - (f2 . x)) by A3, MESFUNC1:def 4;

 then (f2 . x) = -infty or (f1 . x) = +infty by A3, XXREAL_3: 18;

 hence contradiction by MESFUNC5:def 5, MESFUNC5:def 6;

 end;

 hence thesis by A2, FUNCT_2:def 1, MESFUNC5:def 4;

 end;

 end

 theorem :: DBLSEQ_3:7



 Th7: for X be non empty set, x be Element of X, f1,f2 be Function of X, ExtREAL holds (f1 is without-infty & f2 is without-infty implies ((f1 + f2) . x) = ((f1 . x) + (f2 . x))) & (f1 is without+infty & f2 is without+infty implies ((f1 + f2) . x) = ((f1 . x) + (f2 . x))) & (f1 is without-infty & f2 is without+infty implies ((f1 - f2) . x) = ((f1 . x) - (f2 . x))) & (f1 is without+infty & f2 is without-infty implies ((f1 - f2) . x) = ((f1 . x) - (f2 . x)))

 proof

 let X be non empty set, x be Element of X, f1,f2 be Function of X, ExtREAL ;

 hereby

 assume f1 is without-infty & f2 is without-infty;

 then

 reconsider F1 = f1, F2 = f2 as without-infty Function of X, ExtREAL ;

 ( dom (F1 + F2)) = X by FUNCT_2:def 1;

 hence ((f1 + f2) . x) = ((f1 . x) + (f2 . x)) by MESFUNC1:def 3;

 end;

 hereby

 assume f1 is without+infty & f2 is without+infty;

 then

 reconsider F1 = f1, F2 = f2 as without+infty Function of X, ExtREAL ;

 ( dom (F1 + F2)) = X by FUNCT_2:def 1;

 hence ((f1 + f2) . x) = ((f1 . x) + (f2 . x)) by MESFUNC1:def 3;

 end;

 hereby

 assume

 A1: f1 is without-infty & f2 is without+infty;

 then

 reconsider F1 = f1 as without-infty Function of X, ExtREAL ;

 reconsider F2 = f2 as without+infty Function of X, ExtREAL by A1;

 ( dom (F1 - F2)) = X by FUNCT_2:def 1;

 hence ((f1 - f2) . x) = ((f1 . x) - (f2 . x)) by MESFUNC1:def 4;

 end;

 hereby

 assume

 A1: f1 is without+infty & f2 is without-infty;

 then

 reconsider F1 = f1 as without+infty Function of X, ExtREAL ;

 reconsider F2 = f2 as without-infty Function of X, ExtREAL by A1;

 ( dom (F1 - F2)) = X by FUNCT_2:def 1;

 hence ((f1 - f2) . x) = ((f1 . x) - (f2 . x)) by MESFUNC1:def 4;

 end;

 end;



 Lm3: for X be non empty set, f1,f2 be without-infty Function of X, ExtREAL holds (f1 + f2) = (f1 - ( - f2))

 proof

 let X be non empty set, f1,f2 be without-infty Function of X, ExtREAL ;

 now

 let x be Element of X;

 ( dom ( - f2)) = X by FUNCT_2:def 1;

 then (( - f2) . x) = ( - (f2 . x)) by MESFUNC1:def 7;

 then ((f1 - ( - f2)) . x) = ((f1 . x) - ( - (f2 . x))) by Th7;

 hence ((f1 + f2) . x) = ((f1 - ( - f2)) . x) by Th7;

 end;

 hence thesis by FUNCT_2:def 8;

 end;



 Lm4: for X be non empty set, f1,f2 be without+infty Function of X, ExtREAL holds (f1 + f2) = (f1 - ( - f2))

 proof

 let X be non empty set, f1,f2 be without+infty Function of X, ExtREAL ;

 now

 let x be Element of X;

 ( dom ( - f2)) = X by FUNCT_2:def 1;

 then (( - f2) . x) = ( - (f2 . x)) by MESFUNC1:def 7;

 then ((f1 - ( - f2)) . x) = ((f1 . x) - ( - (f2 . x))) by Th7;

 hence ((f1 + f2) . x) = ((f1 - ( - f2)) . x) by Th7;

 end;

 hence thesis by FUNCT_2:def 8;

 end;



 Lm5: for X be non empty set, f1 be without-infty Function of X, ExtREAL , f2 be without+infty Function of X, ExtREAL holds (f1 - f2) = (f1 + ( - f2)) & (f2 - f1) = (f2 + ( - f1))

 proof

 let X be non empty set, f1 be without-infty Function of X, ExtREAL , f2 be without+infty Function of X, ExtREAL ;

 now

 let x be Element of X;

 ( dom ( - f2)) = X by FUNCT_2:def 1;

 then (( - f2) . x) = ( - (f2 . x)) by MESFUNC1:def 7;

 then ((f1 + ( - f2)) . x) = ((f1 . x) - (f2 . x)) by Th7;

 hence ((f1 - f2) . x) = ((f1 + ( - f2)) . x) by Th7;

 end;

 hence (f1 - f2) = (f1 + ( - f2)) by FUNCT_2:def 8;

 now

 let x be Element of X;

 ( dom ( - f1)) = X by FUNCT_2:def 1;

 then (( - f1) . x) = ( - (f1 . x)) by MESFUNC1:def 7;

 then ((f2 + ( - f1)) . x) = ((f2 . x) - (f1 . x)) by Th7;

 hence ((f2 - f1) . x) = ((f2 + ( - f1)) . x) by Th7;

 end;

 hence (f2 - f1) = (f2 + ( - f1)) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:8



 Th8: for X be non empty set, f1,f2 be without-infty Function of X, ExtREAL holds (f1 + f2) = (f1 - ( - f2)) & ( - (f1 + f2)) = (( - f1) - f2)

 proof

 let X be non empty set, f1,f2 be without-infty Function of X, ExtREAL ;

 thus (f1 + f2) = (f1 - ( - f2)) by Lm3;



 A1: ( dom ( - f1)) = X by FUNCT_2:def 1;



 A2: ( dom ( - f2)) = X by FUNCT_2:def 1;



 A3: ( dom ( - (f1 + f2))) = X by FUNCT_2:def 1;

 now

 let x be Element of X;

 (( - (f1 + f2)) . x) = ( - ((f1 + f2) . x)) by A3, MESFUNC1:def 7

 .= ( - ((f1 . x) + (f2 . x))) by Th7

 .= (( - (f1 . x)) - (f2 . x)) by XXREAL_3: 25

 .= ((( - f1) . x) + ( - (f2 . x))) by A1, MESFUNC1:def 7

 .= ((( - f1) . x) + (( - f2) . x)) by A2, MESFUNC1:def 7

 .= ((( - f1) + ( - f2)) . x) by Th7;

 hence (( - (f1 + f2)) . x) = ((( - f1) - f2) . x) by Lm5;

 end;

 hence thesis by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:9



 Th9: for X be non empty set, f1,f2 be without+infty Function of X, ExtREAL holds (f1 + f2) = (f1 - ( - f2)) & ( - (f1 + f2)) = (( - f1) - f2)

 proof

 let X be non empty set, f1,f2 be without+infty Function of X, ExtREAL ;

 thus (f1 + f2) = (f1 - ( - f2)) by Lm4;



 A1: ( dom ( - f1)) = X by FUNCT_2:def 1;



 A2: ( dom ( - f2)) = X by FUNCT_2:def 1;



 A3: ( dom ( - (f1 + f2))) = X by FUNCT_2:def 1;

 now

 let x be Element of X;

 (( - (f1 + f2)) . x) = ( - ((f1 + f2) . x)) by A3, MESFUNC1:def 7

 .= ( - ((f1 . x) + (f2 . x))) by Th7

 .= (( - (f1 . x)) - (f2 . x)) by XXREAL_3: 25

 .= ((( - f1) . x) + ( - (f2 . x))) by A1, MESFUNC1:def 7

 .= ((( - f1) . x) + (( - f2) . x)) by A2, MESFUNC1:def 7

 .= ((( - f1) + ( - f2)) . x) by Th7;

 hence (( - (f1 + f2)) . x) = ((( - f1) - f2) . x) by Lm5;

 end;

 hence thesis by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:10



 Th10: for X be non empty set, f1 be without-infty Function of X, ExtREAL , f2 be without+infty Function of X, ExtREAL holds (f1 - f2) = (f1 + ( - f2)) & (f2 - f1) = (f2 + ( - f1)) & ( - (f1 - f2)) = (( - f1) + f2) & ( - (f2 - f1)) = (( - f2) + f1)

 proof

 let X be non empty set, f1 be without-infty Function of X, ExtREAL , f2 be without+infty Function of X, ExtREAL ;

 thus

 A1: (f1 - f2) = (f1 + ( - f2)) & (f2 - f1) = (f2 + ( - f1)) by Lm5;



 thus ( - (f1 - f2)) = (( - f1) - ( - f2)) by A1, Th8

 .= (( - f1) + ( - ( - f2))) by Lm5

 .= (( - f1) + f2) by Th2;



 thus ( - (f2 - f1)) = (( - f2) - ( - f1)) by A1, Th9

 .= (( - f2) + ( - ( - f1))) by Lm5

 .= (( - f2) + f1) by Th2;

 end;

 definition

 let f be Function of [: NAT , NAT :], ExtREAL , n,m be Nat;

 :: original: .

 redefine

 func f . (n,m) -> Element of ExtREAL ;

 coherence

 proof

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

 (f . (n,m)) in ExtREAL ;

 hence thesis;

 end;

 end

 theorem :: DBLSEQ_3:11



 Th11: for f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds ((f1 + f2) . (n,m)) = ((f1 . (n,m)) + (f2 . (n,m)))

 proof

 let f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat;



 A1: n in NAT & m in NAT by ORDINAL1:def 12;

 then

 reconsider z = [n, m] as Element of [: NAT , NAT :] by ZFMISC_1: 87;

 [n, m] in [: NAT , NAT :] by A1, ZFMISC_1: 87;

 hence thesis by Th7;

 end;

 theorem :: DBLSEQ_3:12

 for f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds ((f1 + f2) . (n,m)) = ((f1 . (n,m)) + (f2 . (n,m)))

 proof

 let f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat;



 A1: n in NAT & m in NAT by ORDINAL1:def 12;

 then

 reconsider z = [n, m] as Element of [: NAT , NAT :] by ZFMISC_1: 87;

 [n, m] in [: NAT , NAT :] by A1, ZFMISC_1: 87;

 hence thesis by Th7;

 end;

 theorem :: DBLSEQ_3:13

 for f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds ((f1 - f2) . (n,m)) = ((f1 . (n,m)) - (f2 . (n,m))) & ((f2 - f1) . (n,m)) = ((f2 . (n,m)) - (f1 . (n,m)))

 proof

 let f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat;



 A1: n in NAT & m in NAT by ORDINAL1:def 12;

 then

 reconsider z = [n, m] as Element of [: NAT , NAT :] by ZFMISC_1: 87;

 [n, m] in [: NAT , NAT :] by A1, ZFMISC_1: 87;

 hence thesis by Th7;

 end;

 theorem :: DBLSEQ_3:14

 for X,Y be non empty set, f1,f2 be without-infty Function of [:X, Y:], ExtREAL holds ( ~ (f1 + f2)) = (( ~ f1) + ( ~ f2))

 proof

 let X,Y be non empty set, f1,f2 be without-infty Function of [:X, Y:], ExtREAL ;

 now

 let z be Element of [:Y, X:];

 consider y,x be object such that

 A1: y in Y & x in X & z = [y, x] by ZFMISC_1:def 2;

 reconsider y as Element of Y by A1;

 reconsider x as Element of X by A1;

 reconsider z1 = [x, y] as Element of [:X, Y:] by ZFMISC_1: 87;

 (( ~ (f1 + f2)) . z) = (( ~ (f1 + f2)) . (y,x)) by A1;

 then (( ~ (f1 + f2)) . z) = ((f1 + f2) . (x,y)) by FUNCT_4:def 8;

 then

 A2: (( ~ (f1 + f2)) . z) = ((f1 . z1) + (f2 . z1)) by Th7;

 (f1 . z1) = (f1 . (x,y)) & (f2 . z1) = (f2 . (x,y));

 then (f1 . z1) = (( ~ f1) . (y,x)) & (f2 . z1) = (( ~ f2) . (y,x)) by FUNCT_4:def 8;

 hence (( ~ (f1 + f2)) . z) = ((( ~ f1) + ( ~ f2)) . z) by A1, A2, Th7;

 end;

 hence thesis by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:15

 for X,Y be non empty set, f1,f2 be without+infty Function of [:X, Y:], ExtREAL holds ( ~ (f1 + f2)) = (( ~ f1) + ( ~ f2))

 proof

 let X,Y be non empty set, f1,f2 be without+infty Function of [:X, Y:], ExtREAL ;

 now

 let z be Element of [:Y, X:];

 consider y,x be object such that

 A1: y in Y & x in X & z = [y, x] by ZFMISC_1:def 2;

 reconsider y as Element of Y by A1;

 reconsider x as Element of X by A1;

 reconsider z1 = [x, y] as Element of [:X, Y:] by ZFMISC_1: 87;

 (( ~ (f1 + f2)) . z) = (( ~ (f1 + f2)) . (y,x)) by A1;

 then (( ~ (f1 + f2)) . z) = ((f1 + f2) . (x,y)) by FUNCT_4:def 8;

 then

 A2: (( ~ (f1 + f2)) . z) = ((f1 . z1) + (f2 . z1)) by Th7;

 (f1 . z1) = (f1 . (x,y)) & (f2 . z1) = (f2 . (x,y));

 then (f1 . z1) = (( ~ f1) . (y,x)) & (f2 . z1) = (( ~ f2) . (y,x)) by FUNCT_4:def 8;

 hence (( ~ (f1 + f2)) . z) = ((( ~ f1) + ( ~ f2)) . z) by A1, A2, Th7;

 end;

 hence thesis by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:16

 for X,Y be non empty set, f1 be without-infty Function of [:X, Y:], ExtREAL , f2 be without+infty Function of [:X, Y:], ExtREAL holds ( ~ (f1 - f2)) = (( ~ f1) - ( ~ f2)) & ( ~ (f2 - f1)) = (( ~ f2) - ( ~ f1))

 proof

 let X,Y be non empty set, f1 be without-infty Function of [:X, Y:], ExtREAL , f2 be without+infty Function of [:X, Y:], ExtREAL ;

 now

 let z be Element of [:Y, X:];

 consider y,x be object such that

 A1: y in Y & x in X & z = [y, x] by ZFMISC_1:def 2;

 reconsider y as Element of Y by A1;

 reconsider x as Element of X by A1;

 reconsider z1 = [x, y] as Element of [:X, Y:] by ZFMISC_1: 87;

 (( ~ (f1 - f2)) . z) = (( ~ (f1 - f2)) . (y,x)) by A1;

 then (( ~ (f1 - f2)) . z) = ((f1 - f2) . (x,y)) by FUNCT_4:def 8;

 then

 A2: (( ~ (f1 - f2)) . z) = ((f1 . z1) - (f2 . z1)) by Th7;

 (f1 . z1) = (f1 . (x,y)) & (f2 . z1) = (f2 . (x,y));

 then (f1 . z1) = (( ~ f1) . (y,x)) & (f2 . z1) = (( ~ f2) . (y,x)) by FUNCT_4:def 8;

 hence (( ~ (f1 - f2)) . z) = ((( ~ f1) - ( ~ f2)) . z) by A1, A2, Th7;

 end;

 hence ( ~ (f1 - f2)) = (( ~ f1) - ( ~ f2)) by FUNCT_2:def 8;

 now

 let z be Element of [:Y, X:];

 consider y,x be object such that

 A1: y in Y & x in X & z = [y, x] by ZFMISC_1:def 2;

 reconsider y as Element of Y by A1;

 reconsider x as Element of X by A1;

 reconsider z1 = [x, y] as Element of [:X, Y:] by ZFMISC_1: 87;

 (( ~ (f2 - f1)) . z) = (( ~ (f2 - f1)) . (y,x)) by A1;

 then (( ~ (f2 - f1)) . z) = ((f2 - f1) . (x,y)) by FUNCT_4:def 8;

 then

 A2: (( ~ (f2 - f1)) . z) = ((f2 . z1) - (f1 . z1)) by Th7;

 (f1 . z1) = (f1 . (x,y)) & (f2 . z1) = (f2 . (x,y));

 then (f1 . z1) = (( ~ f1) . (y,x)) & (f2 . z1) = (( ~ f2) . (y,x)) by FUNCT_4:def 8;

 hence (( ~ (f2 - f1)) . z) = ((( ~ f2) - ( ~ f1)) . z) by A1, A2, Th7;

 end;

 hence ( ~ (f2 - f1)) = (( ~ f2) - ( ~ f1)) by FUNCT_2:def 8;

 end;

 registration

 cluster convergent_to_+infty -> convergent for ExtREAL_sequence;

 correctness by MESFUNC5:def 11;

 cluster convergent_to_-infty -> convergent for ExtREAL_sequence;

 correctness by MESFUNC5:def 11;

 cluster convergent_to_finite_number -> convergent for ExtREAL_sequence;

 correctness by MESFUNC5:def 11;

 end

 registration

 cluster convergent for ExtREAL_sequence;

 existence

 proof

 reconsider z = 0 as Element of ExtREAL by XXREAL_0:def 1;

 reconsider M = ( NAT --> z) as ExtREAL_sequence;

 take M;

 for n be Nat holds (M . n) = 0 by Lm1;

 then M is convergent_to_finite_number by MESFUNC5: 52;

 hence M is convergent;

 end;

 cluster convergent for without-infty ExtREAL_sequence;

 existence

 proof

 reconsider z = 0 as Element of ExtREAL by XXREAL_0:def 1;

 reconsider M = ( NAT --> z) as ExtREAL_sequence;

 reconsider M as without-infty ExtREAL_sequence by Lm2;

 take M;

 for n be Nat holds (M . n) = 0 by Lm1;

 then M is convergent_to_finite_number by MESFUNC5: 52;

 hence M is convergent;

 end;

 cluster convergent for without+infty ExtREAL_sequence;

 existence

 proof

 reconsider z = 0 as Element of ExtREAL by XXREAL_0:def 1;

 reconsider M = ( NAT --> z) as ExtREAL_sequence;

 reconsider M as without+infty ExtREAL_sequence by Lm2;

 take M;

 for n be Nat holds (M . n) = 0 by Lm1;

 then M is convergent_to_finite_number by MESFUNC5: 52;

 hence M is convergent;

 end;

 end



 Lm6: for seq be convergent ExtREAL_sequence holds (seq is convergent_to_finite_number implies ( - seq) is convergent_to_finite_number) & (seq is convergent_to_+infty implies ( - seq) is convergent_to_-infty) & (seq is convergent_to_-infty implies ( - seq) is convergent_to_+infty) & ( - seq) is convergent & ( lim ( - seq)) = ( - ( lim seq))

 proof

 let seq be convergent ExtREAL_sequence;

 P0:

 now

 assume seq is convergent_to_finite_number;

 then

 consider g be Real such that

 A1: ( lim seq) = g & for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq . m) - ( lim seq)).| < p by MESFUNC9: 7;

 reconsider G = ( - g) as R_eal by XXREAL_0:def 1;

 A5:

 now

 let p be Real;

 assume 0 < p;

 then

 consider n be Nat such that

 A2: for m be Nat st n <= m holds |.((seq . m) - ( lim seq)).| < p by A1;

 take n;

 hereby

 let m be Nat;

 assume n <= m;

 then

 A3: |.((seq . m) - g).| < p by A1, A2;

 m in NAT by ORDINAL1:def 12;

 then m in ( dom ( - seq)) by FUNCT_2:def 1;



 then ( - ((( - seq) . m) - G)) = ( - (( - (seq . m)) - G)) by MESFUNC1:def 7

 .= ( - ( - ((seq . m) - g))) by XXREAL_3: 26

 .= ((seq . m) - g);

 hence |.((( - seq) . m) - G).| < p by A3, EXTREAL1: 29;

 end;

 end;

 hence

 A4: ( - seq) is convergent_to_finite_number by MESFUNC5:def 8;

 hence ( - seq) is convergent;

 ( lim ( - seq)) = ( - g) by A4, A5, MESFUNC5:def 12;

 hence ( lim ( - seq)) = ( - ( lim seq)) by A1, XXREAL_3:def 3;

 end;

 thus seq is convergent_to_finite_number implies ( - seq) is convergent_to_finite_number by P0;

 P1:

 now

 assume

 A6: seq is convergent_to_+infty;

 A10:

 now

 let g be Real;

 assume

 A7: g < 0 ;

 reconsider p = ( - g) as ExtReal;

 consider n be Nat such that

 A8: for m be Nat st n <= m holds p <= (seq . m) by A6, A7, MESFUNC5:def 9;

 take n;

 hereby

 let m be Nat;

 assume n <= m;

 then ( - p) >= ( - (seq . m)) by A8, XXREAL_3: 38;

 then

 A9: ( - ( - g)) >= ( - (seq . m)) by XXREAL_3:def 3;

 m in NAT by ORDINAL1:def 12;

 then m in ( dom ( - seq)) by FUNCT_2:def 1;

 hence (( - seq) . m) <= g by A9, MESFUNC1:def 7;

 end;

 end;

 hence ( - seq) is convergent_to_-infty by MESFUNC5:def 10;

 hence ( - seq) is convergent;

 ( lim ( - seq)) = -infty & ( lim seq) = +infty by A6, A10, MESFUNC9: 7, MESFUNC5:def 10;

 hence ( lim ( - seq)) = ( - ( lim seq)) by XXREAL_3: 6;

 end;

 thus seq is convergent_to_+infty implies ( - seq) is convergent_to_-infty by P1;

 P2:

 now

 assume

 A11: seq is convergent_to_-infty;

 A15:

 now

 let g be Real;

 assume

 A12: g > 0 ;

 reconsider p = ( - g) as ExtReal;

 consider n be Nat such that

 A13: for m be Nat st n <= m holds (seq . m) <= p by A11, A12, MESFUNC5:def 10;

 take n;

 hereby

 let m be Nat;

 assume n <= m;

 then ( - (seq . m)) >= ( - p) by A13, XXREAL_3: 38;

 then

 A14: ( - (seq . m)) >= ( - ( - g)) by XXREAL_3:def 3;

 m in NAT by ORDINAL1:def 12;

 then m in ( dom ( - seq)) by FUNCT_2:def 1;

 hence (( - seq) . m) >= g by A14, MESFUNC1:def 7;

 end;

 end;

 hence ( - seq) is convergent_to_+infty by MESFUNC5:def 9;

 hence ( - seq) is convergent;

 ( lim ( - seq)) = +infty & ( lim seq) = -infty by A11, A15, MESFUNC9: 7, MESFUNC5:def 9;

 hence ( lim ( - seq)) = ( - ( lim seq)) by XXREAL_3: 5;

 end;

 thus seq is convergent_to_-infty implies ( - seq) is convergent_to_+infty by P2;

 thus thesis by P0, P1, P2, MESFUNC5:def 11;

 end;

 theorem :: DBLSEQ_3:17



 Th17: for seq be convergent ExtREAL_sequence holds (seq is convergent_to_finite_number iff ( - seq) is convergent_to_finite_number) & (seq is convergent_to_+infty iff ( - seq) is convergent_to_-infty) & (seq is convergent_to_-infty iff ( - seq) is convergent_to_+infty) & ( - seq) is convergent & ( lim ( - seq)) = ( - ( lim seq))

 proof

 let seq be convergent ExtREAL_sequence;

 now

 assume ( - seq) is convergent_to_finite_number;

 then ( - ( - seq)) is convergent_to_finite_number by Lm6;

 hence seq is convergent_to_finite_number by Th2;

 end;

 hence seq is convergent_to_finite_number iff ( - seq) is convergent_to_finite_number by Lm6;

 now

 assume ( - seq) is convergent_to_-infty;

 then ( - ( - seq)) is convergent_to_+infty by Lm6;

 hence seq is convergent_to_+infty by Th2;

 end;

 hence seq is convergent_to_+infty iff ( - seq) is convergent_to_-infty by Lm6;

 now

 assume ( - seq) is convergent_to_+infty;

 then ( - ( - seq)) is convergent_to_-infty by Lm6;

 hence seq is convergent_to_-infty by Th2;

 end;

 hence seq is convergent_to_-infty iff ( - seq) is convergent_to_+infty by Lm6;

 thus ( - seq) is convergent & ( lim ( - seq)) = ( - ( lim seq)) by Lm6;

 end;

 theorem :: DBLSEQ_3:18



 Th18: for seq1,seq2 be without-infty ExtREAL_sequence st seq1 is convergent_to_+infty & seq2 is convergent_to_+infty holds (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty

 proof

 let seq1,seq2 be without-infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_+infty & seq2 is convergent_to_+infty;

 now

 let g be Real;

 assume

 A2: 0 < g;

 then

 consider n1 be Nat such that

 A3: for m be Nat st n1 <= m holds (g / 2) <= (seq1 . m) by A1, MESFUNC5:def 9;

 consider n2 be Nat such that

 A4: for m be Nat st n2 <= m holds (g / 2) <= (seq2 . m) by A1, A2, MESFUNC5:def 9;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A5: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A5, XXREAL_0: 2;

 then (g / 2) <= (seq1 . m) & (g / 2) <= (seq2 . m) by A3, A4;

 then

 A6: ((g / 2) + (g / 2)) <= ((seq1 . m) + (seq2 . m)) by XXREAL_3: 36;

 m is Element of NAT by ORDINAL1:def 12;

 hence g <= ((seq1 + seq2) . m) by A6, Th7;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds g <= ((seq1 + seq2) . m);

 end;

 hence

 A7: (seq1 + seq2) is convergent_to_+infty by MESFUNC5:def 9;

 hence (seq1 + seq2) is convergent;

 thus ( lim (seq1 + seq2)) = +infty by A7, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:19



 Th19: for seq1,seq2 be without-infty ExtREAL_sequence st seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number holds (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty

 proof

 let seq1,seq2 be without-infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number;

 then

 consider S2 be Real such that

 A2: for g be Real st 0 < g holds ex n be Nat st for m be Nat st n <= m holds |.((seq2 . m) - S2) qua ExtReal.| < g by MESFUNC5:def 8;

 now

 let g be Real;

 assume

 A3: 0 < g;

 set G = ( max (1,((2 * g) - S2)));



 A4: 1 <= G & ((2 * g) - S2) <= G by XXREAL_0: 25;

 then

 consider n1 be Nat such that

 A5: for m be Nat st n1 <= m holds G <= (seq1 . m) by A1, MESFUNC5:def 9;

 consider n2 be Nat such that

 A6: for m be Nat st n2 <= m holds |.((seq2 . m) - S2) qua ExtReal.| < g by A2, A3;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A7: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A7, XXREAL_0: 2;

 then

 A8: G <= (seq1 . m) & |.((seq2 . m) - S2) qua ExtReal.| < g by A5, A6;

 reconsider g1 = g as R_eal by XXREAL_0:def 1;

 ( - g1) < ((seq2 . m) - S2) qua ExtReal by A8, EXTREAL1: 21;

 then (( - g1) + S2 qua ExtReal) < (seq2 . m) by XXREAL_3: 53;

 then

 A9: (G + (( - g1) + S2 qua ExtReal)) <= ((seq1 . m) + (seq2 . m)) by A8, XXREAL_3: 36;

 ( - g1) = ( - g) by XXREAL_3:def 3;

 then (( - g1) + S2 qua ExtReal) = (( - g) + S2) by XXREAL_3:def 2;

 then (((2 * g) - S2) + (( - g1) + S2 qua ExtReal)) = (((2 * g) - S2) + (( - g) + S2)) by XXREAL_3:def 2;

 then g <= (G + (( - g1) + S2 qua ExtReal)) by A4, XXREAL_3: 36;

 then

 A10: g <= ((seq1 . m) + (seq2 . m)) by A9, XXREAL_0: 2;

 m is Element of NAT by ORDINAL1:def 12;

 hence g <= ((seq1 + seq2) . m) by A10, Th7;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds g <= ((seq1 + seq2) . m);

 end;

 hence

 A11: (seq1 + seq2) is convergent_to_+infty by MESFUNC5:def 9;

 hence (seq1 + seq2) is convergent;

 thus ( lim (seq1 + seq2)) = +infty by A11, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:20

 for seq1,seq2 be without+infty ExtREAL_sequence st seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number holds (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty

 proof

 let seq1,seq2 be without+infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number;

 then

 consider S2 be Real such that

 A2: for g be Real st 0 < g holds ex n be Nat st for m be Nat st n <= m holds |.((seq2 . m) - S2) qua ExtReal.| < g by MESFUNC5:def 8;

 now

 let g be Real;

 assume

 A3: 0 < g;

 set G = ( max (1,((2 * g) - S2)));



 A4: 1 <= G & ((2 * g) - S2) <= G by XXREAL_0: 25;

 then

 consider n1 be Nat such that

 A5: for m be Nat st n1 <= m holds G <= (seq1 . m) by A1, MESFUNC5:def 9;

 consider n2 be Nat such that

 A6: for m be Nat st n2 <= m holds |.((seq2 . m) - S2) qua ExtReal.| < g by A2, A3;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A7: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A7, XXREAL_0: 2;

 then

 A8: G <= (seq1 . m) & |.((seq2 . m) - S2) qua ExtReal.| < g by A5, A6;

 reconsider g1 = g as R_eal by XXREAL_0:def 1;

 ( - g1) < ((seq2 . m) - S2) qua ExtReal by A8, EXTREAL1: 21;

 then (( - g1) + S2 qua ExtReal) < (seq2 . m) by XXREAL_3: 53;

 then

 A9: (G + (( - g1) + S2 qua ExtReal)) <= ((seq1 . m) + (seq2 . m)) by A8, XXREAL_3: 36;

 ( - g1) = ( - g) by XXREAL_3:def 3;

 then (( - g1) + S2 qua ExtReal) = (( - g) + S2) by XXREAL_3:def 2;

 then (((2 * g) - S2) + (( - g1) + S2 qua ExtReal)) = (((2 * g) - S2) + (( - g) + S2)) by XXREAL_3:def 2;

 then g <= (G + (( - g1) + S2 qua ExtReal)) by A4, XXREAL_3: 36;

 then

 A10: g <= ((seq1 . m) + (seq2 . m)) by A9, XXREAL_0: 2;

 m is Element of NAT by ORDINAL1:def 12;

 hence g <= ((seq1 + seq2) . m) by A10, Th7;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds g <= ((seq1 + seq2) . m);

 end;

 hence

 A11: (seq1 + seq2) is convergent_to_+infty by MESFUNC5:def 9;

 hence (seq1 + seq2) is convergent;

 thus ( lim (seq1 + seq2)) = +infty by A11, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:21



 Th21: for seq1,seq2 be without-infty ExtREAL_sequence st seq1 is convergent_to_-infty & seq2 is convergent_to_-infty holds (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty

 proof

 let seq1,seq2 be without-infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_-infty & seq2 is convergent_to_-infty;

 now

 let g be Real;

 assume

 A2: g < 0 ;

 then

 consider n1 be Nat such that

 A3: for m be Nat st n1 <= m holds (seq1 . m) <= (g / 2) by A1, MESFUNC5:def 10;

 consider n2 be Nat such that

 A4: for m be Nat st n2 <= m holds (seq2 . m) <= (g / 2) by A1, A2, MESFUNC5:def 10;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A5: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A5, XXREAL_0: 2;

 then (seq1 . m) <= (g / 2) & (seq2 . m) <= (g / 2) by A3, A4;

 then

 A6: ((seq1 . m) + (seq2 . m)) <= ((g / 2) + (g / 2)) by XXREAL_3: 36;

 m is Element of NAT by ORDINAL1:def 12;

 hence ((seq1 + seq2) . m) <= g by A6, Th7;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds ((seq1 + seq2) . m) <= g;

 end;

 hence

 A7: (seq1 + seq2) is convergent_to_-infty by MESFUNC5:def 10;

 hence (seq1 + seq2) is convergent;

 thus ( lim (seq1 + seq2)) = -infty by A7, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:22



 Th22: for seq1,seq2 be without-infty ExtREAL_sequence st seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number holds (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty

 proof

 let seq1,seq2 be without-infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number;

 then

 consider S2 be Real such that

 A2: for g be Real st 0 < g holds ex n be Nat st for m be Nat st n <= m holds |.((seq2 . m) - S2) qua ExtReal.| < g by MESFUNC5:def 8;

 now

 let g be Real;

 assume

 A3: g < 0 ;

 set G = ( min (( - 1),((2 * g) - S2)));



 A4: G <= ( - 1) & G <= ((2 * g) - S2) by XXREAL_0: 17;

 then

 consider n1 be Nat such that

 A5: for m be Nat st n1 <= m holds (seq1 . m) <= G by A1, MESFUNC5:def 10;

 consider n2 be Nat such that

 A6: for m be Nat st n2 <= m holds |.((seq2 . m) - S2) qua ExtReal.| < ( - g) by A2, A3;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A7: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A7, XXREAL_0: 2;

 then

 A8: (seq1 . m) <= G & |.((seq2 . m) - S2) qua ExtReal.| < ( - g) by A5, A6;

 reconsider g1 = g as R_eal by XXREAL_0:def 1;



 B1: ( - g1) = ( - g) by XXREAL_3:def 3;

 then ((seq2 . m) - S2) qua ExtReal < ( - g1) by A8, EXTREAL1: 21;

 then (seq2 . m) < (( - g1) + S2 qua ExtReal) by XXREAL_3: 54;

 then

 A9: ((seq1 . m) + (seq2 . m)) <= (G + (( - g1) + S2 qua ExtReal)) by A8, XXREAL_3: 36;

 (( - g1) + S2 qua ExtReal) = (( - g) + S2) by B1, XXREAL_3:def 2;

 then (((2 * g) - S2) + (( - g1) + S2 qua ExtReal)) = (((2 * g) - S2) + (( - g) + S2)) by XXREAL_3:def 2;

 then (G + (( - g1) + S2 qua ExtReal)) <= g by A4, XXREAL_3: 36;

 then

 A10: ((seq1 . m) + (seq2 . m)) <= g by A9, XXREAL_0: 2;

 m is Element of NAT by ORDINAL1:def 12;

 hence ((seq1 + seq2) . m) <= g by A10, Th7;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds ((seq1 + seq2) . m) <= g;

 end;

 hence

 A11: (seq1 + seq2) is convergent_to_-infty by MESFUNC5:def 10;

 hence (seq1 + seq2) is convergent;

 thus ( lim (seq1 + seq2)) = -infty by A11, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:23



 Th23: for seq1,seq2 be without-infty ExtREAL_sequence st seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number holds (seq1 + seq2) is convergent_to_finite_number & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = (( lim seq1) + ( lim seq2))

 proof

 let seq1,seq2 be without-infty ExtREAL_sequence;

 assume

 A1: seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number;



 B2: not seq1 is convergent_to_-infty & not seq1 is convergent_to_+infty & not seq2 is convergent_to_-infty & not seq2 is convergent_to_+infty by A1, MESFUNC5: 50, MESFUNC5: 51;

 consider S1 be Real such that

 A2: ( lim seq1) = S1 & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq1 . m) - ( lim seq1)).| < p) & seq1 is convergent_to_finite_number by A1, B2, MESFUNC5:def 12;

 consider S2 be Real such that

 A3: ( lim seq2) = S2 & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq2 . m) - ( lim seq2)).| < p) & seq2 is convergent_to_finite_number by A1, B2, MESFUNC5:def 12;

 B3:

 now

 let p be Real;

 assume

 A4: 0 < p;

 then

 consider n1 be Nat such that

 A5: for m be Nat st n1 <= m holds |.((seq1 . m) - S1) qua ExtReal.| < (p / 2) by A2;

 consider n2 be Nat such that

 A6: for m be Nat st n2 <= m holds |.((seq2 . m) - S2) qua ExtReal.| < (p / 2) by A3, A4;

 reconsider N1 = n1, N2 = n2 as Element of NAT by ORDINAL1:def 12;

 reconsider n = ( max (N1,N2)) as Nat;



 A7: n1 <= n & n2 <= n by XXREAL_0: 25;

 now

 let m be Nat;

 assume n <= m;

 then n1 <= m & n2 <= m by A7, XXREAL_0: 2;

 then

 A8: |.((seq1 . m) - S1) qua ExtReal.| < (p / 2) & |.((seq2 . m) - S2) qua ExtReal.| < (p / 2) by A5, A6;

 then

 A9: ( |.((seq1 . m) - S1) qua ExtReal.| + |.((seq2 . m) - S2) qua ExtReal.|) < ((p / 2) + (p / 2)) by XXREAL_3: 64;

 |.((seq1 . m) - S1) qua ExtReal.| < +infty by A8, XXREAL_0: 2, XXREAL_0: 4;

 then

 A10: ((seq1 . m) - S1) qua ExtReal in REAL by EXTREAL1: 41;



 A12: (seq1 . m) <> -infty & (seq2 . m) <> -infty & ((seq1 + seq2) . m) <> -infty by MESFUNC5:def 5;



 A13: m is Element of NAT by ORDINAL1:def 12;

 (((seq1 . m) - S1) qua ExtReal + ((seq2 . m) - S2) qua ExtReal) = ((((seq1 . m) - S1) qua ExtReal + (seq2 . m)) - S2) qua ExtReal by A10, XXREAL_3: 30

 .= ((( - S1) qua ExtReal + ((seq1 . m) + (seq2 . m))) - S2) qua ExtReal by A12, XXREAL_3: 29

 .= ((((seq1 + seq2) . m) - S1) qua ExtReal - S2) qua ExtReal by A13, Th7

 .= (((seq1 + seq2) . m) - (S1 qua ExtReal + S2) qua ExtReal) by XXREAL_3: 31;

 then |.(((seq1 + seq2) . m) - (S1 + S2)) qua ExtReal.| <= ( |.((seq1 . m) - S1) qua ExtReal.| + |.((seq2 . m) - S2) qua ExtReal.|) by EXTREAL1: 24;

 hence |.(((seq1 + seq2) . m) - (S1 + S2)) qua ExtReal.| < p by A9, XXREAL_0: 2;

 end;

 hence ex n be Nat st for m be Nat st n <= m holds |.(((seq1 + seq2) . m) - (S1 + S2)) qua ExtReal.| < p;

 end;

 hence

 A14: (seq1 + seq2) is convergent_to_finite_number by MESFUNC5:def 8;

 hence (seq1 + seq2) is convergent;

 for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.(((seq1 + seq2) . m) - (( lim seq1) + ( lim seq2))).| < p

 proof

 let p be Real;

 assume 0 < p;

 then

 consider n be Nat such that

 A17: for m be Nat st n <= m holds |.(((seq1 + seq2) . m) - (S1 + S2)) qua ExtReal.| < p by B3;

 take n;

 hereby

 let m be Nat;

 assume n <= m;

 then |.(((seq1 + seq2) . m) - (S1 + S2)) qua ExtReal.| < p by A17;

 hence |.(((seq1 + seq2) . m) - (( lim seq1) + ( lim seq2))).| < p by A2, A3, XXREAL_3:def 2;

 end;

 end;

 hence ( lim (seq1 + seq2)) = (( lim seq1) + ( lim seq2)) by A2, A3, A14, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:24



 Th24: for seq1,seq2 be without+infty ExtREAL_sequence holds (seq1 is convergent_to_+infty & seq2 is convergent_to_+infty implies (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty ) & (seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty ) & (seq1 is convergent_to_-infty & seq2 is convergent_to_-infty implies (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty ) & (seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty ) & (seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies (seq1 + seq2) is convergent_to_finite_number & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = (( lim seq1) + ( lim seq2)))

 proof

 let seq1,seq2 be without+infty ExtREAL_sequence;

 reconsider f1 = ( - seq1), f2 = ( - seq2) as without-infty ExtREAL_sequence;

 hereby

 assume seq1 is convergent_to_+infty & seq2 is convergent_to_+infty;

 then

 A2: f1 is convergent_to_-infty & f2 is convergent_to_-infty by Th17;

 then

 reconsider F = (f1 + f2) as convergent ExtREAL_sequence by Th21;



 A3: (f1 + f2) is convergent_to_-infty & (f1 + f2) is convergent & ( lim (f1 + f2)) = -infty by A2, Th21;

 (f1 + f2) = (f1 - seq2) by Th10

 .= ( - (seq1 + seq2)) by Th9;

 then (seq1 + seq2) = ( - F) by Th2;

 hence (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty by A3, Th17, XXREAL_3: 5;

 end;

 hereby

 assume seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_-infty & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F = (f1 + f2) as convergent ExtREAL_sequence by Th22;



 A3: (f1 + f2) is convergent_to_-infty & (f1 + f2) is convergent & ( lim (f1 + f2)) = -infty by A2, Th22;

 (f1 + f2) = (f1 - seq2) by Th10

 .= ( - (seq1 + seq2)) by Th9;

 then (seq1 + seq2) = ( - F) by Th2;

 hence (seq1 + seq2) is convergent_to_+infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = +infty by A3, Th17, XXREAL_3: 5;

 end;

 hereby

 assume seq1 is convergent_to_-infty & seq2 is convergent_to_-infty;

 then

 A2: f1 is convergent_to_+infty & f2 is convergent_to_+infty by Th17;

 then

 reconsider F = (f1 + f2) as convergent ExtREAL_sequence by Th18;



 A3: (f1 + f2) is convergent_to_+infty & (f1 + f2) is convergent & ( lim (f1 + f2)) = +infty by A2, Th18;

 (f1 + f2) = (f1 - seq2) by Th10

 .= ( - (seq1 + seq2)) by Th9;

 then (seq1 + seq2) = ( - F) by Th2;

 hence (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty by A3, Th17, XXREAL_3: 6;

 end;

 hereby

 assume seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_+infty & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F = (f1 + f2) as convergent ExtREAL_sequence by Th19;



 A3: (f1 + f2) is convergent_to_+infty & (f1 + f2) is convergent & ( lim (f1 + f2)) = +infty by A2, Th19;

 (f1 + f2) = (f1 - seq2) by Th10

 .= ( - (seq1 + seq2)) by Th9;

 then (seq1 + seq2) = ( - F) by Th2;

 hence (seq1 + seq2) is convergent_to_-infty & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = -infty by A3, Th17, XXREAL_3: 6;

 end;

 hereby

 assume seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_finite_number & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F = (f1 + f2) as convergent ExtREAL_sequence by Th23;



 A3: (f1 + f2) is convergent_to_finite_number & (f1 + f2) is convergent & ( lim (f1 + f2)) = (( lim f1) + ( lim f2)) by A2, Th23;

 (f1 + f2) = (f1 - seq2) by Th10

 .= ( - (seq1 + seq2)) by Th9;

 then

 A4: (seq1 + seq2) = ( - F) by Th2;

 then

 A5: ( lim (seq1 + seq2)) = ( - (( lim f1) + ( lim f2))) by A3, Th17;

 seq1 = ( - f1) & seq2 = ( - f2) by Th2;

 then ( lim seq1) = ( - ( lim f1)) & ( lim seq2) = ( - ( lim f2)) by A2, Th17;

 hence (seq1 + seq2) is convergent_to_finite_number & (seq1 + seq2) is convergent & ( lim (seq1 + seq2)) = (( lim seq1) + ( lim seq2)) by A3, A4, A5, Th17, XXREAL_3: 9;

 end;

 end;

 theorem :: DBLSEQ_3:25

 for seq1 be without-infty ExtREAL_sequence, seq2 be without+infty ExtREAL_sequence holds (seq1 is convergent_to_+infty & seq2 is convergent_to_-infty implies (seq1 - seq2) is convergent_to_+infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_-infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = +infty & ( lim (seq2 - seq1)) = -infty ) & (seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies (seq1 - seq2) is convergent_to_+infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_-infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = +infty & ( lim (seq2 - seq1)) = -infty ) & (seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies (seq1 - seq2) is convergent_to_-infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_+infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = -infty & ( lim (seq2 - seq1)) = +infty ) & (seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies (seq1 - seq2) is convergent_to_finite_number & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_finite_number & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = (( lim seq1) - ( lim seq2)) & ( lim (seq2 - seq1)) = (( lim seq2) - ( lim seq1)))

 proof

 let seq1 be without-infty ExtREAL_sequence, seq2 be without+infty ExtREAL_sequence;

 reconsider f1 = ( - seq1) as without+infty ExtREAL_sequence;

 reconsider f2 = ( - seq2) as without-infty ExtREAL_sequence;

 hereby

 assume

 A1: seq1 is convergent_to_+infty & seq2 is convergent_to_-infty;

 then

 A2: f1 is convergent_to_-infty & f2 is convergent_to_+infty by Th17;

 then

 reconsider F1 = (f1 + seq2), F2 = (seq1 + f2) as convergent ExtREAL_sequence by A1, Th24, Th18;



 A3: (f1 + seq2) is convergent_to_-infty & (f1 + seq2) is convergent & ( lim (f1 + seq2)) = -infty & (seq1 + f2) is convergent_to_+infty & (seq1 + f2) is convergent & ( lim (seq1 + f2)) = +infty by A1, A2, Th24, Th18;



 A4: (seq1 - seq2) = (seq1 + ( - seq2)) by Th10

 .= ( - (seq2 - seq1)) by Th10

 .= ( - (( - seq1) + seq2)) by Th10;

 (seq2 - seq1) = (seq2 + ( - seq1)) by Th10

 .= ( - (seq1 - seq2)) by Th10

 .= ( - (( - seq2) + seq1)) by Th10;

 hence (seq1 - seq2) is convergent_to_+infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_-infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = +infty & ( lim (seq2 - seq1)) = -infty by A3, A4, Th17, XXREAL_3: 5, XXREAL_3: 6;

 end;

 hereby

 assume

 A1: seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_-infty & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F1 = (f1 + seq2), F2 = (seq1 + f2) as convergent ExtREAL_sequence by A1, Th24, Th19;



 A3: (f1 + seq2) is convergent_to_-infty & (f1 + seq2) is convergent & ( lim (f1 + seq2)) = -infty & (seq1 + f2) is convergent_to_+infty & (seq1 + f2) is convergent & ( lim (seq1 + f2)) = +infty by A1, A2, Th24, Th19;



 A4: (seq1 - seq2) = (seq1 + ( - seq2)) by Th10

 .= ( - (seq2 - seq1)) by Th10

 .= ( - (( - seq1) + seq2)) by Th10;

 (seq2 - seq1) = (seq2 + ( - seq1)) by Th10

 .= ( - (seq1 - seq2)) by Th10

 .= ( - (( - seq2) + seq1)) by Th10;

 hence (seq1 - seq2) is convergent_to_+infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_-infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = +infty & ( lim (seq2 - seq1)) = -infty by A3, A4, Th17, XXREAL_3: 5, XXREAL_3: 6;

 end;

 hereby

 assume

 A1: seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_+infty & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F1 = (f1 + seq2), F2 = (seq1 + f2) as convergent ExtREAL_sequence by A1, Th24, Th22;



 A3: (f1 + seq2) is convergent_to_+infty & (f1 + seq2) is convergent & ( lim (f1 + seq2)) = +infty & (seq1 + f2) is convergent_to_-infty & (seq1 + f2) is convergent & ( lim (seq1 + f2)) = -infty by A1, A2, Th24, Th22;



 A4: (seq1 - seq2) = (seq1 + ( - seq2)) by Th10

 .= ( - (seq2 - seq1)) by Th10

 .= ( - (( - seq1) + seq2)) by Th10;

 (seq2 - seq1) = (seq2 + ( - seq1)) by Th10

 .= ( - (seq1 - seq2)) by Th10

 .= ( - (( - seq2) + seq1)) by Th10;

 hence (seq1 - seq2) is convergent_to_-infty & (seq1 - seq2) is convergent & (seq2 - seq1) is convergent_to_+infty & (seq2 - seq1) is convergent & ( lim (seq1 - seq2)) = -infty & ( lim (seq2 - seq1)) = +infty by A3, A4, Th17, XXREAL_3: 5, XXREAL_3: 6;

 end;

 assume

 A1: seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number;

 then

 A2: f1 is convergent_to_finite_number & f2 is convergent_to_finite_number by Th17;

 then

 reconsider F1 = (f1 + seq2), F2 = (seq1 + f2) as convergent ExtREAL_sequence by A1, Th24, Th23;



 A3: (f1 + seq2) is convergent_to_finite_number & (f1 + seq2) is convergent & (seq1 + f2) is convergent_to_finite_number & (seq1 + f2) is convergent & ( lim (f1 + seq2)) = (( lim f1) + ( lim seq2)) & ( lim (seq1 + f2)) = (( lim seq1) + ( lim f2)) by A1, A2, Th24, Th23;



 A4: (seq1 - seq2) = (seq1 + ( - seq2)) by Th10

 .= ( - (seq2 - seq1)) by Th10

 .= ( - (( - seq1) + seq2)) by Th10;

 then

 A5: ( lim (seq1 - seq2)) = ( - (( lim f1) + ( lim seq2))) by A3, Th17;



 A6: (seq2 - seq1) = (seq2 + ( - seq1)) by Th10

 .= ( - (seq1 - seq2)) by Th10

 .= ( - (( - seq2) + seq1)) by Th10;

 then

 A7: ( lim (seq2 - seq1)) = ( - (( lim f2) + ( lim seq1))) by A3, Th17;

 seq1 = ( - f1) & seq2 = ( - f2) by Th2;

 then ( lim seq1) = ( - ( lim f1)) & ( lim seq2) = ( - ( lim f2)) by A2, Th17;

 hence thesis by A3, A4, A5, A6, A7, Th17, XXREAL_3: 9;

 end;

 begin

 theorem :: DBLSEQ_3:26

 for seq1,seq2 be ExtREAL_sequence st seq2 is subsequence of seq1 & seq1 is convergent_to_finite_number holds seq2 is convergent_to_finite_number & ( lim seq1) = ( lim seq2)

 proof

 let seq1,seq2 be ExtREAL_sequence;

 assume that

 A1: seq2 is subsequence of seq1 and

 A2: seq1 is convergent_to_finite_number;

 not seq1 is convergent_to_+infty & not seq1 is convergent_to_-infty by A2, MESFUNC5: 50, MESFUNC5: 51;

 then

 consider g be Real such that

 B3: ( lim seq1) = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq1 . m) - ( lim seq1)).| < p) & seq1 is convergent_to_finite_number by A2, MESFUNC5:def 12;

 reconsider LIM2 = ( lim seq1) as R_eal;

 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 |.((seq2 . m) - g) qua ExtReal.| < p

 proof

 take g;

 hereby

 let p be Real;

 assume 0 < p;

 then

 consider n1 be Nat such that

 A4: for m be Nat st n1 <= m holds |.((seq1 . m) - g) qua ExtReal.| < p by B3;

 take n = n1;

 let m be Nat such that

 A5: n <= m;

 consider Nseq be increasing sequence of NAT such that

 A6: seq2 = (seq1 * Nseq) by A1, VALUED_0:def 17;

 m <= (Nseq . m) by SEQM_3: 14;

 then

 A7: n <= (Nseq . m) by A5, XXREAL_0: 2;

 (seq2 . m) = (seq1 . (Nseq . m)) by A6, FUNCT_2: 15, ORDINAL1:def 12;

 hence |.((seq2 . m) - g) qua ExtReal.| < p by A4, A7;

 end;

 end;

 hence

 A8: seq2 is convergent_to_finite_number by MESFUNC5:def 8;

 for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((seq2 . m) - LIM2).| < p

 proof

 let p be Real;

 assume 0 < p;

 then

 consider n1 be Nat such that

 A10: for m be Nat st n1 <= m holds |.((seq1 . m) - ( lim seq1)).| < p by B3;

 take n = n1;

 let m be Nat such that

 A11: n <= m;

 consider Nseq be increasing sequence of NAT such that

 A12: seq2 = (seq1 * Nseq) by A1, VALUED_0:def 17;

 m <= (Nseq . m) by SEQM_3: 14;

 then

 A13: n <= (Nseq . m) by A11, XXREAL_0: 2;

 (seq2 . m) = (seq1 . (Nseq . m)) by A12, FUNCT_2: 15, ORDINAL1:def 12;

 hence |.((seq2 . m) - LIM2).| < p by A10, A13;

 end;

 hence thesis by A8, B3, MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:27

 for seq1,seq2 be ExtREAL_sequence st seq2 is subsequence of seq1 & seq1 is convergent_to_+infty holds seq2 is convergent_to_+infty & ( lim seq2) = +infty

 proof

 let seq1,seq2 be ExtREAL_sequence;

 assume that

 A1: seq2 is subsequence of seq1 and

 A2: seq1 is convergent_to_+infty;

 now

 let g be Real;

 assume 0 < g;

 then

 consider n1 be Nat such that

 A4: for m be Nat st n1 <= m holds g <= (seq1 . m) by A2, MESFUNC5:def 9;

 take n = n1;

 consider Nseq be increasing sequence of NAT such that

 A5: seq2 = (seq1 * Nseq) by A1, VALUED_0:def 17;

 let m be Nat;

 assume

 A6: n <= m;

 m <= (Nseq . m) by SEQM_3: 14;

 then

 A7: n <= (Nseq . m) by A6, XXREAL_0: 2;

 (seq2 . m) = (seq1 . (Nseq . m)) by A5, FUNCT_2: 15, ORDINAL1:def 12;

 hence g <= (seq2 . m) by A4, A7;

 end;

 hence seq2 is convergent_to_+infty by MESFUNC5:def 9;

 hence thesis by MESFUNC5:def 12;

 end;

 theorem :: DBLSEQ_3:28

 for seq1,seq2 be ExtREAL_sequence st seq2 is subsequence of seq1 & seq1 is convergent_to_-infty holds seq2 is convergent_to_-infty & ( lim seq2) = -infty

 proof

 let seq1,seq2 be ExtREAL_sequence;

 assume that

 A1: seq2 is subsequence of seq1 and

 A2: seq1 is convergent_to_-infty;

 now

 let g be Real;

 assume g < 0 ;

 then

 consider n1 be Nat such that

 A4: for m be Nat st n1 <= m holds (seq1 . m) <= g by A2, MESFUNC5:def 10;

 take n = n1;

 consider Nseq be increasing sequence of NAT such that

 A5: seq2 = (seq1 * Nseq) by A1, VALUED_0:def 17;

 let m be Nat;

 assume

 A6: n <= m;

 m <= (Nseq . m) by SEQM_3: 14;

 then

 A7: n <= (Nseq . m) by A6, XXREAL_0: 2;

 (seq2 . m) = (seq1 . (Nseq . m)) by A5, FUNCT_2: 15, ORDINAL1:def 12;

 hence (seq2 . m) <= g by A4, A7;

 end;

 hence seq2 is convergent_to_-infty by MESFUNC5:def 10;

 hence thesis by MESFUNC5:def 12;

 end;

 begin

 theorem :: DBLSEQ_3:29

 for Rseq be Function of [: NAT , NAT :], REAL st ( lim_in_cod1 Rseq) is convergent holds ( cod1_major_iterated_lim Rseq) = ( lim ( lim_in_cod1 Rseq))

 proof

 let Rseq be Function of [: NAT , NAT :], REAL ;

 assume

 A1: ( lim_in_cod1 Rseq) is convergent;

 then

 consider g be Real such that

 A2: for p be Real st 0 < p holds ex M be Nat st for m be Nat st M <= m holds |.((( lim_in_cod1 Rseq) . m) - g) qua Complex.| < p by SEQ_2:def 6;

 g = ( lim ( lim_in_cod1 Rseq)) by A1, A2, SEQ_2:def 7;

 hence thesis by A1, A2, DBLSEQ_1:def 7;

 end;

 theorem :: DBLSEQ_3:30

 for Rseq be Function of [: NAT , NAT :], REAL st ( lim_in_cod2 Rseq) is convergent holds ( cod2_major_iterated_lim Rseq) = ( lim ( lim_in_cod2 Rseq))

 proof

 let Rseq be Function of [: NAT , NAT :], REAL ;

 assume

 A1: ( lim_in_cod2 Rseq) is convergent;

 then

 consider g be Real such that

 A2: for p be Real st 0 < p holds ex M be Nat st for m be Nat st M <= m holds |.((( lim_in_cod2 Rseq) . m) - g) qua Complex.| < p by SEQ_2:def 6;

 g = ( lim ( lim_in_cod2 Rseq)) by A1, A2, SEQ_2:def 7;

 hence thesis by A1, A2, DBLSEQ_1:def 8;

 end;

 definition

 let Eseq be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def1

 attr Eseq is P-convergent_to_finite_number means ex p be Real st for e be Real st 0 < e holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds |.((Eseq . (n,m)) - p qua ExtReal).| < e;

 :: DBLSEQ_3:def2

 attr Eseq is P-convergent_to_+infty means for g be Real st 0 < g holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds g <= (Eseq . (n,m));

 :: DBLSEQ_3:def3

 attr Eseq is P-convergent_to_-infty means for g be Real st g < 0 holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds (Eseq . (n,m)) <= g;

 end

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def4

 attr f is convergent_in_cod1_to_+infty means for m be Element of NAT holds ( ProjMap2 (f,m)) is convergent_to_+infty;

 :: DBLSEQ_3:def5

 attr f is convergent_in_cod1_to_-infty means for m be Element of NAT holds ( ProjMap2 (f,m)) is convergent_to_-infty;

 :: DBLSEQ_3:def6

 attr f is convergent_in_cod1_to_finite means for m be Element of NAT holds ( ProjMap2 (f,m)) is convergent_to_finite_number;

 :: DBLSEQ_3:def7

 attr f is convergent_in_cod1 means for m be Element of NAT holds ( ProjMap2 (f,m)) is convergent;

 :: DBLSEQ_3:def8

 attr f is convergent_in_cod2_to_+infty means for m be Element of NAT holds ( ProjMap1 (f,m)) is convergent_to_+infty;

 :: DBLSEQ_3:def9

 attr f is convergent_in_cod2_to_-infty means for m be Element of NAT holds ( ProjMap1 (f,m)) is convergent_to_-infty;

 :: DBLSEQ_3:def10

 attr f is convergent_in_cod2_to_finite means for m be Element of NAT holds ( ProjMap1 (f,m)) is convergent_to_finite_number;

 :: DBLSEQ_3:def11

 attr f is convergent_in_cod2 means for m be Element of NAT holds ( ProjMap1 (f,m)) is convergent;

 end

 theorem :: DBLSEQ_3:31

 for f be Function of [: NAT , NAT :], ExtREAL holds (f is convergent_in_cod1_to_+infty or f is convergent_in_cod1_to_-infty or f is convergent_in_cod1_to_finite implies f is convergent_in_cod1) & (f is convergent_in_cod2_to_+infty or f is convergent_in_cod2_to_-infty or f is convergent_in_cod2_to_finite implies f is convergent_in_cod2)

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 hereby

 assume

 A1: f is convergent_in_cod1_to_+infty or f is convergent_in_cod1_to_-infty or f is convergent_in_cod1_to_finite;

 per cases by A1;

 suppose

 A2: f is convergent_in_cod1_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) is convergent_to_+infty by A2;

 hence ( ProjMap2 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod1;

 end;

 suppose

 A3: f is convergent_in_cod1_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) is convergent_to_-infty by A3;

 hence ( ProjMap2 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod1;

 end;

 suppose

 A4: f is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) is convergent_to_finite_number by A4;

 hence ( ProjMap2 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod1;

 end;

 end;

 assume

 A5: f is convergent_in_cod2_to_+infty or f is convergent_in_cod2_to_-infty or f is convergent_in_cod2_to_finite;

 per cases by A5;

 suppose

 A6: f is convergent_in_cod2_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) is convergent_to_+infty by A6;

 hence ( ProjMap1 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod2;

 end;

 suppose

 A7: f is convergent_in_cod2_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) is convergent_to_-infty by A7;

 hence ( ProjMap1 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod2;

 end;

 suppose

 A8: f is convergent_in_cod2_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) is convergent_to_finite_number by A8;

 hence ( ProjMap1 (f,m)) is convergent;

 end;

 hence f is convergent_in_cod2;

 end;

 end;

 theorem :: DBLSEQ_3:32



 Th32: for X,Y,Z be non empty set, F be Function of [:X, Y:], Z, x be Element of X holds ( ProjMap1 (F,x)) = ( ProjMap2 (( ~ F),x))

 proof

 let X,Y,Z be non empty set;

 let F be Function of [:X, Y:], Z;

 let x be Element of X;

 now

 let y be Element of Y;

 (( ProjMap1 (F,x)) . y) = (F . (x,y)) by MESFUNC9:def 6

 .= (( ~ F) . (y,x)) by FUNCT_4:def 8;

 hence (( ProjMap1 (F,x)) . y) = (( ProjMap2 (( ~ F),x)) . y) by MESFUNC9:def 7;

 end;

 hence ( ProjMap1 (F,x)) = ( ProjMap2 (( ~ F),x)) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:33



 Th33: for X,Y,Z be non empty set, F be Function of [:X, Y:], Z, y be Element of Y holds ( ProjMap2 (F,y)) = ( ProjMap1 (( ~ F),y))

 proof

 let X,Y,Z be non empty set;

 let F be Function of [:X, Y:], Z;

 let y be Element of Y;

 now

 let x be Element of X;

 (( ProjMap2 (F,y)) . x) = (F . (x,y)) by MESFUNC9:def 7

 .= (( ~ F) . (y,x)) by FUNCT_4:def 8;

 hence (( ProjMap2 (F,y)) . x) = (( ProjMap1 (( ~ F),y)) . x) by MESFUNC9:def 6;

 end;

 hence ( ProjMap2 (F,y)) = ( ProjMap1 (( ~ F),y)) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:34



 Th34: for X,Y be non empty set, F be Function of [:X, Y:], ExtREAL , x be Element of X holds ( ProjMap1 (( - F),x)) = ( - ( ProjMap1 (F,x)))

 proof

 let X,Y be non empty set, F be Function of [:X, Y:], ExtREAL , x be Element of X;

 now

 let y be Element of Y;

 [x, y] in [:X, Y:] by ZFMISC_1: 87;

 then

 A1: [x, y] in ( dom ( - F)) by FUNCT_2:def 1;



 A2: ( dom ( - ( ProjMap1 (F,x)))) = Y by FUNCT_2:def 1;

 (( ProjMap1 (( - F),x)) . y) = (( - F) . (x,y)) by MESFUNC9:def 6;

 then (( ProjMap1 (( - F),x)) . y) = ( - (F . (x,y))) by A1, MESFUNC1:def 7;

 then (( ProjMap1 (( - F),x)) . y) = ( - (( ProjMap1 (F,x)) . y)) by MESFUNC9:def 6;

 hence (( ProjMap1 (( - F),x)) . y) = (( - ( ProjMap1 (F,x))) . y) by A2, MESFUNC1:def 7;

 end;

 hence ( ProjMap1 (( - F),x)) = ( - ( ProjMap1 (F,x))) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:35



 Th35: for X,Y be non empty set, F be Function of [:X, Y:], ExtREAL , y be Element of Y holds ( ProjMap2 (( - F),y)) = ( - ( ProjMap2 (F,y)))

 proof

 let X,Y be non empty set, F be Function of [:X, Y:], ExtREAL , y be Element of Y;

 now

 let x be Element of X;

 [x, y] in [:X, Y:] by ZFMISC_1: 87;

 then

 A1: [x, y] in ( dom ( - F)) by FUNCT_2:def 1;



 A2: ( dom ( - ( ProjMap2 (F,y)))) = X by FUNCT_2:def 1;

 (( ProjMap2 (( - F),y)) . x) = (( - F) . (x,y)) by MESFUNC9:def 7;

 then (( ProjMap2 (( - F),y)) . x) = ( - (F . (x,y))) by A1, MESFUNC1:def 7;

 then (( ProjMap2 (( - F),y)) . x) = ( - (( ProjMap2 (F,y)) . x)) by MESFUNC9:def 7;

 hence (( ProjMap2 (( - F),y)) . x) = (( - ( ProjMap2 (F,y))) . x) by A2, MESFUNC1:def 7;

 end;

 hence ( ProjMap2 (( - F),y)) = ( - ( ProjMap2 (F,y))) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:36



 Th36: for f be Function of [: NAT , NAT :], ExtREAL holds (f is convergent_in_cod1_to_+infty iff ( ~ f) is convergent_in_cod2_to_+infty) & (f is convergent_in_cod2_to_+infty iff ( ~ f) is convergent_in_cod1_to_+infty) & (f is convergent_in_cod1_to_-infty iff ( ~ f) is convergent_in_cod2_to_-infty) & (f is convergent_in_cod2_to_-infty iff ( ~ f) is convergent_in_cod1_to_-infty) & (f is convergent_in_cod1_to_finite iff ( ~ f) is convergent_in_cod2_to_finite) & (f is convergent_in_cod2_to_finite iff ( ~ f) is convergent_in_cod1_to_finite)

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 now

 hereby

 assume

 A1: f is convergent_in_cod1_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap1 (( ~ f),m)) is convergent_to_+infty by A1;

 end;

 hence ( ~ f) is convergent_in_cod2_to_+infty;

 end;

 assume

 A2: ( ~ f) is convergent_in_cod2_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap2 (f,m)) is convergent_to_+infty by A2;

 end;

 hence f is convergent_in_cod1_to_+infty;

 end;

 hence f is convergent_in_cod1_to_+infty iff ( ~ f) is convergent_in_cod2_to_+infty;

 now

 hereby

 assume

 A3: f is convergent_in_cod2_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap2 (( ~ f),m)) is convergent_to_+infty by A3;

 end;

 hence ( ~ f) is convergent_in_cod1_to_+infty;

 end;

 assume

 A4: ( ~ f) is convergent_in_cod1_to_+infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap1 (f,m)) is convergent_to_+infty by A4;

 end;

 hence f is convergent_in_cod2_to_+infty;

 end;

 hence f is convergent_in_cod2_to_+infty iff ( ~ f) is convergent_in_cod1_to_+infty;

 now

 hereby

 assume

 A5: f is convergent_in_cod1_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap1 (( ~ f),m)) is convergent_to_-infty by A5;

 end;

 hence ( ~ f) is convergent_in_cod2_to_-infty;

 end;

 assume

 A6: ( ~ f) is convergent_in_cod2_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap2 (f,m)) is convergent_to_-infty by A6;

 end;

 hence f is convergent_in_cod1_to_-infty;

 end;

 hence f is convergent_in_cod1_to_-infty iff ( ~ f) is convergent_in_cod2_to_-infty;

 now

 hereby

 assume

 A7: f is convergent_in_cod2_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap2 (( ~ f),m)) is convergent_to_-infty by A7;

 end;

 hence ( ~ f) is convergent_in_cod1_to_-infty;

 end;

 assume

 A8: ( ~ f) is convergent_in_cod1_to_-infty;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap1 (f,m)) is convergent_to_-infty by A8;

 end;

 hence f is convergent_in_cod2_to_-infty;

 end;

 hence f is convergent_in_cod2_to_-infty iff ( ~ f) is convergent_in_cod1_to_-infty;

 now

 hereby

 assume

 A9: f is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap1 (( ~ f),m)) is convergent_to_finite_number by A9;

 end;

 hence ( ~ f) is convergent_in_cod2_to_finite;

 end;

 assume

 A10: ( ~ f) is convergent_in_cod2_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) = ( ProjMap1 (( ~ f),m)) by Th33;

 hence ( ProjMap2 (f,m)) is convergent_to_finite_number by A10;

 end;

 hence f is convergent_in_cod1_to_finite;

 end;

 hence f is convergent_in_cod1_to_finite iff ( ~ f) is convergent_in_cod2_to_finite;

 now

 hereby

 assume

 A11: f is convergent_in_cod2_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap2 (( ~ f),m)) is convergent_to_finite_number by A11;

 end;

 hence ( ~ f) is convergent_in_cod1_to_finite;

 end;

 assume

 A12: ( ~ f) is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence ( ProjMap1 (f,m)) is convergent_to_finite_number by A12;

 end;

 hence f is convergent_in_cod2_to_finite;

 end;

 hence f is convergent_in_cod2_to_finite iff ( ~ f) is convergent_in_cod1_to_finite;

 end;

 theorem :: DBLSEQ_3:37

 for f be Function of [: NAT , NAT :], ExtREAL holds (f is convergent_in_cod1_to_+infty iff ( - f) is convergent_in_cod1_to_-infty) & (f is convergent_in_cod1_to_-infty iff ( - f) is convergent_in_cod1_to_+infty) & (f is convergent_in_cod1_to_finite iff ( - f) is convergent_in_cod1_to_finite) & (f is convergent_in_cod2_to_+infty iff ( - f) is convergent_in_cod2_to_-infty) & (f is convergent_in_cod2_to_-infty iff ( - f) is convergent_in_cod2_to_+infty) & (f is convergent_in_cod2_to_finite iff ( - f) is convergent_in_cod2_to_finite)

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 now

 hereby

 assume

 A1: f is convergent_in_cod1_to_+infty;

 now

 let m be Element of NAT ;



 A2: ( ProjMap2 (f,m)) is convergent_to_+infty by A1;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 hence ( ProjMap2 (( - f),m)) is convergent_to_-infty by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod1_to_-infty;

 end;

 assume

 A3: ( - f) is convergent_in_cod1_to_-infty;

 now

 let m be Element of NAT ;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 then ( - ( ProjMap2 (f,m))) is convergent_to_-infty by A3;

 then ( - ( - ( ProjMap2 (f,m)))) is convergent_to_+infty by Th17;

 hence ( ProjMap2 (f,m)) is convergent_to_+infty by Th2;

 end;

 hence f is convergent_in_cod1_to_+infty;

 end;

 hence f is convergent_in_cod1_to_+infty iff ( - f) is convergent_in_cod1_to_-infty;

 now

 hereby

 assume

 A1: f is convergent_in_cod1_to_-infty;

 now

 let m be Element of NAT ;



 A2: ( ProjMap2 (f,m)) is convergent_to_-infty by A1;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 hence ( ProjMap2 (( - f),m)) is convergent_to_+infty by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod1_to_+infty;

 end;

 assume

 A3: ( - f) is convergent_in_cod1_to_+infty;

 now

 let m be Element of NAT ;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 then ( - ( ProjMap2 (f,m))) is convergent_to_+infty by A3;

 then ( - ( - ( ProjMap2 (f,m)))) is convergent_to_-infty by Th17;

 hence ( ProjMap2 (f,m)) is convergent_to_-infty by Th2;

 end;

 hence f is convergent_in_cod1_to_-infty;

 end;

 hence f is convergent_in_cod1_to_-infty iff ( - f) is convergent_in_cod1_to_+infty;

 now

 hereby

 assume

 A1: f is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;



 A2: ( ProjMap2 (f,m)) is convergent_to_finite_number by A1;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 hence ( ProjMap2 (( - f),m)) is convergent_to_finite_number by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod1_to_finite;

 end;

 assume

 A3: ( - f) is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 ( - ( ProjMap2 (f,m))) = ( ProjMap2 (( - f),m)) by Th35;

 then ( - ( ProjMap2 (f,m))) is convergent_to_finite_number by A3;

 then ( - ( - ( ProjMap2 (f,m)))) is convergent_to_finite_number by Th17;

 hence ( ProjMap2 (f,m)) is convergent_to_finite_number by Th2;

 end;

 hence f is convergent_in_cod1_to_finite;

 end;

 hence f is convergent_in_cod1_to_finite iff ( - f) is convergent_in_cod1_to_finite;

 now

 hereby

 assume

 A1: f is convergent_in_cod2_to_+infty;

 now

 let m be Element of NAT ;



 A2: ( ProjMap1 (f,m)) is convergent_to_+infty by A1;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 hence ( ProjMap1 (( - f),m)) is convergent_to_-infty by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod2_to_-infty;

 end;

 assume

 A3: ( - f) is convergent_in_cod2_to_-infty;

 now

 let m be Element of NAT ;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 then ( - ( ProjMap1 (f,m))) is convergent_to_-infty by A3;

 then ( - ( - ( ProjMap1 (f,m)))) is convergent_to_+infty by Th17;

 hence ( ProjMap1 (f,m)) is convergent_to_+infty by Th2;

 end;

 hence f is convergent_in_cod2_to_+infty;

 end;

 hence f is convergent_in_cod2_to_+infty iff ( - f) is convergent_in_cod2_to_-infty;

 now

 hereby

 assume

 A1: f is convergent_in_cod2_to_-infty;

 now

 let m be Element of NAT ;



 A2: ( ProjMap1 (f,m)) is convergent_to_-infty by A1;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 hence ( ProjMap1 (( - f),m)) is convergent_to_+infty by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod2_to_+infty;

 end;

 assume

 A3: ( - f) is convergent_in_cod2_to_+infty;

 now

 let m be Element of NAT ;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 then ( - ( ProjMap1 (f,m))) is convergent_to_+infty by A3;

 then ( - ( - ( ProjMap1 (f,m)))) is convergent_to_-infty by Th17;

 hence ( ProjMap1 (f,m)) is convergent_to_-infty by Th2;

 end;

 hence f is convergent_in_cod2_to_-infty;

 end;

 hence f is convergent_in_cod2_to_-infty iff ( - f) is convergent_in_cod2_to_+infty;

 now

 hereby

 assume

 A1: f is convergent_in_cod2_to_finite;

 now

 let m be Element of NAT ;



 A2: ( ProjMap1 (f,m)) is convergent_to_finite_number by A1;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 hence ( ProjMap1 (( - f),m)) is convergent_to_finite_number by A2, Th17;

 end;

 hence ( - f) is convergent_in_cod2_to_finite;

 end;

 assume

 A3: ( - f) is convergent_in_cod2_to_finite;

 now

 let m be Element of NAT ;

 ( - ( ProjMap1 (f,m))) = ( ProjMap1 (( - f),m)) by Th34;

 then ( - ( ProjMap1 (f,m))) is convergent_to_finite_number by A3;

 then ( - ( - ( ProjMap1 (f,m)))) is convergent_to_finite_number by Th17;

 hence ( ProjMap1 (f,m)) is convergent_to_finite_number by Th2;

 end;

 hence f is convergent_in_cod2_to_finite;

 end;

 hence f is convergent_in_cod2_to_finite iff ( - f) is convergent_in_cod2_to_finite;

 end;

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def12

 func lim_in_cod1 f -> ExtREAL_sequence means

 : D1DEF5: for m be Element of NAT holds (it . m) = ( lim ( ProjMap2 (f,m)));

 existence

 proof

 defpred P[ Element of NAT , set] means $2 = ( lim ( ProjMap2 (f,$1)));



 A1: for m be Element of NAT holds ex n be Element of ExtREAL st P[m, n];

 consider IT be Function of NAT , ExtREAL such that

 A2: for m be Element of NAT holds P[m, (IT . m)] from FUNCT_2:sch 3( A1);

 take IT;

 thus thesis by A2;

 end;

 uniqueness

 proof

 let f1,f2 be Function of NAT , ExtREAL ;

 assume that

 A3: for m be Element of NAT holds (f1 . m) = ( lim ( ProjMap2 (f,m))) and

 A4: for m be Element of NAT holds (f2 . m) = ( lim ( ProjMap2 (f,m)));

 now

 let m be Element of NAT ;



 thus (f1 . m) = ( lim ( ProjMap2 (f,m))) by A3

 .= (f2 . m) by A4;

 end;

 hence f1 = f2 by FUNCT_2: 63;

 end;

 :: DBLSEQ_3:def13

 func lim_in_cod2 f -> ExtREAL_sequence means

 : D1DEF6: for n be Element of NAT holds (it . n) = ( lim ( ProjMap1 (f,n)));

 existence

 proof

 defpred P[ Element of NAT , set] means $2 = ( lim ( ProjMap1 (f,$1)));



 A1: for m be Element of NAT holds ex n be Element of ExtREAL st P[m, n];

 consider IT be Function of NAT , ExtREAL such that

 A2: for m be Element of NAT holds P[m, (IT . m)] from FUNCT_2:sch 3( A1);

 take IT;

 thus thesis by A2;

 end;

 uniqueness

 proof

 let f1,f2 be Function of NAT , ExtREAL ;

 assume that

 A3: for n be Element of NAT holds (f1 . n) = ( lim ( ProjMap1 (f,n))) and

 A4: for n be Element of NAT holds (f2 . n) = ( lim ( ProjMap1 (f,n)));

 now

 let n be Element of NAT ;



 thus (f1 . n) = ( lim ( ProjMap1 (f,n))) by A3

 .= (f2 . n) by A4;

 end;

 hence f1 = f2 by FUNCT_2: 63;

 end;

 end

 theorem :: DBLSEQ_3:38



 Th38: for f be Function of [: NAT , NAT :], ExtREAL holds ( lim_in_cod1 f) = ( lim_in_cod2 ( ~ f)) & ( lim_in_cod2 f) = ( lim_in_cod1 ( ~ f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 now

 let n be Element of NAT ;

 (( lim_in_cod1 f) . n) = ( lim ( ProjMap2 (f,n))) by D1DEF5

 .= ( lim ( ProjMap1 (( ~ f),n))) by Th33;

 hence (( lim_in_cod1 f) . n) = (( lim_in_cod2 ( ~ f)) . n) by D1DEF6;

 end;

 hence ( lim_in_cod1 f) = ( lim_in_cod2 ( ~ f)) by FUNCT_2:def 8;

 now

 let n be Element of NAT ;

 (( lim_in_cod2 f) . n) = ( lim ( ProjMap1 (f,n))) by D1DEF6

 .= ( lim ( ProjMap2 (( ~ f),n))) by Th32;

 hence (( lim_in_cod2 f) . n) = (( lim_in_cod1 ( ~ f)) . n) by D1DEF5;

 end;

 hence ( lim_in_cod2 f) = ( lim_in_cod1 ( ~ f)) by FUNCT_2:def 8;

 end;

 registration

 let X,Y be non empty set, F be without+infty Function of [:X, Y:], ExtREAL , x be Element of X;

 cluster ( ProjMap1 (F,x)) -> without+infty;

 correctness

 proof

 now

 let y be object;

 per cases ;

 suppose not y in ( dom ( ProjMap1 (F,x)));

 hence (( ProjMap1 (F,x)) . y) < +infty by FUNCT_1:def 2;

 end;

 suppose y in ( dom ( ProjMap1 (F,x)));

 then

 reconsider y1 = y as Element of Y;

 (( ProjMap1 (F,x)) . y) = (F . (x,y1)) by MESFUNC9:def 6;

 hence (( ProjMap1 (F,x)) . y) < +infty by MESFUNC5:def 6;

 end;

 end;

 hence thesis by MESFUNC5:def 6;

 end;

 end

 registration

 let X,Y be non empty set, F be without+infty Function of [:X, Y:], ExtREAL , y be Element of Y;

 cluster ( ProjMap2 (F,y)) -> without+infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose not x in ( dom ( ProjMap2 (F,y)));

 hence (( ProjMap2 (F,y)) . x) < +infty by FUNCT_1:def 2;

 end;

 suppose x in ( dom ( ProjMap2 (F,y)));

 then

 reconsider x1 = x as Element of X;

 (( ProjMap2 (F,y)) . x) = (F . (x1,y)) by MESFUNC9:def 7;

 hence (( ProjMap2 (F,y)) . x) < +infty by MESFUNC5:def 6;

 end;

 end;

 hence thesis by MESFUNC5:def 6;

 end;

 end

 registration

 let X,Y be non empty set, F be without-infty Function of [:X, Y:], ExtREAL , x be Element of X;

 cluster ( ProjMap1 (F,x)) -> without-infty;

 correctness

 proof

 now

 let y be object;

 per cases ;

 suppose not y in ( dom ( ProjMap1 (F,x)));

 hence -infty < (( ProjMap1 (F,x)) . y) by FUNCT_1:def 2;

 end;

 suppose y in ( dom ( ProjMap1 (F,x)));

 then

 reconsider y1 = y as Element of Y;

 (( ProjMap1 (F,x)) . y) = (F . (x,y1)) by MESFUNC9:def 6;

 hence -infty < (( ProjMap1 (F,x)) . y) by MESFUNC5:def 5;

 end;

 end;

 hence thesis by MESFUNC5:def 5;

 end;

 end

 registration

 let X,Y be non empty set, F be without-infty Function of [:X, Y:], ExtREAL , y be Element of Y;

 cluster ( ProjMap2 (F,y)) -> without-infty;

 correctness

 proof

 now

 let x be object;

 per cases ;

 suppose not x in ( dom ( ProjMap2 (F,y)));

 hence -infty < (( ProjMap2 (F,y)) . x) by FUNCT_1:def 2;

 end;

 suppose x in ( dom ( ProjMap2 (F,y)));

 then

 reconsider x1 = x as Element of X;

 (( ProjMap2 (F,y)) . x) = (F . (x1,y)) by MESFUNC9:def 7;

 hence -infty < (( ProjMap2 (F,y)) . x) by MESFUNC5:def 5;

 end;

 end;

 hence thesis by MESFUNC5:def 5;

 end;

 end

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def14

 func Partial_Sums_in_cod2 f -> Function of [: NAT , NAT :], ExtREAL means

 : DefCSM: for n,m be Nat holds (it . (n, 0 )) = (f . (n, 0 )) & (it . (n,(m + 1))) = ((it . (n,m)) + (f . (n,(m + 1))));

 existence

 proof

 deffunc F0( Element of NAT ) = (f . ($1, 0 ));

 consider f0 be Function of NAT , ExtREAL such that

 A1: for n be Element of NAT holds (f0 . n) = F0(n) from FUNCT_2:sch 4;

 deffunc F( Element of ExtREAL , Nat, Nat) = ($1 + (f . ($2,($3 + 1))));

 consider IT be Function of [: NAT , NAT :], ExtREAL such that

 A2: for a be Element of NAT holds (IT . (a, 0 )) = (f0 . a) & for n be Nat holds (IT . (a,(n + 1))) = F(.,a,n) from DBLSEQ_2:sch 1;

 take IT;

 hereby

 let n,m be Nat;



 A3: n in NAT & m in NAT by ORDINAL1:def 12;

 then (IT . (n, 0 )) = (f0 . n) by A2;

 hence (IT . (n, 0 )) = (f . (n, 0 )) & (IT . (n,(m + 1))) = ((IT . (n,m)) + (f . (n,(m + 1)))) by A1, A2, A3;

 end;

 end;

 uniqueness

 proof

 let f1,f2 be Function of [: NAT , NAT :], ExtREAL ;

 assume that

 A1: for n,m be natural number holds (f1 . (n, 0 )) = (f . (n, 0 )) & (f1 . (n,(m + 1))) = ((f1 . (n,m)) + (f . (n,(m + 1)))) and

 A2: for n,m be natural number holds (f2 . (n, 0 )) = (f . (n, 0 )) & (f2 . (n,(m + 1))) = ((f2 . (n,m)) + (f . (n,(m + 1))));



 A3: ( dom f1) = [: NAT , NAT :] & ( dom f2) = [: NAT , NAT :] by FUNCT_2:def 1;

 for n,m be object st n in NAT & m in NAT holds (f1 . (n,m)) = (f2 . (n,m))

 proof

 let n,m be object;

 assume n in NAT & m in NAT ;

 then

 reconsider n1 = n, m1 = m as Element of NAT ;

 defpred P[ Nat] means (f1 . (n1,$1)) = (f2 . (n1,$1));

 (f1 . (n1, 0 )) = (f . (n1, 0 )) by A1;

 then

 A4: P[ 0 ] by A2;



 A5: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A6: P[k];

 (f1 . (n1,(k + 1))) = ((f1 . (n1,k)) + (f . (n1,(k + 1)))) by A1;

 hence (f1 . (n1,(k + 1))) = (f2 . (n1,(k + 1))) by A2, A6;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

 then P[m1];

 hence (f1 . (n,m)) = (f2 . (n,m));

 end;

 hence thesis by A3, FUNCT_3: 6;

 end;

 end

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def15

 func Partial_Sums_in_cod1 f -> Function of [: NAT , NAT :], ExtREAL means

 : DefRSM: for n,m be Nat holds (it . ( 0 ,m)) = (f . ( 0 ,m)) & (it . ((n + 1),m)) = ((it . (n,m)) + (f . ((n + 1),m)));

 existence

 proof

 deffunc F0( Element of NAT ) = (f . ( 0 ,$1));

 consider f0 be Function of NAT , ExtREAL such that

 A1: for n be Element of NAT holds (f0 . n) = F0(n) from FUNCT_2:sch 4;

 deffunc F( Element of ExtREAL , Nat, Nat) = ($1 + (f . (($3 + 1),$2)));

 consider IT be Function of [: NAT , NAT :], ExtREAL such that

 A2: for a be Element of NAT holds (IT . ( 0 ,a)) = (f0 . a) & for n be Nat holds (IT . ((n + 1),a)) = F(.,a,n) from DBLSEQ_2:sch 3;

 take IT;

 hereby

 let n,m be Nat;



 A3: n in NAT & m in NAT by ORDINAL1:def 12;

 then (IT . ( 0 ,m)) = (f0 . m) by A2;

 hence (IT . ( 0 ,m)) = (f . ( 0 ,m)) & (IT . ((n + 1),m)) = ((IT . (n,m)) + (f . ((n + 1),m))) by A1, A2, A3;

 end;

 end;

 uniqueness

 proof

 let f1,f2 be Function of [: NAT , NAT :], ExtREAL ;

 assume that

 A1: for n,m be natural number holds (f1 . ( 0 ,m)) = (f . ( 0 ,m)) & (f1 . ((n + 1),m)) = ((f1 . (n,m)) + (f . ((n + 1),m))) and

 A2: for n,m be natural number holds (f2 . ( 0 ,m)) = (f . ( 0 ,m)) & (f2 . ((n + 1),m)) = ((f2 . (n,m)) + (f . ((n + 1),m)));



 A3: ( dom f1) = [: NAT , NAT :] & ( dom f2) = [: NAT , NAT :] by FUNCT_2:def 1;

 for n,m be object st n in NAT & m in NAT holds (f1 . (n,m)) = (f2 . (n,m))

 proof

 let n,m be object;

 assume n in NAT & m in NAT ;

 then

 reconsider n1 = n, m1 = m as Element of NAT ;

 defpred P[ Nat] means (f1 . ($1,m1)) = (f2 . ($1,m1));

 (f1 . ( 0 ,m1)) = (f . ( 0 ,m1)) by A1;

 then

 A4: P[ 0 ] by A2;



 A5: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A6: P[k];

 (f1 . ((k + 1),m1)) = ((f1 . (k,m1)) + (f . ((k + 1),m1))) by A1;

 hence (f1 . ((k + 1),m1)) = (f2 . ((k + 1),m1)) by A2, A6;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

 then P[n1];

 hence (f1 . (n,m)) = (f2 . (n,m));

 end;

 hence thesis by A3, FUNCT_3: 6;

 end;

 end

 registration

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod2 f) -> without-infty;

 correctness

 proof

 for x be object holds -infty < (( Partial_Sums_in_cod2 f) . x)

 proof

 let x be object;

 per cases ;

 suppose x in ( dom ( Partial_Sums_in_cod2 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & x = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) > -infty ;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then

 A2: P[ 0 ] by MESFUNC5:def 5;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . (n,(k + 1))) > -infty by MESFUNC5:def 5;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence P[(k + 1)] by A4, A5, XXREAL_3: 17, XXREAL_0: 6;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then P[m];

 hence -infty < (( Partial_Sums_in_cod2 f) . x) by A1;

 end;

 suppose not x in ( dom ( Partial_Sums_in_cod2 f));

 hence -infty < (( Partial_Sums_in_cod2 f) . x) by FUNCT_1:def 2;

 end;

 end;

 hence ( Partial_Sums_in_cod2 f) is without-infty by MESFUNC5:def 5;

 end;

 end

 registration

 let f be without+infty Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod2 f) -> without+infty;

 correctness

 proof

 for x be object holds +infty > (( Partial_Sums_in_cod2 f) . x)

 proof

 let x be object;

 per cases ;

 suppose x in ( dom ( Partial_Sums_in_cod2 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & x = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) < +infty ;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then

 A2: P[ 0 ] by MESFUNC5:def 6;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . (n,(k + 1))) < +infty by MESFUNC5:def 6;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence P[(k + 1)] by A4, A5, XXREAL_3: 16, XXREAL_0: 4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then P[m];

 hence +infty > (( Partial_Sums_in_cod2 f) . x) by A1;

 end;

 suppose not x in ( dom ( Partial_Sums_in_cod2 f));

 hence +infty > (( Partial_Sums_in_cod2 f) . x) by FUNCT_1:def 2;

 end;

 end;

 hence ( Partial_Sums_in_cod2 f) is without+infty by MESFUNC5:def 6;

 end;

 end

 registration

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod2 f) -> nonnegative;

 correctness

 proof

 for z be object st z in ( dom ( Partial_Sums_in_cod2 f)) holds 0. <= (( Partial_Sums_in_cod2 f) . z)

 proof

 let z be object;

 assume z in ( dom ( Partial_Sums_in_cod2 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) >= 0 ;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then

 A2: P[ 0 ] by SUPINF_2: 51;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . (n,(k + 1))) >= 0 by SUPINF_2: 51;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence (( Partial_Sums_in_cod2 f) . (n,(k + 1))) >= 0 by A5, A4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then (( Partial_Sums_in_cod2 f) . (n,m)) >= 0 ;

 hence 0. <= (( Partial_Sums_in_cod2 f) . z) by A1;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 end

 registration

 let f be nonpositive Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod2 f) -> nonpositive;

 correctness

 proof

 for z be set st z in ( dom ( Partial_Sums_in_cod2 f)) holds 0. >= (( Partial_Sums_in_cod2 f) . z)

 proof

 let z be set;

 assume z in ( dom ( Partial_Sums_in_cod2 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) <= 0 ;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then

 A2: P[ 0 ] by MESFUNC5: 8;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . (n,(k + 1))) <= 0 by MESFUNC5: 8;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence (( Partial_Sums_in_cod2 f) . (n,(k + 1))) <= 0 by A5, A4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then (( Partial_Sums_in_cod2 f) . (n,m)) <= 0 ;

 hence 0. >= (( Partial_Sums_in_cod2 f) . z) by A1;

 end;

 hence thesis by MESFUNC5: 9;

 end;

 end

 registration

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod1 f) -> without-infty;

 correctness

 proof

 for x be object holds -infty < (( Partial_Sums_in_cod1 f) . x)

 proof

 let x be object;

 per cases ;

 suppose x in ( dom ( Partial_Sums_in_cod1 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & x = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,m)) > -infty ;

 (( Partial_Sums_in_cod1 f) . ( 0 ,m)) = (f . ( 0 ,m)) by DefRSM;

 then

 A2: P[ 0 ] by MESFUNC5:def 5;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . ((k + 1),m)) > -infty by MESFUNC5:def 5;

 (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM;

 hence P[(k + 1)] by A4, A5, XXREAL_3: 17, XXREAL_0: 6;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then P[n];

 hence -infty < (( Partial_Sums_in_cod1 f) . x) by A1;

 end;

 suppose not x in ( dom ( Partial_Sums_in_cod1 f));

 hence -infty < (( Partial_Sums_in_cod1 f) . x) by FUNCT_1:def 2;

 end;

 end;

 hence ( Partial_Sums_in_cod1 f) is without-infty by MESFUNC5:def 5;

 end;

 end

 registration

 let f be without+infty Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod1 f) -> without+infty;

 correctness

 proof

 for x be object holds +infty > (( Partial_Sums_in_cod1 f) . x)

 proof

 let x be object;

 per cases ;

 suppose x in ( dom ( Partial_Sums_in_cod1 f));

 then

 consider n,m be object such that

 A1: n in NAT & m in NAT & x = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,m)) < +infty ;

 (( Partial_Sums_in_cod1 f) . ( 0 ,m)) = (f . ( 0 ,m)) by DefRSM;

 then

 A2: P[ 0 ] by MESFUNC5:def 6;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . ((k + 1),m)) < +infty by MESFUNC5:def 6;

 (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM;

 hence P[(k + 1)] by A4, A5, XXREAL_3: 16, XXREAL_0: 4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then P[n];

 hence +infty > (( Partial_Sums_in_cod1 f) . x) by A1;

 end;

 suppose not x in ( dom ( Partial_Sums_in_cod1 f));

 hence +infty > (( Partial_Sums_in_cod1 f) . x) by FUNCT_1:def 2;

 end;

 end;

 hence ( Partial_Sums_in_cod1 f) is without+infty by MESFUNC5:def 6;

 end;

 end

 registration

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod1 f) -> nonnegative;

 correctness

 proof

 for z be object st z in ( dom ( Partial_Sums_in_cod1 f)) holds 0. <= (( Partial_Sums_in_cod1 f) . z)

 proof

 let z be object;

 assume z in ( dom ( Partial_Sums_in_cod1 f));

 then

 consider m,n be object such that

 A1: m in NAT & n in NAT & z = [m, n] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,n)) >= 0 ;

 (( Partial_Sums_in_cod1 f) . ( 0 ,n)) = (f . ( 0 ,n)) by DefRSM;

 then

 A2: P[ 0 ] by SUPINF_2: 51;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . ((k + 1),n)) >= 0 by SUPINF_2: 51;

 (( Partial_Sums_in_cod1 f) . ((k + 1),n)) = ((( Partial_Sums_in_cod1 f) . (k,n)) + (f . ((k + 1),n))) by DefRSM;

 hence (( Partial_Sums_in_cod1 f) . ((k + 1),n)) >= 0 by A5, A4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then (( Partial_Sums_in_cod1 f) . (m,n)) >= 0 ;

 hence 0. <= (( Partial_Sums_in_cod1 f) . z) by A1;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 end

 registration

 let f be nonpositive Function of [: NAT , NAT :], ExtREAL ;

 cluster ( Partial_Sums_in_cod1 f) -> nonpositive;

 correctness

 proof

 for z be set st z in ( dom ( Partial_Sums_in_cod1 f)) holds 0. >= (( Partial_Sums_in_cod1 f) . z)

 proof

 let z be set;

 assume z in ( dom ( Partial_Sums_in_cod1 f));

 then

 consider m,n be object such that

 A1: m in NAT & n in NAT & z = [m, n] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,n)) <= 0 ;

 (( Partial_Sums_in_cod1 f) . ( 0 ,n)) = (f . ( 0 ,n)) by DefRSM;

 then

 A2: P[ 0 ] by MESFUNC5: 8;



 A3: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P[k];



 A5: (f . ((k + 1),n)) <= 0 by MESFUNC5: 8;

 (( Partial_Sums_in_cod1 f) . ((k + 1),n)) = ((( Partial_Sums_in_cod1 f) . (k,n)) + (f . ((k + 1),n))) by DefRSM;

 hence (( Partial_Sums_in_cod1 f) . ((k + 1),n)) <= 0 by A5, A4;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

 then (( Partial_Sums_in_cod1 f) . (m,n)) <= 0 ;

 hence 0. >= (( Partial_Sums_in_cod1 f) . z) by A1;

 end;

 hence thesis by MESFUNC5: 9;

 end;

 end

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def16

 func Partial_Sums f -> Function of [: NAT , NAT :], ExtREAL equals ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f));

 correctness ;

 end

 theorem :: DBLSEQ_3:39



 Th39: for f be Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds (( Partial_Sums_in_cod1 f) . (n,m)) = (( Partial_Sums_in_cod2 ( ~ f)) . (m,n)) & (( Partial_Sums_in_cod2 f) . (n,m)) = (( Partial_Sums_in_cod1 ( ~ f)) . (m,n))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 let n,m be Nat;

 reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def 12;

 defpred P1[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,m)) = (( Partial_Sums_in_cod2 ( ~ f)) . (m,$1));

 (( Partial_Sums_in_cod1 f) . ( 0 ,m)) = (f . ( 0 ,m)) by DefRSM

 .= (( ~ f) . (m1, 0 )) by FUNCT_4:def 8;

 then

 A2: P1[ 0 ] by DefCSM;



 A3: for k be Nat st P1[k] holds P1[(k + 1)]

 proof

 let k be Nat;

 assume

 A4: P1[k];

 (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM

 .= ((( Partial_Sums_in_cod2 ( ~ f)) . (m,k)) + (( ~ f) . (m1,(k + 1)))) by A4, FUNCT_4:def 8;

 hence P1[(k + 1)] by DefCSM;

 end;

 for k be Nat holds P1[k] from NAT_1:sch 2( A2, A3);

 hence (( Partial_Sums_in_cod1 f) . (n,m)) = (( Partial_Sums_in_cod2 ( ~ f)) . (m,n));

 defpred P2[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) = (( Partial_Sums_in_cod1 ( ~ f)) . ($1,n));

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM

 .= (( ~ f) . ( 0 ,n1)) by FUNCT_4:def 8;

 then

 A5: P2[ 0 ] by DefRSM;



 A6: for k be Nat st P2[k] holds P2[(k + 1)]

 proof

 let k be Nat;

 assume

 A7: P2[k];

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM

 .= ((( Partial_Sums_in_cod1 ( ~ f)) . (k,n)) + (( ~ f) . ((k + 1),n1))) by A7, FUNCT_4:def 8;

 hence P2[(k + 1)] by DefRSM;

 end;

 for k be Nat holds P2[k] from NAT_1:sch 2( A5, A6);

 hence (( Partial_Sums_in_cod2 f) . (n,m)) = (( Partial_Sums_in_cod1 ( ~ f)) . (m,n));

 end;

 theorem :: DBLSEQ_3:40



 Th40: for f be Function of [: NAT , NAT :], ExtREAL holds ( ~ ( Partial_Sums_in_cod1 f)) = ( Partial_Sums_in_cod2 ( ~ f)) & ( ~ ( Partial_Sums_in_cod2 f)) = ( Partial_Sums_in_cod1 ( ~ f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 now

 let z be Element of [: NAT , NAT :];

 consider n,m be object such that

 A1: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A1;

 (( Partial_Sums_in_cod2 ( ~ f)) . z) = (( Partial_Sums_in_cod2 ( ~ f)) . (n,m)) by A1

 .= (( Partial_Sums_in_cod1 f) . (m,n)) by Th39

 .= (( ~ ( Partial_Sums_in_cod1 f)) . (n,m)) by FUNCT_4:def 8;

 hence (( Partial_Sums_in_cod2 ( ~ f)) . z) = (( ~ ( Partial_Sums_in_cod1 f)) . z) by A1;

 end;

 hence ( ~ ( Partial_Sums_in_cod1 f)) = ( Partial_Sums_in_cod2 ( ~ f)) by FUNCT_2:def 8;

 now

 let z be Element of [: NAT , NAT :];

 consider n,m be object such that

 A2: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A2;

 (( Partial_Sums_in_cod1 ( ~ f)) . z) = (( Partial_Sums_in_cod1 ( ~ f)) . (n,m)) by A2

 .= (( Partial_Sums_in_cod2 f) . (m,n)) by Th39

 .= (( ~ ( Partial_Sums_in_cod2 f)) . (n,m)) by FUNCT_4:def 8;

 hence (( Partial_Sums_in_cod1 ( ~ f)) . z) = (( ~ ( Partial_Sums_in_cod2 f)) . z) by A2;

 end;

 hence ( ~ ( Partial_Sums_in_cod2 f)) = ( Partial_Sums_in_cod1 ( ~ f)) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:41

 for f be Function of [: NAT , NAT :], ExtREAL , g be ext-real-valued Function, n be Nat st (for k be Nat holds (( Partial_Sums_in_cod1 f) . (n,k)) = (g . k)) holds (for k be Nat holds (( Partial_Sums f) . (n,k)) = (( Partial_Sums g) . k)) & (( lim_in_cod2 ( Partial_Sums f)) . n) = ( Sum g)

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , g be ext-real-valued Function, n be Nat;

 assume

 A1: for k be Nat holds (( Partial_Sums_in_cod1 f) . (n,k)) = (g . k);

 A4:

 now

 let k be Nat;

 defpred P[ Nat] means (( Partial_Sums f) . (n,$1)) = (( Partial_Sums g) . $1);

 (( Partial_Sums f) . (n, 0 )) = (( Partial_Sums_in_cod1 f) . (n, 0 )) by DefCSM

 .= (g . 0 ) by A1;

 then

 A2: P[ 0 ] by MESFUNC9:def 1;



 A3: for m be Nat st P[m] holds P[(m + 1)]

 proof

 let m be Nat;

 assume

 A4: P[m];

 (( Partial_Sums f) . (n,(m + 1))) = ((( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)) . (n,m)) + (( Partial_Sums_in_cod1 f) . (n,(m + 1)))) by DefCSM

 .= ((( Partial_Sums g) . m) + (g . (m + 1))) by A1, A4;

 hence P[(m + 1)] by MESFUNC9:def 1;

 end;

 for m be Nat holds P[m] from NAT_1:sch 2( A2, A3);

 hence (( Partial_Sums f) . (n,k)) = (( Partial_Sums g) . k);

 end;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 now

 let k be Element of NAT ;

 (( ProjMap1 (( Partial_Sums f),n1)) . k) = (( Partial_Sums f) . (n,k)) by MESFUNC9:def 6;

 hence (( ProjMap1 (( Partial_Sums f),n1)) . k) = (( Partial_Sums g) . k) by A4;

 end;

 then ( ProjMap1 (( Partial_Sums f),n1)) = ( Partial_Sums g) by FUNCT_2:def 8;

 then (( lim_in_cod2 ( Partial_Sums f)) . n1) = ( lim ( Partial_Sums g)) by D1DEF6;

 hence thesis by A4, MESFUNC9:def 3;

 end;

 theorem :: DBLSEQ_3:42



 Th42: for f be Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums_in_cod2 ( - f)) = ( - ( Partial_Sums_in_cod2 f)) & ( Partial_Sums_in_cod1 ( - f)) = ( - ( Partial_Sums_in_cod1 f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;



 A1: ( dom ( - ( Partial_Sums_in_cod2 f))) = [: NAT , NAT :] & ( dom ( - ( Partial_Sums_in_cod1 f))) = [: NAT , NAT :] by FUNCT_2:def 1;



 A2: ( dom ( - f)) = [: NAT , NAT :] by FUNCT_2:def 1;

 for z be Element of [: NAT , NAT :] holds (( - ( Partial_Sums_in_cod2 f)) . z) = (( Partial_Sums_in_cod2 ( - f)) . z)

 proof

 let z be Element of [: NAT , NAT :];

 consider n,m be object such that

 A3: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A3;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 ( - f)) . (n,$1)) = ( - (( Partial_Sums_in_cod2 f) . (n,$1)));

 reconsider z0 = [n, 0 ] as Element of [: NAT , NAT :] by ZFMISC_1: 87;



 A4: [n, 0 ] in [: NAT , NAT :] by ZFMISC_1: 87;

 (( Partial_Sums_in_cod2 ( - f)) . (n, 0 )) = (( - f) . (n, 0 )) by DefCSM

 .= ( - (f . (n, 0 ))) by A4, A2, MESFUNC1:def 7;

 then

 A5: P[ 0 ] by DefCSM;



 A6: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A7: P[k];



 A8: [n, (k + 1)] in [: NAT , NAT :] by ZFMISC_1: 87;



 thus (( Partial_Sums_in_cod2 ( - f)) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 ( - f)) . (n,k)) + (( - f) . (n,(k + 1)))) by DefCSM

 .= (( - (( Partial_Sums_in_cod2 f) . (n,k))) - (f . (n,(k + 1)))) by A7, A8, A2, MESFUNC1:def 7

 .= ( - ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1))))) by XXREAL_3: 25

 .= ( - (( Partial_Sums_in_cod2 f) . (n,(k + 1)))) by DefCSM;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A5, A6);

 then (( Partial_Sums_in_cod2 ( - f)) . (n,m)) = ( - (( Partial_Sums_in_cod2 f) . (n,m)));

 hence thesis by A3, A1, MESFUNC1:def 7;

 end;

 hence ( Partial_Sums_in_cod2 ( - f)) = ( - ( Partial_Sums_in_cod2 f)) by FUNCT_2:def 8;

 for z be Element of [: NAT , NAT :] holds (( - ( Partial_Sums_in_cod1 f)) . z) = (( Partial_Sums_in_cod1 ( - f)) . z)

 proof

 let z be Element of [: NAT , NAT :];

 consider n,m be object such that

 A3: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A3;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 ( - f)) . ($1,m)) = ( - (( Partial_Sums_in_cod1 f) . ($1,m)));

 reconsider z0 = [ 0 , m] as Element of [: NAT , NAT :] by ZFMISC_1: 87;



 A4: [ 0 , m] in [: NAT , NAT :] by ZFMISC_1: 87;

 (( Partial_Sums_in_cod1 ( - f)) . ( 0 ,m)) = (( - f) . ( 0 ,m)) by DefRSM

 .= ( - (f . ( 0 ,m))) by A4, A2, MESFUNC1:def 7;

 then

 A5: P[ 0 ] by DefRSM;



 A6: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A7: P[k];



 A8: [(k + 1), m] in [: NAT , NAT :] by ZFMISC_1: 87;



 thus (( Partial_Sums_in_cod1 ( - f)) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 ( - f)) . (k,m)) + (( - f) . ((k + 1),m))) by DefRSM

 .= (( - (( Partial_Sums_in_cod1 f) . (k,m))) - (f . ((k + 1),m))) by A7, A8, A2, MESFUNC1:def 7

 .= ( - ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m)))) by XXREAL_3: 25

 .= ( - (( Partial_Sums_in_cod1 f) . ((k + 1),m))) by DefRSM;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A5, A6);

 then (( Partial_Sums_in_cod1 ( - f)) . (n,m)) = ( - (( Partial_Sums_in_cod1 f) . (n,m)));

 hence thesis by A3, A1, MESFUNC1:def 7;

 end;

 hence ( Partial_Sums_in_cod1 ( - f)) = ( - ( Partial_Sums_in_cod1 f)) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:43



 Th43: for f be Function of [: NAT , NAT :], ExtREAL , m,n be Element of NAT holds (( Partial_Sums_in_cod1 f) . (m,n)) = (( Partial_Sums ( ProjMap2 (f,n))) . m) & (( Partial_Sums_in_cod2 f) . (m,n)) = (( Partial_Sums ( ProjMap1 (f,m))) . n)

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 let m,n be Element of NAT ;

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,n)) = (( Partial_Sums ( ProjMap2 (f,n))) . $1);

 (( Partial_Sums ( ProjMap2 (f,n))) . 0 ) = (( ProjMap2 (f,n)) . 0 ) by MESFUNC9:def 1

 .= (f . ( 0 ,n)) by MESFUNC9:def 7;

 then

 a1: P[ 0 ] by DefRSM;



 a2: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume P[k];



 then (( Partial_Sums_in_cod1 f) . ((k + 1),n)) = ((( Partial_Sums ( ProjMap2 (f,n))) . k) + (f . ((k + 1),n))) by DefRSM

 .= ((( Partial_Sums ( ProjMap2 (f,n))) . k) + (( ProjMap2 (f,n)) . (k + 1))) by MESFUNC9:def 7;

 hence P[(k + 1)] by MESFUNC9:def 1;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( a1, a2);

 hence (( Partial_Sums_in_cod1 f) . (m,n)) = (( Partial_Sums ( ProjMap2 (f,n))) . m);

 defpred Q[ Nat] means (( Partial_Sums_in_cod2 f) . (m,$1)) = (( Partial_Sums ( ProjMap1 (f,m))) . $1);

 (( Partial_Sums ( ProjMap1 (f,m))) . 0 ) = (( ProjMap1 (f,m)) . 0 ) by MESFUNC9:def 1

 .= (f . (m, 0 )) by MESFUNC9:def 6;

 then

 a3: Q[ 0 ] by DefCSM;



 a4: for k be Nat st Q[k] holds Q[(k + 1)]

 proof

 let k be Nat;

 assume Q[k];



 then (( Partial_Sums_in_cod2 f) . (m,(k + 1))) = ((( Partial_Sums ( ProjMap1 (f,m))) . k) + (f . (m,(k + 1)))) by DefCSM

 .= ((( Partial_Sums ( ProjMap1 (f,m))) . k) + (( ProjMap1 (f,m)) . (k + 1))) by MESFUNC9:def 6;

 hence Q[(k + 1)] by MESFUNC9:def 1;

 end;

 for k be Nat holds Q[k] from NAT_1:sch 2( a3, a4);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:44



 Th44: for f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums_in_cod2 (f1 + f2)) = (( Partial_Sums_in_cod2 f1) + ( Partial_Sums_in_cod2 f2)) & ( Partial_Sums_in_cod1 (f1 + f2)) = (( Partial_Sums_in_cod1 f1) + ( Partial_Sums_in_cod1 f2))

 proof

 let f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL ;

 set CS1 = ( Partial_Sums_in_cod2 f1);

 set CS2 = ( Partial_Sums_in_cod2 f2);

 set CS12 = ( Partial_Sums_in_cod2 (f1 + f2));

 set RS1 = ( Partial_Sums_in_cod1 f1);

 set RS2 = ( Partial_Sums_in_cod1 f2);

 set RS12 = ( Partial_Sums_in_cod1 (f1 + f2));

 now

 let n be Element of NAT , m be Element of NAT ;

 defpred C[ Nat] means (CS12 . (n,$1)) = ((CS1 . (n,$1)) + (CS2 . (n,$1)));

 (CS12 . (n, 0 )) = ((f1 + f2) . (n, 0 )) by DefCSM

 .= ((f1 . (n, 0 )) + (f2 . (n, 0 ))) by Th11

 .= ((CS1 . (n, 0 )) + (f2 . (n, 0 ))) by DefCSM;

 then

 a1: C[ 0 ] by DefCSM;



 a2: for k be Nat st C[k] holds C[(k + 1)]

 proof

 let k be Nat;

 assume

 a3: C[k];



 X1: (CS1 . (n,k)) <> -infty & (CS2 . (n,k)) <> -infty & (f1 . (n,(k + 1))) <> -infty & (f2 . (n,(k + 1))) <> -infty & ((f1 + f2) . (n,(k + 1))) <> -infty by MESFUNC5:def 5;

 then

 X2: ((CS2 . (n,k)) + (f2 . (n,(k + 1)))) <> -infty by XXREAL_3: 17;

 (CS12 . (n,(k + 1))) = ((CS12 . (n,k)) + ((f1 + f2) . (n,(k + 1)))) by DefCSM

 .= ((CS1 . (n,k)) + (((f1 + f2) . (n,(k + 1))) + (CS2 . (n,k)))) by a3, X1, XXREAL_3: 29

 .= ((CS1 . (n,k)) + (((f1 . (n,(k + 1))) + (f2 . (n,(k + 1)))) + (CS2 . (n,k)))) by Th11

 .= ((CS1 . (n,k)) + ((f1 . (n,(k + 1))) + ((CS2 . (n,k)) + (f2 . (n,(k + 1)))))) by X1, XXREAL_3: 29

 .= (((CS1 . (n,k)) + (f1 . (n,(k + 1)))) + ((CS2 . (n,k)) + (f2 . (n,(k + 1))))) by X1, X2, XXREAL_3: 29

 .= ((CS1 . (n,(k + 1))) + ((CS2 . (n,k)) + (f2 . (n,(k + 1))))) by DefCSM;

 hence C[(k + 1)] by DefCSM;

 end;

 for k be Nat holds C[k] from NAT_1:sch 2( a1, a2);

 then C[m];

 hence (CS12 . (n,m)) = ((CS1 + CS2) . (n,m)) by Th11;

 end;

 hence CS12 = (CS1 + CS2) by BINOP_1: 2;

 now

 let n,m be Element of NAT ;

 defpred R[ Nat] means (RS12 . ($1,m)) = ((RS1 . ($1,m)) + (RS2 . ($1,m)));

 (RS12 . ( 0 ,m)) = ((f1 + f2) . ( 0 ,m)) by DefRSM

 .= ((f1 . ( 0 ,m)) + (f2 . ( 0 ,m))) by Th11

 .= ((RS1 . ( 0 ,m)) + (f2 . ( 0 ,m))) by DefRSM;

 then

 a4: R[ 0 ] by DefRSM;



 a5: for k be Nat st R[k] holds R[(k + 1)]

 proof

 let k be Nat;

 assume

 a6: R[k];



 X3: (RS1 . (k,m)) <> -infty & (RS2 . (k,m)) <> -infty & (f1 . ((k + 1),m)) <> -infty & (f2 . ((k + 1),m)) <> -infty & ((f1 + f2) . ((k + 1),m)) <> -infty by MESFUNC5:def 5;

 then

 X4: ((RS2 . (k,m)) + (f2 . ((k + 1),m))) <> -infty by XXREAL_3: 17;

 (RS12 . ((k + 1),m)) = ((RS12 . (k,m)) + ((f1 + f2) . ((k + 1),m))) by DefRSM

 .= ((RS1 . (k,m)) + (((f1 + f2) . ((k + 1),m)) + (RS2 . (k,m)))) by a6, X3, XXREAL_3: 29

 .= ((RS1 . (k,m)) + (((f1 . ((k + 1),m)) + (f2 . ((k + 1),m))) + (RS2 . (k,m)))) by Th11

 .= ((RS1 . (k,m)) + ((f1 . ((k + 1),m)) + ((RS2 . (k,m)) + (f2 . ((k + 1),m))))) by X3, XXREAL_3: 29

 .= (((RS1 . (k,m)) + (f1 . ((k + 1),m))) + ((RS2 . (k,m)) + (f2 . ((k + 1),m)))) by X3, X4, XXREAL_3: 29

 .= ((RS1 . ((k + 1),m)) + ((RS2 . (k,m)) + (f2 . ((k + 1),m)))) by DefRSM;

 hence R[(k + 1)] by DefRSM;

 end;

 for k be Nat holds R[k] from NAT_1:sch 2( a4, a5);

 then R[n];

 hence (RS12 . (n,m)) = ((RS1 + RS2) . (n,m)) by Th11;

 end;

 hence RS12 = (RS1 + RS2) by BINOP_1: 2;

 end;

 theorem :: DBLSEQ_3:45



 Th45: for f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums_in_cod2 (f1 + f2)) = (( Partial_Sums_in_cod2 f1) + ( Partial_Sums_in_cod2 f2)) & ( Partial_Sums_in_cod1 (f1 + f2)) = (( Partial_Sums_in_cod1 f1) + ( Partial_Sums_in_cod1 f2))

 proof

 let f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL ;

 reconsider F1 = ( - f1), F2 = ( - f2) as without-infty Function of [: NAT , NAT :], ExtREAL ;

 (F1 + F2) = (( - f1) - f2) by Th10

 .= ( - (f1 + f2)) by Th9;

 then

 A1: ( - (F1 + F2)) = (f1 + f2) by Th2;



 then ( Partial_Sums_in_cod2 (f1 + f2)) = ( - ( Partial_Sums_in_cod2 (F1 + F2))) by Th42

 .= ( - (( Partial_Sums_in_cod2 F1) + ( Partial_Sums_in_cod2 F2))) by Th44

 .= ( - (( - ( Partial_Sums_in_cod2 f1)) + ( Partial_Sums_in_cod2 F2))) by Th42

 .= ( - (( - ( Partial_Sums_in_cod2 f1)) + ( - ( Partial_Sums_in_cod2 f2)))) by Th42

 .= (( - ( - ( Partial_Sums_in_cod2 f1))) - ( - ( Partial_Sums_in_cod2 f2))) by Th8

 .= (( Partial_Sums_in_cod2 f1) - ( - ( Partial_Sums_in_cod2 f2))) by Th2

 .= (( Partial_Sums_in_cod2 f1) + ( - ( - ( Partial_Sums_in_cod2 f2)))) by Th10;

 hence ( Partial_Sums_in_cod2 (f1 + f2)) = (( Partial_Sums_in_cod2 f1) + ( Partial_Sums_in_cod2 f2)) by Th2;

 ( Partial_Sums_in_cod1 (f1 + f2)) = ( - ( Partial_Sums_in_cod1 (F1 + F2))) by A1, Th42

 .= ( - (( Partial_Sums_in_cod1 F1) + ( Partial_Sums_in_cod1 F2))) by Th44

 .= ( - (( - ( Partial_Sums_in_cod1 f1)) + ( Partial_Sums_in_cod1 F2))) by Th42

 .= ( - (( - ( Partial_Sums_in_cod1 f1)) + ( - ( Partial_Sums_in_cod1 f2)))) by Th42

 .= (( - ( - ( Partial_Sums_in_cod1 f1))) - ( - ( Partial_Sums_in_cod1 f2))) by Th8

 .= (( Partial_Sums_in_cod1 f1) - ( - ( Partial_Sums_in_cod1 f2))) by Th2

 .= (( Partial_Sums_in_cod1 f1) + ( - ( - ( Partial_Sums_in_cod1 f2)))) by Th10;

 hence thesis by Th2;

 end;

 theorem :: DBLSEQ_3:46



 Th46: for f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums_in_cod2 (f1 - f2)) = (( Partial_Sums_in_cod2 f1) - ( Partial_Sums_in_cod2 f2)) & ( Partial_Sums_in_cod1 (f1 - f2)) = (( Partial_Sums_in_cod1 f1) - ( Partial_Sums_in_cod1 f2)) & ( Partial_Sums_in_cod2 (f2 - f1)) = (( Partial_Sums_in_cod2 f2) - ( Partial_Sums_in_cod2 f1)) & ( Partial_Sums_in_cod1 (f2 - f1)) = (( Partial_Sums_in_cod1 f2) - ( Partial_Sums_in_cod1 f1))

 proof

 let f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL ;

 ( Partial_Sums_in_cod2 (f1 - f2)) = ( Partial_Sums_in_cod2 (f1 + ( - f2))) by Th10

 .= (( Partial_Sums_in_cod2 f1) + ( Partial_Sums_in_cod2 ( - f2))) by Th44

 .= (( Partial_Sums_in_cod2 f1) + ( - ( Partial_Sums_in_cod2 f2))) by Th42;

 hence ( Partial_Sums_in_cod2 (f1 - f2)) = (( Partial_Sums_in_cod2 f1) - ( Partial_Sums_in_cod2 f2)) by Th10;

 ( Partial_Sums_in_cod1 (f1 - f2)) = ( Partial_Sums_in_cod1 (f1 + ( - f2))) by Th10

 .= (( Partial_Sums_in_cod1 f1) + ( Partial_Sums_in_cod1 ( - f2))) by Th44

 .= (( Partial_Sums_in_cod1 f1) + ( - ( Partial_Sums_in_cod1 f2))) by Th42;

 hence ( Partial_Sums_in_cod1 (f1 - f2)) = (( Partial_Sums_in_cod1 f1) - ( Partial_Sums_in_cod1 f2)) by Th10;

 ( Partial_Sums_in_cod2 (f2 - f1)) = ( Partial_Sums_in_cod2 (f2 + ( - f1))) by Th10

 .= (( Partial_Sums_in_cod2 f2) + ( Partial_Sums_in_cod2 ( - f1))) by Th45

 .= (( Partial_Sums_in_cod2 f2) + ( - ( Partial_Sums_in_cod2 f1))) by Th42;

 hence ( Partial_Sums_in_cod2 (f2 - f1)) = (( Partial_Sums_in_cod2 f2) - ( Partial_Sums_in_cod2 f1)) by Th10;

 ( Partial_Sums_in_cod1 (f2 - f1)) = ( Partial_Sums_in_cod1 (f2 + ( - f1))) by Th10

 .= (( Partial_Sums_in_cod1 f2) + ( Partial_Sums_in_cod1 ( - f1))) by Th45

 .= (( Partial_Sums_in_cod1 f2) + ( - ( Partial_Sums_in_cod1 f1))) by Th42;

 hence ( Partial_Sums_in_cod1 (f2 - f1)) = (( Partial_Sums_in_cod1 f2) - ( Partial_Sums_in_cod1 f1)) by Th10;

 end;

 theorem :: DBLSEQ_3:47



 Th47: for f be without-infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds (( Partial_Sums f) . ((n + 1),m)) = ((( Partial_Sums_in_cod2 f) . ((n + 1),m)) + (( Partial_Sums f) . (n,m))) & (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,(m + 1))) = ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,m)))

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 let n,m be Nat;

 set RPS = ( Partial_Sums f);

 set CPS = ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f));

 set ROW = ( Partial_Sums_in_cod1 f);

 set COL = ( Partial_Sums_in_cod2 f);

 defpred P[ Nat] means (RPS . ((n + 1),$1)) = ((COL . ((n + 1),$1)) + (RPS . (n,$1)));



 a1: (RPS . (n, 0 )) = (ROW . (n, 0 )) by DefCSM;

 (RPS . ((n + 1), 0 )) = (ROW . ((n + 1), 0 )) by DefCSM

 .= ((ROW . (n, 0 )) + (f . ((n + 1), 0 ))) by DefRSM;

 then

 a3: P[ 0 ] by a1, DefCSM;



 a4: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A5: P[k];



 a6: (COL . ((n + 1),(k + 1))) = ((COL . ((n + 1),k)) + (f . ((n + 1),(k + 1)))) by DefCSM;



 X1: (COL . ((n + 1),k)) <> -infty & (f . ((n + 1),(k + 1))) <> -infty & (RPS . (n,k)) <> -infty & (ROW . (n,(k + 1))) <> -infty & (RPS . ((n + 1),k)) <> -infty by MESFUNC5:def 5;

 then

 X2: ((COL . ((n + 1),k)) + (f . ((n + 1),(k + 1)))) <> -infty by XXREAL_3: 17;

 (RPS . (n,(k + 1))) = ((RPS . (n,k)) + (ROW . (n,(k + 1)))) by DefCSM;



 then ((COL . ((n + 1),(k + 1))) + (RPS . (n,(k + 1)))) = ((((COL . ((n + 1),k)) + (f . ((n + 1),(k + 1)))) + (RPS . (n,k))) + (ROW . (n,(k + 1)))) by a6, X1, X2, XXREAL_3: 29

 .= ((((COL . ((n + 1),k)) + (RPS . (n,k))) + (f . ((n + 1),(k + 1)))) + (ROW . (n,(k + 1)))) by X1, XXREAL_3: 29

 .= ((RPS . ((n + 1),k)) + ((f . ((n + 1),(k + 1))) + (ROW . (n,(k + 1))))) by A5, X1, XXREAL_3: 29

 .= ((RPS . ((n + 1),k)) + (ROW . ((n + 1),(k + 1)))) by DefRSM;

 hence P[(k + 1)] by DefCSM;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( a3, a4);

 hence (RPS . ((n + 1),m)) = ((COL . ((n + 1),m)) + (RPS . (n,m)));

 defpred Q[ Nat] means (CPS . ($1,(m + 1))) = ((ROW . ($1,(m + 1))) + (CPS . ($1,m)));



 b1: (CPS . ( 0 ,m)) = (COL . ( 0 ,m)) by DefRSM;

 (CPS . ( 0 ,(m + 1))) = (COL . ( 0 ,(m + 1))) by DefRSM

 .= ((COL . ( 0 ,m)) + (f . ( 0 ,(m + 1)))) by DefCSM;

 then

 b3: Q[ 0 ] by b1, DefRSM;



 b4: for k be Nat st Q[k] holds Q[(k + 1)]

 proof

 let k be Nat;

 assume

 B5: Q[k];



 b6: (ROW . ((k + 1),(m + 1))) = ((ROW . (k,(m + 1))) + (f . ((k + 1),(m + 1)))) by DefRSM;



 X3: (ROW . (k,(m + 1))) <> -infty & (f . ((k + 1),(m + 1))) <> -infty & (CPS . (k,m)) <> -infty & (COL . ((k + 1),m)) <> -infty & (CPS . (k,(m + 1))) <> -infty by MESFUNC5:def 5;

 then

 X4: ((ROW . (k,(m + 1))) + (f . ((k + 1),(m + 1)))) <> -infty by XXREAL_3: 17;

 (CPS . ((k + 1),m)) = ((CPS . (k,m)) + (COL . ((k + 1),m))) by DefRSM;



 then ((ROW . ((k + 1),(m + 1))) + (CPS . ((k + 1),m))) = ((((ROW . (k,(m + 1))) + (f . ((k + 1),(m + 1)))) + (CPS . (k,m))) + (COL . ((k + 1),m))) by b6, X3, X4, XXREAL_3: 29

 .= ((((ROW . (k,(m + 1))) + (CPS . (k,m))) + (f . ((k + 1),(m + 1)))) + (COL . ((k + 1),m))) by X3, XXREAL_3: 29

 .= ((CPS . (k,(m + 1))) + ((f . ((k + 1),(m + 1))) + (COL . ((k + 1),m)))) by B5, X3, XXREAL_3: 29

 .= ((CPS . (k,(m + 1))) + (COL . ((k + 1),(m + 1)))) by DefCSM;

 hence Q[(k + 1)] by DefRSM;

 end;

 for k be Nat holds Q[k] from NAT_1:sch 2( b3, b4);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:48

 for f be without+infty Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds (( Partial_Sums f) . ((n + 1),m)) = ((( Partial_Sums_in_cod2 f) . ((n + 1),m)) + (( Partial_Sums f) . (n,m))) & (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,(m + 1))) = ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,m)))

 proof

 let f be without+infty Function of [: NAT , NAT :], ExtREAL ;

 let n,m be Nat;

 reconsider g = ( - f) as without-infty Function of [: NAT , NAT :], ExtREAL ;



 A2: ( dom ( - ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 g)))) = [: NAT , NAT :] & ( dom ( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g)))) = [: NAT , NAT :] by FUNCT_2:def 1;



 A4: ( dom ( - ( Partial_Sums_in_cod2 g))) = [: NAT , NAT :] & ( dom ( - ( Partial_Sums_in_cod1 g))) = [: NAT , NAT :] by FUNCT_2:def 1;



 A5: ( dom ( - ( Partial_Sums g))) = [: NAT , NAT :] by FUNCT_2:def 1;

 n in NAT & m in NAT by ORDINAL1:def 12;

 then

 A3: [(n + 1), m] in [: NAT , NAT :] & [n, m] in [: NAT , NAT :] & [n, (m + 1)] in [: NAT , NAT :] by ZFMISC_1: 87;



 A1: ( Partial_Sums f) = ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 ( - g))) by Th2

 .= ( Partial_Sums_in_cod2 ( - ( Partial_Sums_in_cod1 g))) by Th42

 .= ( - ( Partial_Sums g)) by Th42;



 thus (( Partial_Sums f) . ((n + 1),m)) = ( - (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 g)) . ((n + 1),m))) by A1, A2, A3, MESFUNC1:def 7

 .= ( - ((( Partial_Sums_in_cod2 g) . ((n + 1),m)) + (( Partial_Sums g) . (n,m)))) by Th47

 .= (( - (( Partial_Sums_in_cod2 g) . ((n + 1),m))) - (( Partial_Sums g) . (n,m))) by XXREAL_3: 25

 .= ((( - ( Partial_Sums_in_cod2 g)) . ((n + 1),m)) + ( - (( Partial_Sums g) . (n,m)))) by A3, A4, MESFUNC1:def 7

 .= ((( Partial_Sums_in_cod2 ( - g)) . ((n + 1),m)) + ( - (( Partial_Sums g) . (n,m)))) by Th42

 .= ((( Partial_Sums_in_cod2 f) . ((n + 1),m)) + ( - (( Partial_Sums g) . (n,m)))) by Th2

 .= ((( Partial_Sums_in_cod2 f) . ((n + 1),m)) + (( Partial_Sums f) . (n,m))) by A1, A3, A5, MESFUNC1:def 7;



 thus (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,(m + 1))) = (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( - g))) . (n,(m + 1))) by Th2

 .= (( Partial_Sums_in_cod1 ( - ( Partial_Sums_in_cod2 g))) . (n,(m + 1))) by Th42

 .= (( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g))) . (n,(m + 1))) by Th42

 .= ( - (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g)) . (n,(m + 1)))) by A3, A2, MESFUNC1:def 7

 .= ( - ((( Partial_Sums_in_cod1 g) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g)) . (n,m)))) by Th47

 .= (( - (( Partial_Sums_in_cod1 g) . (n,(m + 1)))) - (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g)) . (n,m))) by XXREAL_3: 25

 .= ((( - ( Partial_Sums_in_cod1 g)) . (n,(m + 1))) - (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g)) . (n,m))) by A4, A3, MESFUNC1:def 7

 .= ((( - ( Partial_Sums_in_cod1 g)) . (n,(m + 1))) + (( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g))) . (n,m))) by A3, A2, MESFUNC1:def 7

 .= ((( Partial_Sums_in_cod1 ( - g)) . (n,(m + 1))) + (( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g))) . (n,m))) by Th42

 .= ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g))) . (n,m))) by Th2

 .= ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( - ( Partial_Sums_in_cod2 g))) . (n,m))) by Th42

 .= ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( - g))) . (n,m))) by Th42

 .= ((( Partial_Sums_in_cod1 f) . (n,(m + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,m))) by Th2;

 end;



 Lm7: for f be without-infty Function of [: NAT , NAT :], ExtREAL , m,n be Nat holds (( Partial_Sums f) . (m,n)) = (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,n))

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL , m,n be Nat;

 defpred P1[ Nat] means for m be Nat holds (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,$1)) = (( Partial_Sums f) . (m,$1));

 defpred P2[ Nat] means (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ($1, 0 )) = (( Partial_Sums f) . ($1, 0 ));



 Y3: (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ( 0 , 0 )) = (( Partial_Sums_in_cod2 f) . ( 0 , 0 )) by DefRSM

 .= (f . ( 0 , 0 )) by DefCSM;

 (( Partial_Sums f) . ( 0 , 0 )) = (( Partial_Sums_in_cod1 f) . ( 0 , 0 )) by DefCSM

 .= (f . ( 0 , 0 )) by DefRSM;

 then

 Y1: P2[ 0 ] by Y3;



 Y2: for i be Nat st P2[i] holds P2[(i + 1)]

 proof

 let i be Nat;

 assume

 Y3: P2[i];



 Y4: (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ((i + 1), 0 )) = ((( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i, 0 )) + (( Partial_Sums_in_cod2 f) . ((i + 1), 0 ))) by DefRSM

 .= ((( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i, 0 )) + (f . ((i + 1), 0 ))) by DefCSM;

 (( Partial_Sums f) . ((i + 1), 0 )) = (( Partial_Sums_in_cod1 f) . ((i + 1), 0 )) by DefCSM

 .= ((( Partial_Sums_in_cod1 f) . (i, 0 )) + (f . ((i + 1), 0 ))) by DefRSM

 .= ((( Partial_Sums f) . (i, 0 )) + (f . ((i + 1), 0 ))) by DefCSM;

 hence P2[(i + 1)] by Y3, Y4;

 end;

 for n be Nat holds P2[n] from NAT_1:sch 2( Y1, Y2);

 then

 X1: P1[ 0 ];



 X2: for j be Nat st P1[j] holds P1[(j + 1)]

 proof

 let j be Nat;

 assume

 Z3: P1[j];

 for m be Nat holds (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,(j + 1))) = (( Partial_Sums f) . (m,(j + 1)))

 proof

 let n be Nat;

 defpred P3[ Nat] means (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ($1,(j + 1))) = (( Partial_Sums f) . ($1,(j + 1)));



 Z4: (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ( 0 ,(j + 1))) = (( Partial_Sums_in_cod2 f) . ( 0 ,(j + 1))) by DefRSM

 .= ((( Partial_Sums_in_cod2 f) . ( 0 ,j)) + (f . ( 0 ,(j + 1)))) by DefCSM

 .= ((( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ( 0 ,j)) + (f . ( 0 ,(j + 1)))) by DefRSM;

 (( Partial_Sums f) . ( 0 ,(j + 1))) = ((( Partial_Sums f) . ( 0 ,j)) + (( Partial_Sums_in_cod1 f) . ( 0 ,(j + 1)))) by DefCSM

 .= ((( Partial_Sums f) . ( 0 ,j)) + (f . ( 0 ,(j + 1)))) by DefRSM

 .= ((( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ( 0 ,j)) + (f . ( 0 ,(j + 1)))) by Z3;

 then

 Z1: P3[ 0 ] by Z4;



 Z2: for i be Nat st P3[i] holds P3[(i + 1)]

 proof

 let i be Nat;

 assume P3[i];



 W1: (( Partial_Sums f) . (i,j)) <> -infty & (( Partial_Sums_in_cod1 f) . (i,(j + 1))) <> -infty & (( Partial_Sums_in_cod2 f) . ((i + 1),j)) <> -infty & (f . ((i + 1),(j + 1))) <> -infty & (( Partial_Sums_in_cod1 f) . ((i + 1),(j + 1))) <> -infty & (( Partial_Sums_in_cod2 f) . ((i + 1),j)) <> -infty by MESFUNC5:def 5;

 then

 W2: ((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod1 f) . (i,(j + 1)))) <> -infty by XXREAL_3: 17;



 Z6: (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i,j)) = (( Partial_Sums f) . (i,j)) by Z3;

 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ((i + 1),(j + 1))) = ((( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i,(j + 1))) + (( Partial_Sums_in_cod2 f) . ((i + 1),(j + 1)))) by DefRSM

 .= (((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod1 f) . (i,(j + 1)))) + (( Partial_Sums_in_cod2 f) . ((i + 1),(j + 1)))) by Z6, Th47

 .= (((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod1 f) . (i,(j + 1)))) + ((( Partial_Sums_in_cod2 f) . ((i + 1),j)) + (f . ((i + 1),(j + 1))))) by DefCSM

 .= ((((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod1 f) . (i,(j + 1)))) + (f . ((i + 1),(j + 1)))) + (( Partial_Sums_in_cod2 f) . ((i + 1),j))) by W1, W2, XXREAL_3: 29

 .= (((( Partial_Sums f) . (i,j)) + ((( Partial_Sums_in_cod1 f) . (i,(j + 1))) + (f . ((i + 1),(j + 1))))) + (( Partial_Sums_in_cod2 f) . ((i + 1),j))) by W1, XXREAL_3: 29

 .= (((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod1 f) . ((i + 1),(j + 1)))) + (( Partial_Sums_in_cod2 f) . ((i + 1),j))) by DefRSM

 .= (((( Partial_Sums f) . (i,j)) + (( Partial_Sums_in_cod2 f) . ((i + 1),j))) + (( Partial_Sums_in_cod1 f) . ((i + 1),(j + 1)))) by W1, XXREAL_3: 29

 .= ((( Partial_Sums f) . ((i + 1),j)) + (( Partial_Sums_in_cod1 f) . ((i + 1),(j + 1)))) by Th47;

 hence thesis by DefCSM;

 end;

 for n be Nat holds P3[n] from NAT_1:sch 2( Z1, Z2);

 hence thesis;

 end;

 hence P1[(j + 1)];

 end;

 for m be Nat holds P1[m] from NAT_1:sch 2( X1, X2);

 hence thesis;

 end;



 Lm8: for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums f) = ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f))

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 now

 let x be Element of [: NAT , NAT :];

 consider n,m be object such that

 A1: n in NAT & m in NAT & x = [n, m] by ZFMISC_1:def 2;

 reconsider n1 = n, m1 = m as Nat by A1;

 (( Partial_Sums f) . (n1,m1)) = (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n1,m1)) by Lm7;

 hence (( Partial_Sums f) . x) = (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . x) by A1;

 end;

 hence thesis by FUNCT_2: 63;

 end;



 Lm9: for f be without+infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums f) = ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f))

 proof

 let f be without+infty Function of [: NAT , NAT :], ExtREAL ;

 reconsider g = ( - f) as without-infty Function of [: NAT , NAT :], ExtREAL ;



 A1: ( Partial_Sums f) = ( Partial_Sums ( - g)) by Th2

 .= ( Partial_Sums_in_cod2 ( - ( Partial_Sums_in_cod1 g))) by Th42

 .= ( - ( Partial_Sums g)) by Th42;

 ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) = ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( - g))) by Th2

 .= ( Partial_Sums_in_cod1 ( - ( Partial_Sums_in_cod2 g))) by Th42

 .= ( - ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 g))) by Th42;

 hence thesis by A1, Lm8;

 end;

 theorem :: DBLSEQ_3:49

 for f be Function of [: NAT , NAT :], ExtREAL st f is without-infty or f is without+infty holds ( Partial_Sums f) = ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) by Lm8, Lm9;

 theorem :: DBLSEQ_3:50

 for f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums (f1 + f2)) = (( Partial_Sums f1) + ( Partial_Sums f2))

 proof

 let f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL ;

 ( Partial_Sums (f1 + f2)) = ( Partial_Sums_in_cod2 (( Partial_Sums_in_cod1 f1) + ( Partial_Sums_in_cod1 f2))) by Th44;

 hence thesis by Th44;

 end;

 theorem :: DBLSEQ_3:51

 for f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums (f1 + f2)) = (( Partial_Sums f1) + ( Partial_Sums f2))

 proof

 let f1,f2 be without+infty Function of [: NAT , NAT :], ExtREAL ;

 ( Partial_Sums (f1 + f2)) = ( Partial_Sums_in_cod2 (( Partial_Sums_in_cod1 f1) + ( Partial_Sums_in_cod1 f2))) by Th45;

 hence thesis by Th45;

 end;

 theorem :: DBLSEQ_3:52

 for f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums (f1 - f2)) = (( Partial_Sums f1) - ( Partial_Sums f2)) & ( Partial_Sums (f2 - f1)) = (( Partial_Sums f2) - ( Partial_Sums f1))

 proof

 let f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be without+infty Function of [: NAT , NAT :], ExtREAL ;

 ( Partial_Sums (f1 - f2)) = ( Partial_Sums_in_cod2 (( Partial_Sums_in_cod1 f1) - ( Partial_Sums_in_cod1 f2))) by Th46;

 hence ( Partial_Sums (f1 - f2)) = (( Partial_Sums f1) - ( Partial_Sums f2)) by Th46;

 ( Partial_Sums (f2 - f1)) = ( Partial_Sums_in_cod2 (( Partial_Sums_in_cod1 f2) - ( Partial_Sums_in_cod1 f1))) by Th46;

 hence ( Partial_Sums (f2 - f1)) = (( Partial_Sums f2) - ( Partial_Sums f1)) by Th46;

 end;

 theorem :: DBLSEQ_3:53

 for f be Function of [: NAT , NAT :], ExtREAL , k be Element of NAT holds ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) = ( Partial_Sums ( ProjMap2 (f,k))) & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) = ( Partial_Sums ( ProjMap1 (f,k)))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , k be Element of NAT ;

 now

 let n be Element of NAT ;

 (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) = (( Partial_Sums_in_cod1 f) . (n,k)) by MESFUNC9:def 7;

 hence (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) = (( Partial_Sums ( ProjMap2 (f,k))) . n) by Th43;

 end;

 hence ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) = ( Partial_Sums ( ProjMap2 (f,k))) by FUNCT_2:def 8;

 now

 let n be Element of NAT ;

 (( ProjMap1 (( Partial_Sums_in_cod2 f),k)) . n) = (( Partial_Sums_in_cod2 f) . (k,n)) by MESFUNC9:def 6;

 hence (( ProjMap1 (( Partial_Sums_in_cod2 f),k)) . n) = (( Partial_Sums ( ProjMap1 (f,k))) . n) by Th43;

 end;

 hence ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) = ( Partial_Sums ( ProjMap1 (f,k))) by FUNCT_2:def 8;

 end;

 theorem :: DBLSEQ_3:54



 Th54: for f be Function of [: NAT , NAT :], ExtREAL st f is without-infty or f is without+infty holds ( ProjMap1 (( Partial_Sums f), 0 )) = ( ProjMap1 (( Partial_Sums_in_cod2 f), 0 )) & ( ProjMap2 (( Partial_Sums f), 0 )) = ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 ))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A0: f is without-infty or f is without+infty;

 A1:

 now

 let m be Element of NAT ;

 (( ProjMap1 (( Partial_Sums f), 0 )) . m) = (( Partial_Sums f) . ( 0 ,m)) by MESFUNC9:def 6

 .= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . ( 0 ,m)) by A0, Lm8, Lm9

 .= (( Partial_Sums_in_cod2 f) . ( 0 ,m)) by DefRSM;

 hence (( ProjMap1 (( Partial_Sums f), 0 )) . m) = (( ProjMap1 (( Partial_Sums_in_cod2 f), 0 )) . m) by MESFUNC9:def 6;

 end;

 now

 let n be Element of NAT ;

 (( ProjMap2 (( Partial_Sums f), 0 )) . n) = (( Partial_Sums f) . (n, 0 )) by MESFUNC9:def 7

 .= (( Partial_Sums_in_cod1 f) . (n, 0 )) by DefCSM;

 hence (( ProjMap2 (( Partial_Sums f), 0 )) . n) = (( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) . n) by MESFUNC9:def 7;

 end;

 hence thesis by A1, FUNCT_2: 63;

 end;

 theorem :: DBLSEQ_3:55

 for C,D be non empty set, F1,F2 be without-infty Function of [:C, D:], ExtREAL , c be Element of C holds ( ProjMap1 ((F1 + F2),c)) = (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c)))

 proof

 let C,D be non empty set;

 let F1,F2 be without-infty Function of [:C, D:], ExtREAL ;

 let c be Element of C;



 A2: ( dom ( ProjMap1 ((F1 + F2),c))) = D & ( dom ( ProjMap1 (F1,c))) = D & ( dom ( ProjMap1 (F2,c))) = D by FUNCT_2:def 1;

 { -infty } misses ( rng ( ProjMap1 (F1,c))) & { -infty } misses ( rng ( ProjMap1 (F2,c))) by ZFMISC_1: 50, MESFUNC5:def 3;

 then (( ProjMap1 (F1,c)) " { -infty }) = {} & (( ProjMap1 (F2,c)) " { -infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap1 ((F1 + F2),c))) = ((( dom ( ProjMap1 (F1,c))) /\ ( dom ( ProjMap1 (F2,c)))) \ (((( ProjMap1 (F1,c)) " { -infty }) /\ (( ProjMap1 (F2,c)) " { +infty })) \/ ((( ProjMap1 (F1,c)) " { +infty }) /\ (( ProjMap1 (F2,c)) " { -infty })))) by A2;

 then

 A5: ( dom ( ProjMap1 ((F1 + F2),c))) = ( dom (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c)))) by MESFUNC1:def 3;

 for d be object st d in ( dom ( ProjMap1 ((F1 + F2),c))) holds (( ProjMap1 ((F1 + F2),c)) . d) = ((( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c))) . d)

 proof

 let d be object;

 assume

 A3: d in ( dom ( ProjMap1 ((F1 + F2),c)));

 then

 A4: (( ProjMap1 ((F1 + F2),c)) . d) = ((F1 + F2) . (c,d)) & (( ProjMap1 (F1,c)) . d) = (F1 . (c,d)) & (( ProjMap1 (F2,c)) . d) = (F2 . (c,d)) by MESFUNC9:def 6;

 reconsider d1 = d as Element of D by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap1 ((F1 + F2),c)) . d) = ((( ProjMap1 (F1,c)) . d) + (( ProjMap1 (F2,c)) . d)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 3;

 end;

 hence ( ProjMap1 ((F1 + F2),c)) = (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c))) by A5, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:56

 for C,D be non empty set, F1,F2 be without-infty Function of [:C, D:], ExtREAL , d be Element of D holds ( ProjMap2 ((F1 + F2),d)) = (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d)))

 proof

 let C,D be non empty set;

 let F1,F2 be without-infty Function of [:C, D:], ExtREAL ;

 let d be Element of D;



 A2: ( dom ( ProjMap2 ((F1 + F2),d))) = C & ( dom ( ProjMap2 (F1,d))) = C & ( dom ( ProjMap2 (F2,d))) = C by FUNCT_2:def 1;

 { -infty } misses ( rng ( ProjMap2 (F1,d))) & { -infty } misses ( rng ( ProjMap2 (F2,d))) by ZFMISC_1: 50, MESFUNC5:def 3;

 then (( ProjMap2 (F1,d)) " { -infty }) = {} & (( ProjMap2 (F2,d)) " { -infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap2 ((F1 + F2),d))) = ((( dom ( ProjMap2 (F1,d))) /\ ( dom ( ProjMap2 (F2,d)))) \ (((( ProjMap2 (F1,d)) " { -infty }) /\ (( ProjMap2 (F2,d)) " { +infty })) \/ ((( ProjMap2 (F1,d)) " { +infty }) /\ (( ProjMap2 (F2,d)) " { -infty })))) by A2;

 then

 A5: ( dom ( ProjMap2 ((F1 + F2),d))) = ( dom (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d)))) by MESFUNC1:def 3;

 for c be object st c in ( dom ( ProjMap2 ((F1 + F2),d))) holds (( ProjMap2 ((F1 + F2),d)) . c) = ((( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d))) . c)

 proof

 let c be object;

 assume

 A3: c in ( dom ( ProjMap2 ((F1 + F2),d)));

 then

 A4: (( ProjMap2 ((F1 + F2),d)) . c) = ((F1 + F2) . (c,d)) & (( ProjMap2 (F1,d)) . c) = (F1 . (c,d)) & (( ProjMap2 (F2,d)) . c) = (F2 . (c,d)) by MESFUNC9:def 7;

 reconsider c1 = c as Element of C by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap2 ((F1 + F2),d)) . c) = ((( ProjMap2 (F1,d)) . c) + (( ProjMap2 (F2,d)) . c)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 3;

 end;

 hence ( ProjMap2 ((F1 + F2),d)) = (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d))) by A5, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:57

 for C,D be non empty set, F1,F2 be without+infty Function of [:C, D:], ExtREAL , c be Element of C holds ( ProjMap1 ((F1 + F2),c)) = (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c)))

 proof

 let C,D be non empty set;

 let F1,F2 be without+infty Function of [:C, D:], ExtREAL ;

 let c be Element of C;



 A2: ( dom ( ProjMap1 ((F1 + F2),c))) = D & ( dom ( ProjMap1 (F1,c))) = D & ( dom ( ProjMap1 (F2,c))) = D by FUNCT_2:def 1;

 { +infty } misses ( rng ( ProjMap1 (F1,c))) & { +infty } misses ( rng ( ProjMap1 (F2,c))) by ZFMISC_1: 50, MESFUNC5:def 4;

 then (( ProjMap1 (F1,c)) " { +infty }) = {} & (( ProjMap1 (F2,c)) " { +infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap1 ((F1 + F2),c))) = ((( dom ( ProjMap1 (F1,c))) /\ ( dom ( ProjMap1 (F2,c)))) \ (((( ProjMap1 (F1,c)) " { -infty }) /\ (( ProjMap1 (F2,c)) " { +infty })) \/ ((( ProjMap1 (F1,c)) " { +infty }) /\ (( ProjMap1 (F2,c)) " { -infty })))) by A2;

 then

 A5: ( dom ( ProjMap1 ((F1 + F2),c))) = ( dom (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c)))) by MESFUNC1:def 3;

 for d be object st d in ( dom ( ProjMap1 ((F1 + F2),c))) holds (( ProjMap1 ((F1 + F2),c)) . d) = ((( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c))) . d)

 proof

 let d be object;

 assume

 A3: d in ( dom ( ProjMap1 ((F1 + F2),c)));

 then

 A4: (( ProjMap1 ((F1 + F2),c)) . d) = ((F1 + F2) . (c,d)) & (( ProjMap1 (F1,c)) . d) = (F1 . (c,d)) & (( ProjMap1 (F2,c)) . d) = (F2 . (c,d)) by MESFUNC9:def 6;

 reconsider d1 = d as Element of D by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap1 ((F1 + F2),c)) . d) = ((( ProjMap1 (F1,c)) . d) + (( ProjMap1 (F2,c)) . d)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 3;

 end;

 hence ( ProjMap1 ((F1 + F2),c)) = (( ProjMap1 (F1,c)) + ( ProjMap1 (F2,c))) by A5, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:58

 for C,D be non empty set, F1,F2 be without+infty Function of [:C, D:], ExtREAL , d be Element of D holds ( ProjMap2 ((F1 + F2),d)) = (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d)))

 proof

 let C,D be non empty set;

 let F1,F2 be without+infty Function of [:C, D:], ExtREAL ;

 let d be Element of D;



 A2: ( dom ( ProjMap2 ((F1 + F2),d))) = C & ( dom ( ProjMap2 (F1,d))) = C & ( dom ( ProjMap2 (F2,d))) = C by FUNCT_2:def 1;

 { +infty } misses ( rng ( ProjMap2 (F1,d))) & { +infty } misses ( rng ( ProjMap2 (F2,d))) by ZFMISC_1: 50, MESFUNC5:def 4;

 then (( ProjMap2 (F1,d)) " { +infty }) = {} & (( ProjMap2 (F2,d)) " { +infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap2 ((F1 + F2),d))) = ((( dom ( ProjMap2 (F1,d))) /\ ( dom ( ProjMap2 (F2,d)))) \ (((( ProjMap2 (F1,d)) " { -infty }) /\ (( ProjMap2 (F2,d)) " { +infty })) \/ ((( ProjMap2 (F1,d)) " { +infty }) /\ (( ProjMap2 (F2,d)) " { -infty })))) by A2;

 then

 A5: ( dom ( ProjMap2 ((F1 + F2),d))) = ( dom (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d)))) by MESFUNC1:def 3;

 for c be object st c in ( dom ( ProjMap2 ((F1 + F2),d))) holds (( ProjMap2 ((F1 + F2),d)) . c) = ((( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d))) . c)

 proof

 let c be object;

 assume

 A3: c in ( dom ( ProjMap2 ((F1 + F2),d)));

 then

 A4: (( ProjMap2 ((F1 + F2),d)) . c) = ((F1 + F2) . (c,d)) & (( ProjMap2 (F1,d)) . c) = (F1 . (c,d)) & (( ProjMap2 (F2,d)) . c) = (F2 . (c,d)) by MESFUNC9:def 7;

 reconsider c1 = c as Element of C by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap2 ((F1 + F2),d)) . c) = ((( ProjMap2 (F1,d)) . c) + (( ProjMap2 (F2,d)) . c)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 3;

 end;

 hence ( ProjMap2 ((F1 + F2),d)) = (( ProjMap2 (F1,d)) + ( ProjMap2 (F2,d))) by A5, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:59

 for C,D be non empty set, F1 be without-infty Function of [:C, D:], ExtREAL , F2 be without+infty Function of [:C, D:], ExtREAL , c be Element of C holds ( ProjMap1 ((F1 - F2),c)) = (( ProjMap1 (F1,c)) - ( ProjMap1 (F2,c))) & ( ProjMap1 ((F2 - F1),c)) = (( ProjMap1 (F2,c)) - ( ProjMap1 (F1,c)))

 proof

 let C,D be non empty set;

 let F1 be without-infty Function of [:C, D:], ExtREAL ;

 let F2 be without+infty Function of [:C, D:], ExtREAL ;

 let c be Element of C;



 A2: ( dom ( ProjMap1 ((F1 - F2),c))) = D & ( dom ( ProjMap1 ((F2 - F1),c))) = D & ( dom ( ProjMap1 (F1,c))) = D & ( dom ( ProjMap1 (F2,c))) = D by FUNCT_2:def 1;

 { -infty } misses ( rng ( ProjMap1 (F1,c))) & { +infty } misses ( rng ( ProjMap1 (F2,c))) by ZFMISC_1: 50, MESFUNC5:def 3, MESFUNC5:def 4;

 then

 B1: (( ProjMap1 (F1,c)) " { -infty }) = {} & (( ProjMap1 (F2,c)) " { +infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap1 ((F1 - F2),c))) = ((( dom ( ProjMap1 (F1,c))) /\ ( dom ( ProjMap1 (F2,c)))) \ (((( ProjMap1 (F1,c)) " { -infty }) /\ (( ProjMap1 (F2,c)) " { -infty })) \/ ((( ProjMap1 (F1,c)) " { +infty }) /\ (( ProjMap1 (F2,c)) " { +infty })))) by A2;

 then

 A5: ( dom ( ProjMap1 ((F1 - F2),c))) = ( dom (( ProjMap1 (F1,c)) - ( ProjMap1 (F2,c)))) by MESFUNC1:def 4;

 ( dom ( ProjMap1 ((F2 - F1),c))) = ((( dom ( ProjMap1 (F2,c))) /\ ( dom ( ProjMap1 (F1,c)))) \ (((( ProjMap1 (F2,c)) " { +infty }) /\ (( ProjMap1 (F1,c)) " { +infty })) \/ ((( ProjMap1 (F2,c)) " { -infty }) /\ (( ProjMap1 (F1,c)) " { -infty })))) by A2, B1;

 then

 A6: ( dom ( ProjMap1 ((F2 - F1),c))) = ( dom (( ProjMap1 (F2,c)) - ( ProjMap1 (F1,c)))) by MESFUNC1:def 4;

 for d be object st d in ( dom ( ProjMap1 ((F1 - F2),c))) holds (( ProjMap1 ((F1 - F2),c)) . d) = ((( ProjMap1 (F1,c)) - ( ProjMap1 (F2,c))) . d)

 proof

 let d be object;

 assume

 A3: d in ( dom ( ProjMap1 ((F1 - F2),c)));

 then

 A4: (( ProjMap1 ((F1 - F2),c)) . d) = ((F1 - F2) . (c,d)) & (( ProjMap1 (F1,c)) . d) = (F1 . (c,d)) & (( ProjMap1 (F2,c)) . d) = (F2 . (c,d)) by MESFUNC9:def 6;

 reconsider d1 = d as Element of D by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap1 ((F1 - F2),c)) . d) = ((( ProjMap1 (F1,c)) . d) - (( ProjMap1 (F2,c)) . d)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 4;

 end;

 hence ( ProjMap1 ((F1 - F2),c)) = (( ProjMap1 (F1,c)) - ( ProjMap1 (F2,c))) by A5, FUNCT_1: 2;

 for d be object st d in ( dom ( ProjMap1 ((F2 - F1),c))) holds (( ProjMap1 ((F2 - F1),c)) . d) = ((( ProjMap1 (F2,c)) - ( ProjMap1 (F1,c))) . d)

 proof

 let d be object;

 assume

 A3: d in ( dom ( ProjMap1 ((F2 - F1),c)));

 then

 A4: (( ProjMap1 ((F2 - F1),c)) . d) = ((F2 - F1) . (c,d)) & (( ProjMap1 (F1,c)) . d) = (F1 . (c,d)) & (( ProjMap1 (F2,c)) . d) = (F2 . (c,d)) by MESFUNC9:def 6;

 reconsider d1 = d as Element of D by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap1 ((F2 - F1),c)) . d) = ((( ProjMap1 (F2,c)) . d) - (( ProjMap1 (F1,c)) . d)) by A4, Th7;

 hence thesis by A3, A6, MESFUNC1:def 4;

 end;

 hence ( ProjMap1 ((F2 - F1),c)) = (( ProjMap1 (F2,c)) - ( ProjMap1 (F1,c))) by A6, FUNCT_1: 2;

 end;

 theorem :: DBLSEQ_3:60

 for C,D be non empty set, F1 be without-infty Function of [:C, D:], ExtREAL , F2 be without+infty Function of [:C, D:], ExtREAL , d be Element of D holds ( ProjMap2 ((F1 - F2),d)) = (( ProjMap2 (F1,d)) - ( ProjMap2 (F2,d))) & ( ProjMap2 ((F2 - F1),d)) = (( ProjMap2 (F2,d)) - ( ProjMap2 (F1,d)))

 proof

 let C,D be non empty set;

 let F1 be without-infty Function of [:C, D:], ExtREAL ;

 let F2 be without+infty Function of [:C, D:], ExtREAL ;

 let d be Element of D;



 A2: ( dom ( ProjMap2 ((F1 - F2),d))) = C & ( dom ( ProjMap2 ((F2 - F1),d))) = C & ( dom ( ProjMap2 (F1,d))) = C & ( dom ( ProjMap2 (F2,d))) = C by FUNCT_2:def 1;

 { -infty } misses ( rng ( ProjMap2 (F1,d))) & { +infty } misses ( rng ( ProjMap2 (F2,d))) by ZFMISC_1: 50, MESFUNC5:def 3, MESFUNC5:def 4;

 then

 B1: (( ProjMap2 (F1,d)) " { -infty }) = {} & (( ProjMap2 (F2,d)) " { +infty }) = {} by RELAT_1: 138;

 then ( dom ( ProjMap2 ((F1 - F2),d))) = ((( dom ( ProjMap2 (F1,d))) /\ ( dom ( ProjMap2 (F2,d)))) \ (((( ProjMap2 (F1,d)) " { -infty }) /\ (( ProjMap2 (F2,d)) " { -infty })) \/ ((( ProjMap2 (F1,d)) " { +infty }) /\ (( ProjMap2 (F2,d)) " { +infty })))) by A2;

 then

 A5: ( dom ( ProjMap2 ((F1 - F2),d))) = ( dom (( ProjMap2 (F1,d)) - ( ProjMap2 (F2,d)))) by MESFUNC1:def 4;

 ( dom ( ProjMap2 ((F2 - F1),d))) = ((( dom ( ProjMap2 (F2,d))) /\ ( dom ( ProjMap2 (F1,d)))) \ (((( ProjMap2 (F2,d)) " { +infty }) /\ (( ProjMap2 (F1,d)) " { +infty })) \/ ((( ProjMap2 (F2,d)) " { -infty }) /\ (( ProjMap2 (F1,d)) " { -infty })))) by A2, B1;

 then

 A6: ( dom ( ProjMap2 ((F2 - F1),d))) = ( dom (( ProjMap2 (F2,d)) - ( ProjMap2 (F1,d)))) by MESFUNC1:def 4;

 for c be object st c in ( dom ( ProjMap2 ((F1 - F2),d))) holds (( ProjMap2 ((F1 - F2),d)) . c) = ((( ProjMap2 (F1,d)) - ( ProjMap2 (F2,d))) . c)

 proof

 let c be object;

 assume

 A3: c in ( dom ( ProjMap2 ((F1 - F2),d)));

 then

 A4: (( ProjMap2 ((F1 - F2),d)) . c) = ((F1 - F2) . (c,d)) & (( ProjMap2 (F1,d)) . c) = (F1 . (c,d)) & (( ProjMap2 (F2,d)) . c) = (F2 . (c,d)) by MESFUNC9:def 7;

 reconsider c1 = c as Element of C by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap2 ((F1 - F2),d)) . c) = ((( ProjMap2 (F1,d)) . c) - (( ProjMap2 (F2,d)) . c)) by A4, Th7;

 hence thesis by A3, A5, MESFUNC1:def 4;

 end;

 hence ( ProjMap2 ((F1 - F2),d)) = (( ProjMap2 (F1,d)) - ( ProjMap2 (F2,d))) by A5, FUNCT_1: 2;

 for c be object st c in ( dom ( ProjMap2 ((F2 - F1),d))) holds (( ProjMap2 ((F2 - F1),d)) . c) = ((( ProjMap2 (F2,d)) - ( ProjMap2 (F1,d))) . c)

 proof

 let c be object;

 assume

 A3: c in ( dom ( ProjMap2 ((F2 - F1),d)));

 then

 A4: (( ProjMap2 ((F2 - F1),d)) . c) = ((F2 - F1) . (c,d)) & (( ProjMap2 (F1,d)) . c) = (F1 . (c,d)) & (( ProjMap2 (F2,d)) . c) = (F2 . (c,d)) by MESFUNC9:def 7;

 reconsider c1 = c as Element of C by A3;

 [c, d] in [:C, D:] by A3, ZFMISC_1:def 2;

 then (( ProjMap2 ((F2 - F1),d)) . c) = ((( ProjMap2 (F2,d)) . c) - (( ProjMap2 (F1,d)) . c)) by A4, Th7;

 hence thesis by A3, A6, MESFUNC1:def 4;

 end;

 hence ( ProjMap2 ((F2 - F1),d)) = (( ProjMap2 (F2,d)) - ( ProjMap2 (F1,d))) by A6, FUNCT_1: 2;

 end;

 begin

 theorem :: DBLSEQ_3:61

 for seq be nonnegative ExtREAL_sequence st not ( Partial_Sums seq) is convergent_to_+infty holds for n be Nat holds (seq . n) is Real

 proof

 let seq be nonnegative ExtREAL_sequence;

 assume

 A2: not ( Partial_Sums seq) is convergent_to_+infty;

 given N be Nat such that

 A3: not (seq . N) is Real;

 not (seq . N) in REAL by A3;

 then

 A4: (seq . N) = +infty or (seq . N) = -infty by XXREAL_0: 14;



 A6: ( Partial_Sums seq) is non-decreasing by MESFUNC9: 16;

 now

 let g be Real;

 assume 0 < g;

 take N;

 hereby

 let m be Nat;

 assume

 A7: N <= m;

 per cases ;

 suppose N = 0 ;

 then (( Partial_Sums seq) . N) = (seq . N) by MESFUNC9:def 1;

 then

 A9: g <= (( Partial_Sums seq) . N) by A4, SUPINF_2: 51, XXREAL_0: 3;

 (( Partial_Sums seq) . N) <= (( Partial_Sums seq) . m) by A7, MESFUNC9: 16, RINFSUP2: 7;

 hence g <= (( Partial_Sums seq) . m) by A9, XXREAL_0: 2;

 end;

 suppose N <> 0 ;

 then

 consider N1 be Nat such that

 A11: N = (N1 + 1) by NAT_1: 6;



 A12: (( Partial_Sums seq) . N1) >= 0 by SUPINF_2: 51;

 (( Partial_Sums seq) . N) = ((( Partial_Sums seq) . N1) + (seq . N)) by A11, MESFUNC9:def 1

 .= +infty by A4, SUPINF_2: 51, XXREAL_0: 4, A12, XXREAL_3: 39;

 then (( Partial_Sums seq) . m) = +infty by A6, A7, RINFSUP2: 7, XXREAL_0: 4;

 hence g <= (( Partial_Sums seq) . m) by XXREAL_0: 3;

 end;

 end;

 end;

 hence contradiction by A2, MESFUNC5:def 9;

 end;

 theorem :: DBLSEQ_3:62



 Th62: for seq be nonnegative ExtREAL_sequence st seq is non-decreasing holds seq is convergent_to_+infty or seq is convergent_to_finite_number

 proof

 let seq be nonnegative ExtREAL_sequence;

 assume

 A1: seq is non-decreasing;

 now

 assume seq is convergent_to_-infty;

 then

 consider N be Nat such that

 A4: for n be Nat st N <= n holds (seq . n) <= ( - 1) by MESFUNC5:def 10;

 (seq . N) <= ( - 1) & (seq . N) >= 0 by SUPINF_2: 51, A4;

 hence contradiction;

 end;

 hence thesis by A1, RINFSUP2: 37, MESFUNC5:def 11;

 end;



 Lm10: for f be Function of [: NAT , NAT :], ExtREAL , n be Element of NAT st f is nonnegative holds ( ProjMap1 (f,n)) is nonnegative & ( ProjMap2 (f,n)) is nonnegative

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , n be Element of NAT ;

 assume

 A1: f is nonnegative;

 now

 let m be object;

 assume m in ( dom ( ProjMap1 (f,n)));

 then

 reconsider m1 = m as Element of NAT ;

 (( ProjMap1 (f,n)) . m1) = (f . (n,m1)) by MESFUNC9:def 6;

 hence (( ProjMap1 (f,n)) . m) >= 0 by A1, SUPINF_2: 51;

 end;

 hence ( ProjMap1 (f,n)) is nonnegative by SUPINF_2: 52;

 now

 let m be object;

 assume m in ( dom ( ProjMap2 (f,n)));

 then

 reconsider m1 = m as Element of NAT ;

 (( ProjMap2 (f,n)) . m1) = (f . (m1,n)) by MESFUNC9:def 7;

 hence (( ProjMap2 (f,n)) . m) >= 0 by A1, SUPINF_2: 51;

 end;

 hence ( ProjMap2 (f,n)) is nonnegative by SUPINF_2: 52;

 end;

 registration

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , n be Element of NAT ;

 cluster ( ProjMap1 (f,n)) -> nonnegative;

 correctness by Lm10;

 cluster ( ProjMap2 (f,n)) -> nonnegative;

 correctness by Lm10;

 end

 theorem :: DBLSEQ_3:63



 Th63: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT holds ( ProjMap1 (( Partial_Sums_in_cod2 f),m)) is non-decreasing

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT ;

 set PS = ( ProjMap1 (( Partial_Sums_in_cod2 f),m));

 for n,j be Nat st j <= n holds (PS . j) <= (PS . n)

 proof

 let n,j be Nat;

 defpred Q[ Nat] means (PS . j) <= (PS . $1);



 A6: for k be Nat holds (PS . k) <= (PS . (k + 1))

 proof

 let k be Nat;

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

 (PS . (k + 1)) = (( Partial_Sums_in_cod2 f) . (m,(k + 1))) by MESFUNC9:def 6

 .= ((( Partial_Sums_in_cod2 f) . (m,k1)) + (f . (m,(k + 1)))) by DefCSM

 .= ((PS . k) + (f . (m,(k + 1)))) by MESFUNC9:def 6;

 hence thesis by SUPINF_2: 51, XXREAL_3: 39;

 end;



 A8: for k be Nat st k >= j & (for l be Nat st l >= j & l < k holds Q[l]) holds Q[k]

 proof

 let k be Nat;

 assume that

 A9: k >= j and

 A10: for l be Nat st l >= j & l < k holds Q[l];

 now

 assume k > j;

 then

 A11: k >= (j + 1) by NAT_1: 13;

 per cases by A11, XXREAL_0: 1;

 suppose k = (j + 1);

 hence thesis by A6;

 end;

 suppose

 A12: k > (j + 1);

 then

 reconsider l = (k - 1) as Element of NAT by NAT_1: 20;

 k < (k + 1) by NAT_1: 13;

 then

 A13: k > l by XREAL_1: 19;

 k = (l + 1);

 then

 A14: (PS . l) <= (PS . k) by A6;

 (PS . j) <= (PS . l) by A10, A13, A12, XREAL_1: 19;

 hence thesis by A14, XXREAL_0: 2;

 end;

 end;

 hence thesis by A9, XXREAL_0: 1;

 end;



 A15: for k be Nat st k >= j holds Q[k] from NAT_1:sch 9( A8);

 assume j <= n;

 hence thesis by A15;

 end;

 hence thesis by RINFSUP2: 7;

 end;

 theorem :: DBLSEQ_3:64



 Th64: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , n be Element of NAT holds ( ProjMap2 (( Partial_Sums_in_cod1 f),n)) is non-decreasing

 proof

 let f be nonnegative without-infty Function of [: NAT , NAT :], ExtREAL , n be Element of NAT ;



 A1: ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),n)) is non-decreasing by Th63;

 ( Partial_Sums_in_cod2 ( ~ f)) = ( ~ ( Partial_Sums_in_cod1 f)) by Th40;

 hence ( ProjMap2 (( Partial_Sums_in_cod1 f),n)) is non-decreasing by A1, Th33;

 end;

 registration

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT ;

 cluster ( ProjMap1 (( Partial_Sums_in_cod2 f),m)) -> non-decreasing;

 correctness by Th63;

 cluster ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) -> non-decreasing;

 correctness by Th64;

 end

 theorem :: DBLSEQ_3:65



 Th65: for f be nonnegative Function of [: NAT , NAT :], ExtREAL holds (f is convergent_in_cod1 implies ( lim_in_cod1 f) is nonnegative) & (f is convergent_in_cod2 implies ( lim_in_cod2 f) is nonnegative)

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 hereby

 assume

 A2: f is convergent_in_cod1;

 now

 let n be object;

 assume n in ( dom ( lim_in_cod1 f));

 then

 reconsider n1 = n as Element of NAT ;



 A4: (( lim_in_cod1 f) . n) = ( lim ( ProjMap2 (f,n1))) by D1DEF5;

 for k be Nat holds 0 <= (( ProjMap2 (f,n1)) . k) by SUPINF_2: 51;

 hence (( lim_in_cod1 f) . n) >= 0 by A2, A4, MESFUNC9: 10;

 end;

 hence ( lim_in_cod1 f) is nonnegative by SUPINF_2: 52;

 end;

 assume

 A2: f is convergent_in_cod2;

 now

 let n be object;

 assume n in ( dom ( lim_in_cod2 f));

 then

 reconsider n1 = n as Element of NAT ;



 A4: (( lim_in_cod2 f) . n) = ( lim ( ProjMap1 (f,n1))) by D1DEF6;

 for k be Nat holds 0 <= (( ProjMap1 (f,n1)) . k) by SUPINF_2: 51;

 hence (( lim_in_cod2 f) . n) >= 0 by A2, A4, MESFUNC9: 10;

 end;

 hence thesis by SUPINF_2: 52;

 end;

 theorem :: DBLSEQ_3:66

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums_in_cod1 f) is convergent_in_cod1 & ( Partial_Sums_in_cod2 f) is convergent_in_cod2 by RINFSUP2: 37;

 theorem :: DBLSEQ_3:67

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT st not ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_+infty holds for n be Nat holds (f . (n,m)) is Real

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 let m be Element of NAT ;

 assume

 A2: not ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_+infty;

 given N be Nat such that

 A3: not (f . (N,m)) is Real;

 not (f . (N,m)) in REAL by A3;

 then

 A4: (f . (N,m)) = +infty or (f . (N,m)) = -infty by XXREAL_0: 14;

 reconsider N1 = N as Element of NAT by ORDINAL1:def 12;

 now

 let g be Real;

 assume 0 < g;

 take N;

 hereby

 let k be Nat;

 assume

 A7: N <= k;

 per cases ;

 suppose

 A8: N = 0 ;

 (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . N) = (( Partial_Sums_in_cod1 f) . (N1,m)) by MESFUNC9:def 7

 .= (f . (N,m)) by A8, DefRSM;

 then

 A9: g <= (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . N) by A4, SUPINF_2: 51, XXREAL_0: 3;

 (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . N) <= (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . k) by A7, RINFSUP2: 7;

 hence g <= (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . k) by A9, XXREAL_0: 2;

 end;

 suppose N <> 0 ;

 then

 consider N2 be Nat such that

 A11: N = (N2 + 1) by NAT_1: 6;

 reconsider N3 = N2 as Element of NAT by ORDINAL1:def 12;



 A12: (( Partial_Sums_in_cod1 f) . (N3,m)) >= 0 by SUPINF_2: 51;

 (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . N1) = (( Partial_Sums_in_cod1 f) . (N1,m)) by MESFUNC9:def 7

 .= ((( Partial_Sums_in_cod1 f) . (N2,m)) + (f . (N1,m))) by A11, DefRSM;

 then (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . N1) = +infty by A4, SUPINF_2: 51, XXREAL_0: 4, A12, XXREAL_3: 39;

 then (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . k) = +infty by XXREAL_0: 4, A7, RINFSUP2: 7;

 hence g <= (( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . k) by XXREAL_0: 3;

 end;

 end;

 end;

 hence contradiction by A2, MESFUNC5:def 9;

 end;

 theorem :: DBLSEQ_3:68

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT st not ( ProjMap1 (( Partial_Sums_in_cod2 f),m)) is convergent_to_+infty holds for n be Nat holds (f . (m,n)) is Real

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 let m be Element of NAT ;

 assume

 A2: not ( ProjMap1 (( Partial_Sums_in_cod2 f),m)) is convergent_to_+infty;

 given N be Nat such that

 A3: not (f . (m,N)) is Real;

 not (f . (m,N)) in REAL by A3;

 then

 A4: (f . (m,N)) = +infty or (f . (m,N)) = -infty by XXREAL_0: 14;

 reconsider N1 = N as Element of NAT by ORDINAL1:def 12;

 now

 let g be Real;

 assume 0 < g;

 take N;

 hereby

 let k be Nat;

 assume

 A7: N <= k;

 per cases ;

 suppose

 A8: N = 0 ;

 (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . N) = (( Partial_Sums_in_cod2 f) . (m,N1)) by MESFUNC9:def 6

 .= (f . (m,N)) by A8, DefCSM;

 then

 A9: g <= (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . N) by A4, SUPINF_2: 51, XXREAL_0: 3;

 (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . N) <= (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . k) by A7, RINFSUP2: 7;

 hence g <= (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . k) by A9, XXREAL_0: 2;

 end;

 suppose N <> 0 ;

 then

 consider N2 be Nat such that

 A11: N = (N2 + 1) by NAT_1: 6;

 reconsider N3 = N2 as Element of NAT by ORDINAL1:def 12;



 A12: (( Partial_Sums_in_cod2 f) . (m,N3)) >= 0 by SUPINF_2: 51;

 (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . N1) = (( Partial_Sums_in_cod2 f) . (m,N1)) by MESFUNC9:def 6

 .= ((( Partial_Sums_in_cod2 f) . (m,N2)) + (f . (m,N1))) by A11, DefCSM;

 then (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . N1) = +infty by A4, SUPINF_2: 51, XXREAL_0: 4, A12, XXREAL_3: 39;

 then (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . k) = +infty by XXREAL_0: 4, A7, RINFSUP2: 7;

 hence g <= (( ProjMap1 (( Partial_Sums_in_cod2 f),m)) . k) by XXREAL_0: 3;

 end;

 end;

 end;

 hence contradiction by A2, MESFUNC5:def 9;

 end;

 theorem :: DBLSEQ_3:69

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat st (for i be Nat st i <= n holds (f . (i,m)) is Real) holds (( Partial_Sums_in_cod1 f) . (n,m)) < +infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat;

 assume

 A2: for i be Nat st i <= n holds (f . (i,m)) is Real;

 defpred P[ Nat] means $1 <= n implies (( Partial_Sums_in_cod1 f) . ($1,m)) < +infty ;

 (( Partial_Sums_in_cod1 f) . ( 0 ,m)) = (f . ( 0 ,m)) by DefRSM;

 then (( Partial_Sums_in_cod1 f) . ( 0 ,m)) is Real by A2;

 then

 A4: P[ 0 ] by XXREAL_0: 9, XREAL_0:def 1;



 A5: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A6: P[k];

 now

 assume

 A7: (k + 1) <= n;

 then

 A8: (f . ((k + 1),m)) is Real & (f . ((k + 1),m)) >= 0 by A2, SUPINF_2: 51;

 (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM;

 hence (( Partial_Sums_in_cod1 f) . ((k + 1),m)) < +infty by A6, A7, A8, NAT_1: 13, XXREAL_3: 16, XXREAL_0: 4;

 end;

 hence P[(k + 1)];

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:70

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat st (for i be Nat st i <= m holds (f . (n,i)) is Real) holds (( Partial_Sums_in_cod2 f) . (n,m)) < +infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat;

 assume

 A2: for i be Nat st i <= m holds (f . (n,i)) is Real;

 defpred P[ Nat] means $1 <= m implies (( Partial_Sums_in_cod2 f) . (n,$1)) < +infty ;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then (( Partial_Sums_in_cod2 f) . (n, 0 )) is Real by A2;

 then

 A4: P[ 0 ] by XXREAL_0: 9, XREAL_0:def 1;



 A5: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A6: P[k];

 now

 assume

 A7: (k + 1) <= m;

 then

 A8: (f . (n,(k + 1))) is Real & (f . (n,(k + 1))) >= 0 by A2, SUPINF_2: 51;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence (( Partial_Sums_in_cod2 f) . (n,(k + 1))) < +infty by A6, A7, A8, NAT_1: 13, XXREAL_3: 16, XXREAL_0: 4;

 end;

 hence P[(k + 1)];

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:71

 for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums f) is convergent_in_cod1_to_-infty implies ex m be Element of NAT st ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_-infty

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: ( Partial_Sums f) is convergent_in_cod1_to_-infty;



 A3: ( ProjMap2 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)), 0 )) = ( ProjMap2 (( Partial_Sums f), 0 ))

 .= ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) by Th54;

 assume for m be Element of NAT holds not ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_-infty;

 hence contradiction by A1, A3;

 end;

 theorem :: DBLSEQ_3:72



 Th72: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat holds (for k be Element of NAT st k <= m holds not ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty) iff (for k be Element of NAT st k <= m holds ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) < +infty )

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat;

 hereby

 assume

 A1: for k be Element of NAT st k <= m holds not ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty;

 hereby

 let k be Element of NAT ;

 assume k <= m;

 then not ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty by A1;

 then ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_finite_number by Th62;

 then ex g be Real st ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) = g & for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((( ProjMap1 (( Partial_Sums_in_cod2 f),k)) . m) - ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k)))).| < p by MESFUNC9: 7;

 hence ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) < +infty by XREAL_0:def 1, XXREAL_0: 9;

 end;

 end;

 assume

 A2: for k be Element of NAT st k <= m holds ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) < +infty ;

 now

 let k be Element of NAT ;

 assume k <= m;

 then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) < +infty by A2;

 hence not ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty by MESFUNC9: 7;

 end;

 hence thesis;

 end;

 theorem :: DBLSEQ_3:73

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat holds (for k be Element of NAT st k <= m holds not ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty) iff (for k be Element of NAT st k <= m holds ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) < +infty )

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat;

 hereby

 assume

 A1: for k be Element of NAT st k <= m holds not ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty;

 hereby

 let k be Element of NAT ;

 assume k <= m;

 then not ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty by A1;

 then ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_finite_number by Th62;

 then ex g be Real st ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) = g & for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . m) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k)))).| < p by MESFUNC9: 7;

 hence ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) < +infty by XREAL_0:def 1, XXREAL_0: 9;

 end;

 end;

 assume

 A2: for k be Element of NAT st k <= m holds ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) < +infty ;

 now

 let k be Element of NAT ;

 assume k <= m;

 then ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) < +infty by A2;

 hence not ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty by MESFUNC9: 7;

 end;

 hence thesis;

 end;

 theorem :: DBLSEQ_3:74



 Th74: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat holds (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = +infty iff ex k be Element of NAT st k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat;

 hereby

 assume

 A2: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = +infty ;

 now

 assume

 A3: for k be Element of NAT holds k > m or not ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty;

 defpred P[ Nat] means $1 <= m implies (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . $1) <> +infty ;

 (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . 0 ) = (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . 0 ) by MESFUNC9:def 1

 .= ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f), 0 ))) by D1DEF6;

 then

 A5: P[ 0 ] by A3, Th72;



 A6: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A7: P[k];

 assume

 A8: (k + 1) <= m;



 A10: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . (k + 1)) = ((( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . k) + (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . (k + 1))) by MESFUNC9:def 1;

 now

 assume (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . (k + 1)) = +infty ;

 then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),(k + 1)))) = +infty by D1DEF6;

 hence contradiction by A3, A8, Th72;

 end;

 hence (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . (k + 1)) <> +infty by A7, A8, NAT_1: 13, A10, XXREAL_3: 16;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A5, A6);

 hence contradiction by A2;

 end;

 hence ex k be Element of NAT st k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty;

 end;

 given k be Element of NAT such that

 B1: k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty;

 ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),k))) = +infty by B1, MESFUNC9: 7;

 then

 B2: (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . k) = +infty by D1DEF6;



 B3: ( Partial_Sums_in_cod2 f) is convergent_in_cod2 by RINFSUP2: 37;

 then ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) is nonnegative by Th65;

 then

 B5: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . k) >= +infty by B2, Th4;

 ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) is nonnegative by B3, Th65;

 then (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . k) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) by B1, RINFSUP2: 7, MESFUNC9: 16;

 hence (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = +infty by B5, XXREAL_0: 2, XXREAL_0: 4;

 end;

 theorem :: DBLSEQ_3:75

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat holds (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . m) = +infty iff ex k be Element of NAT st k <= m & ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Nat;

 hereby

 assume

 A1: (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . m) = +infty ;

 ( lim_in_cod1 ( Partial_Sums_in_cod1 f)) = ( lim_in_cod2 ( ~ ( Partial_Sums_in_cod1 f))) by Th38

 .= ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f))) by Th40;

 then

 consider k be Element of NAT such that

 A2: k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),k)) is convergent_to_+infty by A1, Th74;

 ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),k)) = ( ProjMap2 (( ~ ( Partial_Sums_in_cod2 ( ~ f))),k)) by Th32

 .= ( ProjMap2 (( Partial_Sums_in_cod1 ( ~ ( ~ f))),k)) by Th40

 .= ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) by DBLSEQ_2: 7;

 hence ex k be Element of NAT st k <= m & ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty by A2;

 end;

 given k be Element of NAT such that

 A3: k <= m & ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty;



 A4: ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) = ( ProjMap2 (( Partial_Sums_in_cod1 ( ~ ( ~ f))),k)) by DBLSEQ_2: 7

 .= ( ProjMap2 (( ~ ( Partial_Sums_in_cod2 ( ~ f))),k)) by Th40

 .= ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),k)) by Th32;

 ( lim_in_cod1 ( Partial_Sums_in_cod1 f)) = ( lim_in_cod2 ( ~ ( Partial_Sums_in_cod1 f))) by Th38

 .= ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f))) by Th40;

 hence (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . m) = +infty by A3, A4, Th74;

 end;

 theorem :: DBLSEQ_3:76



 Th76: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat holds (( Partial_Sums_in_cod1 f) . (n,m)) >= (f . (n,m)) & (( Partial_Sums_in_cod2 f) . (n,m)) >= (f . (n,m))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , n,m be Nat;

 defpred P[ Nat] means $1 <= n implies (( Partial_Sums_in_cod1 f) . ($1,m)) >= (f . ($1,m));



 A2: P[ 0 ] by DefRSM;



 A5: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat such that P[k];

 assume (k + 1) <= n;

 (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM;

 hence (( Partial_Sums_in_cod1 f) . ((k + 1),m)) >= (f . ((k + 1),m)) by SUPINF_2: 51, XXREAL_3: 39;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A2, A5);

 hence (( Partial_Sums_in_cod1 f) . (n,m)) >= (f . (n,m));

 defpred Q[ Nat] means $1 <= m implies (( Partial_Sums_in_cod2 f) . (n,$1)) >= (f . (n,$1));



 A2: Q[ 0 ] by DefCSM;



 A5: for k be Nat st Q[k] holds Q[(k + 1)]

 proof

 let k be Nat such that Q[k];

 assume (k + 1) <= m;

 (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 hence (( Partial_Sums_in_cod2 f) . (n,(k + 1))) >= (f . (n,(k + 1))) by SUPINF_2: 51, XXREAL_3: 39;

 end;

 for k be Nat holds Q[k] from NAT_1:sch 2( A2, A5);

 hence (( Partial_Sums_in_cod2 f) . (n,m)) >= (f . (n,m));

 end;

 theorem :: DBLSEQ_3:77



 Th77: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT st (ex k be Element of NAT st k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty) holds ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)) is convergent_to_+infty & ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m))) = +infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT ;

 given k be Element of NAT such that

 A2: k <= m & ( ProjMap1 (( Partial_Sums_in_cod2 f),k)) is convergent_to_+infty;



 A5: for g be Real st 0 < g holds ex N be Nat st for n be Nat st N <= n holds g <= (( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)) . n)

 proof

 let g be Real;

 assume 0 < g;

 then

 consider N be Nat such that

 A4: for n be Nat st N <= n holds g <= (( ProjMap1 (( Partial_Sums_in_cod2 f),k)) . n) by A2, MESFUNC5:def 9;

 now

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 assume N <= n;

 then g <= (( ProjMap1 (( Partial_Sums_in_cod2 f),k)) . n) by A4;

 then

 A7: g <= (( Partial_Sums_in_cod2 f) . (k,n1)) by MESFUNC9:def 6;

 (( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),n1)) . k) <= (( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),n1)) . m) by A2, RINFSUP2: 7;

 then (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (k,n1)) <= (( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),n1)) . m) by MESFUNC9:def 7;

 then

 A10: (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (k,n1)) <= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,n1)) by MESFUNC9:def 7;

 (( Partial_Sums_in_cod2 f) . (k,n1)) <= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (k,n1)) by Th76;

 then

 A9: (( Partial_Sums_in_cod2 f) . (k,n1)) <= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,n)) by A10, XXREAL_0: 2;

 (( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)) . n1) = (( Partial_Sums f) . (m,n)) by MESFUNC9:def 6

 .= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (m,n)) by Lm8;

 hence g <= (( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)) . n) by A7, A9, XXREAL_0: 2;

 end;

 hence thesis;

 end;

 hence ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)) is convergent_to_+infty by MESFUNC5:def 9;

 thus ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m))) = +infty by A5, MESFUNC5:def 9, MESFUNC9: 7;

 end;

 theorem :: DBLSEQ_3:78

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT st (ex k be Element of NAT st k <= m & ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty) holds ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),m)) is convergent_to_+infty & ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),m))) = +infty

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , m be Element of NAT ;

 given k be Element of NAT such that

 A1: k <= m & ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_+infty;



 A3: ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) = ( ProjMap2 (( Partial_Sums_in_cod1 ( ~ ( ~ f))),k)) by DBLSEQ_2: 7

 .= ( ProjMap2 (( ~ ( Partial_Sums_in_cod2 ( ~ f))),k)) by Th40

 .= ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),k)) by Th32;

 ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 ( ~ f))),m)) = ( ProjMap2 (( ~ ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 ( ~ f)))),m)) by Th32

 .= ( ProjMap2 (( Partial_Sums_in_cod1 ( ~ ( Partial_Sums_in_cod1 ( ~ f)))),m)) by Th40

 .= ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( ~ ( ~ f)))),m)) by Th40

 .= ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),m)) by DBLSEQ_2: 7;

 hence thesis by A3, A1, Th77;

 end;

 theorem :: DBLSEQ_3:79



 Th79: for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums f) is convergent_in_cod1_to_finite iff ( Partial_Sums_in_cod1 f) is convergent_in_cod1_to_finite

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 hereby

 assume

 A1: ( Partial_Sums f) is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 defpred P[ Nat] means for k be Element of NAT st k = $1 holds ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_finite_number;

 now

 let k be Element of NAT ;

 assume k = 0 ;

 then ( ProjMap2 (( Partial_Sums f),k)) = ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) by Th54;

 hence ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_finite_number by A1;

 end;

 then

 A3: P[ 0 ];



 A4: for m1 be Nat st P[m1] holds P[(m1 + 1)]

 proof

 let m1 be Nat;

 reconsider m = m1 as Element of NAT by ORDINAL1:def 12;

 assume

 X1: P[m1];

 hereby

 let k be Element of NAT ;

 assume

 B2: k = (m1 + 1);

 then

 reconsider k1 = (k - 1) as Element of NAT by NAT_1: 11, NAT_1: 21;



 F1: ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_finite_number by X1;

 not ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_+infty & not ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_-infty by X1, MESFUNC5: 50, MESFUNC5: 51;

 then

 consider G0 be Real such that

 F3: ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),m))) = G0 & (for e be Real st 0 < e holds ex N be Nat st for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),m)))).| < e) & ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_finite_number by F1, MESFUNC5:def 12;

 consider G1 be Real such that

 E3: ( lim ( ProjMap2 (( Partial_Sums f),m))) = G1 & (for e be Real st 0 < e holds ex N be Nat st for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))).| < e) by A1, MESFUNC9: 7;

 consider G2 be Real such that

 E4: ( lim ( ProjMap2 (( Partial_Sums f),k))) = G2 & (for e be Real st 0 < e holds ex N be Nat st for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))).| < e) by A1, MESFUNC9: 7;



 E7: ( - G1) = ( - ( lim ( ProjMap2 (( Partial_Sums f),m)))) & ( - G2) = ( - ( lim ( ProjMap2 (( Partial_Sums f),k)))) by E3, E4, XXREAL_3:def 3;

 then

 E5: (G2 + ( - G1)) = (( lim ( ProjMap2 (( Partial_Sums f),k))) + ( - ( lim ( ProjMap2 (( Partial_Sums f),m))))) by E4, XXREAL_3:def 2;

 now

 let e be Real;

 assume

 B6: 0 < e;

 then

 consider N1 be Nat such that

 B7: for n be Nat st n >= N1 holds |.((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))).| < (e / 2) by E3;

 consider N2 be Nat such that

 B8: for n be Nat st n >= N2 holds |.((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))).| < (e / 2) by B6, E4;

 consider N0 be Nat such that

 B10: for n be Nat st n >= N0 holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),m)))).| < e by B6, F3;

 reconsider N = ( max (( max (N1,N2)),N0)) as Nat by TARSKI: 1;

 take N;

 hereby

 let n be Nat;

 assume

 B9: n >= N;

 N >= ( max (N1,N2)) & N >= N0 & ( max (N1,N2)) >= N1 & ( max (N1,N2)) >= N2 by XXREAL_0: 25;

 then N >= N1 & N >= N2 & N >= N0 by XXREAL_0: 2;

 then

 K0: n >= N1 & n >= N2 & n >= N0 by B9, XXREAL_0: 2;

 then |.((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))).| < (e / 2) & |.((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))).| < (e / 2) & |.((( ProjMap2 (( Partial_Sums_in_cod1 f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),m)))).| < e by B7, B8, B10;

 then

 B12: ( |.((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))).| + |.((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))).|) < ((e / 2) + (e / 2)) by XXREAL_3: 64;

 K2:

 now

 assume (( ProjMap2 (( Partial_Sums f),m)) . n) = +infty ;

 then ((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))) = +infty by E3, XXREAL_3: 13;

 hence contradiction by K0, B7, XXREAL_0: 3, EXTREAL1: 30;

 end;

 KK2:

 now

 assume (( ProjMap2 (( Partial_Sums f),m)) . n) = -infty ;

 then ((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))) = -infty by E3, XXREAL_3: 14;

 hence contradiction by K0, B7, XXREAL_0: 3, EXTREAL1: 30;

 end;

 K5:

 now

 assume (( ProjMap2 (( Partial_Sums f),k)) . n) = +infty ;

 then ((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))) = +infty by E4, XXREAL_3: 13;

 hence contradiction by K0, B8, XXREAL_0: 3, EXTREAL1: 30;

 end;

 KK5:

 now

 assume (( ProjMap2 (( Partial_Sums f),k)) . n) = -infty ;

 then ((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))) = -infty by E4, XXREAL_3: 14;

 hence contradiction by K0, B8, XXREAL_0: 3, EXTREAL1: 30;

 end;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 XX2: (( ProjMap2 (( Partial_Sums f),k)) . n) = (( Partial_Sums f) . (n1,k)) by MESFUNC9:def 7

 .= ((( Partial_Sums f) . (n1,m)) + (( Partial_Sums_in_cod1 f) . (n1,k))) by B2, DefCSM

 .= ((( ProjMap2 (( Partial_Sums f),m)) . n) + (( Partial_Sums_in_cod1 f) . (n1,k))) by MESFUNC9:def 7

 .= ((( ProjMap2 (( Partial_Sums f),m)) . n) + (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n)) by MESFUNC9:def 7;

 (( ProjMap2 (( Partial_Sums f),k)) . n) in REAL by K5, KK5, XXREAL_0: 14;

 then

 reconsider r4 = (( ProjMap2 (( Partial_Sums f),k)) . n) as Real;

 (( ProjMap2 (( Partial_Sums f),m)) . n) in REAL by K2, KK2, XXREAL_0: 14;

 then

 reconsider r5 = (( ProjMap2 (( Partial_Sums f),m)) . n) as Real;

 r4 = (r5 + (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n)) by XX2;

 then (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) <> +infty & (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) <> -infty ;

 then (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) in REAL by XXREAL_0: 14;

 then

 reconsider r1 = (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) as Real;



 T1: ((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) - (( lim ( ProjMap2 (( Partial_Sums f),k))) - ( lim ( ProjMap2 (( Partial_Sums f),m))))) = ((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) - (G2 - G1)) by E5

 .= (r1 + ( - (G2 - G1))) by XXREAL_3:def 2;



 T2: (r5 + r1) = ((( ProjMap2 (( Partial_Sums f),m)) . n) + (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n)) by XXREAL_3:def 2;



 E8: ((( ProjMap2 (( Partial_Sums f),k)) . n) + ( - ( lim ( ProjMap2 (( Partial_Sums f),k))))) = (r4 + ( - G2)) & ((( ProjMap2 (( Partial_Sums f),m)) . n) + ( - ( lim ( ProjMap2 (( Partial_Sums f),m))))) = (r5 + ( - G1)) by E7, XXREAL_3:def 2;

 then ( - ((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m))))) = ( - (r5 - G1)) by XXREAL_3:def 3;

 then (((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))) + ( - ((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))))) = ((r4 - G2) + ( - (r5 - G1))) by E8, XXREAL_3:def 2;

 then

 T3: (((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))) - ((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m))))) = ((r4 - G2) - (r5 - G1));

 |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) - (( lim ( ProjMap2 (( Partial_Sums f),k))) - ( lim ( ProjMap2 (( Partial_Sums f),m))))).| <= ( |.((( ProjMap2 (( Partial_Sums f),k)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),k)))).| + |.((( ProjMap2 (( Partial_Sums f),m)) . n) - ( lim ( ProjMap2 (( Partial_Sums f),m)))).|) by T1, T2, XX2, T3, EXTREAL1: 32;

 hence |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . n) - (G2 - G1)).| < e by B12, E5, XXREAL_0: 2;

 end;

 end;

 hence ( ProjMap2 (( Partial_Sums_in_cod1 f),k)) is convergent_to_finite_number by MESFUNC5:def 8;

 end;

 end;

 for m1 be Nat holds P[m1] from NAT_1:sch 2( A3, A4);

 hence ( ProjMap2 (( Partial_Sums_in_cod1 f),m)) is convergent_to_finite_number;

 end;

 hence ( Partial_Sums_in_cod1 f) is convergent_in_cod1_to_finite;

 end;

 assume

 C0: ( Partial_Sums_in_cod1 f) is convergent_in_cod1_to_finite;

 now

 let m be Element of NAT ;

 defpred P[ Nat] means for k be Element of NAT st k = $1 holds ( ProjMap2 (( Partial_Sums f),k)) is convergent_to_finite_number;

 ( ProjMap2 (( Partial_Sums f), 0 )) = ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) by Th54;

 then

 C1: P[ 0 ] by C0;



 C2: for m be Nat st P[m] holds P[(m + 1)]

 proof

 let m be Nat;

 assume

 C3: P[m];

 reconsider m1 = m as Element of NAT by ORDINAL1:def 12;

 hereby

 let k be Element of NAT ;

 assume

 C4: k = (m + 1);

 then

 reconsider k1 = (k - 1) as Element of NAT by NAT_1: 11, NAT_1: 21;

 reconsider f1 = ( ProjMap2 (( Partial_Sums f),m1)), f2 = ( ProjMap2 (( Partial_Sums_in_cod1 f),(m1 + 1))) as without-infty ExtREAL_sequence;

 for n be Element of NAT holds (( ProjMap2 (( Partial_Sums f),k)) . n) = ((( ProjMap2 (( Partial_Sums f),m1)) + ( ProjMap2 (( Partial_Sums_in_cod1 f),(m1 + 1)))) . n)

 proof

 let n be Element of NAT ;

 (( ProjMap2 (( Partial_Sums f),k)) . n) = (( Partial_Sums f) . (n,(m1 + 1))) by C4, MESFUNC9:def 7

 .= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,(m1 + 1))) by Lm8

 .= ((( Partial_Sums_in_cod1 f) . (n,(m1 + 1))) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n,m1))) by Th47

 .= ((( Partial_Sums_in_cod1 f) . (n,(m1 + 1))) + (( Partial_Sums f) . (n,m1))) by Lm8

 .= ((( ProjMap2 (( Partial_Sums_in_cod1 f),(m1 + 1))) . n) + (( Partial_Sums f) . (n,m1))) by MESFUNC9:def 7

 .= ((( ProjMap2 (( Partial_Sums_in_cod1 f),(m1 + 1))) . n) + (( ProjMap2 (( Partial_Sums f),m1)) . n)) by MESFUNC9:def 7;

 hence thesis by Th7;

 end;

 then

 C5: ( ProjMap2 (( Partial_Sums f),k)) = (f1 + f2) by FUNCT_2:def 8;

 ( ProjMap2 (( Partial_Sums f),m1)) is convergent_to_finite_number & ( ProjMap2 (( Partial_Sums_in_cod1 f),(m1 + 1))) is convergent_to_finite_number by C3, C0;

 hence ( ProjMap2 (( Partial_Sums f),k)) is convergent_to_finite_number by C5, Th23;

 end;

 end;

 for m be Nat holds P[m] from NAT_1:sch 2( C1, C2);

 hence ( ProjMap2 (( Partial_Sums f),m)) is convergent_to_finite_number;

 end;

 hence thesis;

 end;

 theorem :: DBLSEQ_3:80



 Th80: for f be nonnegative Function of [: NAT , NAT :], ExtREAL st ( Partial_Sums f) is convergent_in_cod1_to_finite holds for m be Element of NAT holds (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . m) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),m)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: ( Partial_Sums f) is convergent_in_cod1_to_finite;

 then

 A2: ( Partial_Sums_in_cod1 f) is convergent_in_cod1_to_finite by Th79;

 let m be Element of NAT ;

 defpred P[ Nat] means for k be Element of NAT st k <= $1 holds (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . k) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),k)));

 now

 let k be Element of NAT ;

 assume k <= 0 ;

 then

 A5: k = 0 ;



 A6: (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . 0 ) = (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . 0 ) by MESFUNC9:def 1

 .= ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 ))) by D1DEF5;

 consider G be Real such that

 A7: ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 ))) = G & for p be Real st 0 < p holds ex N be Nat st for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) . n) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )))).| < p by A2, MESFUNC9: 7;

 reconsider G1 = G as R_eal by XXREAL_0:def 1;

 ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)), 0 )) = ( ProjMap2 (( Partial_Sums f), 0 )) by Lm8;

 then

 A8: ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)), 0 )) is convergent_to_finite_number by A1;

 for p be Real st 0 < p holds ex N be Nat st for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)), 0 )) . n) - G1).| < p

 proof

 let p be Real;

 assume 0 < p;

 then

 consider N be Nat such that

 A11: for n be Nat st N <= n holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) . n) - G1).| < p by A7;

 now

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 assume

 A12: N <= n;

 (( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)), 0 )) . n1) = (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (n, 0 )) by MESFUNC9:def 7

 .= (( Partial_Sums f) . (n, 0 )) by Lm8

 .= (( ProjMap2 (( Partial_Sums f), 0 )) . n1) by MESFUNC9:def 7

 .= (( ProjMap2 (( Partial_Sums_in_cod1 f), 0 )) . n1) by Th54;

 hence |.((( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)), 0 )) . n) - G1).| < p by A11, A12;

 end;

 hence thesis;

 end;

 hence (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . k) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),k))) by A5, A6, A7, A8, MESFUNC5:def 12;

 end;

 then

 A13: P[ 0 ];



 A14: for n be Nat st P[n] holds P[(n + 1)]

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 assume

 A15: P[n];

 now

 let k be Element of NAT ;

 assume

 A16: k <= (n + 1);

 per cases ;

 suppose k < (n + 1);

 then k <= n by NAT_1: 13;

 hence (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . k) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),k))) by A15;

 end;

 suppose k >= (n + 1);

 then

 A17: k = (n + 1) by A16, XXREAL_0: 1;



 then

 A18: (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . k) = ((( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . n) + (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . (n + 1))) by MESFUNC9:def 1

 .= (( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),n1))) + (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . (n + 1))) by A15

 .= (( lim ( ProjMap2 (( Partial_Sums f),n1))) + (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . (n + 1))) by Lm8

 .= (( lim ( ProjMap2 (( Partial_Sums f),n1))) + ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k)))) by A17, D1DEF5;

 consider Gn be Real such that

 A19: ( lim ( ProjMap2 (( Partial_Sums f),n1))) = Gn & for p be Real st 0 < p holds ex I be Nat st for i be Nat st I <= i holds |.((( ProjMap2 (( Partial_Sums f),n1)) . i) - ( lim ( ProjMap2 (( Partial_Sums f),n1)))).| < p by A1, MESFUNC9: 7;

 consider Gn1 be Real such that

 A20: ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k))) = Gn1 & for p be Real st 0 < p holds ex I be Nat st for i be Nat st I <= i holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k)))).| < p by A2, MESFUNC9: 7;

 reconsider Gna = Gn, Gn1a = Gn1 as R_eal by XXREAL_0:def 1;

 set G = (Gna + Gn1a);



 A21: ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),k))) = (( lim_in_cod1 ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f))) . k) by D1DEF5

 .= (( lim_in_cod1 ( Partial_Sums f)) . k) by Lm8

 .= ( lim ( ProjMap2 (( Partial_Sums f),k))) by D1DEF5;



 A22: ( ProjMap2 (( Partial_Sums f),k)) is convergent_to_finite_number by A1;

 for p be Real st 0 < p holds ex I be Nat st for i be Nat st I <= i holds |.((( ProjMap2 (( Partial_Sums f),k)) . i) - G).| < p

 proof

 let p be Real;

 assume

 A24: 0 < p;

 then

 consider I1 be Nat such that

 A25: for i be Nat st I1 <= i holds |.((( ProjMap2 (( Partial_Sums f),n1)) . i) - ( lim ( ProjMap2 (( Partial_Sums f),n1)))).| < (p / 2) by A19;

 consider I2 be Nat such that

 A26: for i be Nat st I2 <= i holds |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),k)))).| < (p / 2) by A20, A24;

 reconsider I = ( max (I1,I2)) as Nat by XXREAL_0: 16;



 A27: I >= I1 & I >= I2 by XXREAL_0: 25;

 now

 let i be Nat;

 reconsider i1 = i as Element of NAT by ORDINAL1:def 12;

 assume I <= i;

 then I1 <= i & I2 <= i by A27, XXREAL_0: 2;

 then |.((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gn).| < (p / 2) & |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1).| < (p / 2) by A19, A20, A25, A26;

 then

 A28: ( |.((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gn).| + |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1).|) < ((p / 2) + (p / 2)) by XXREAL_3: 64;



 A29: (( ProjMap2 (( Partial_Sums f),k)) . i1) = (( Partial_Sums f) . (i,k)) by MESFUNC9:def 7

 .= (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i,k)) by Lm8

 .= ((( Partial_Sums_in_cod1 f) . (i,k)) + (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) . (i,n))) by A17, Th47

 .= ((( Partial_Sums f) . (i,n)) + (( Partial_Sums_in_cod1 f) . (i,k))) by Lm8

 .= ((( ProjMap2 (( Partial_Sums f),n1)) . i1) + (( Partial_Sums_in_cod1 f) . (i,k))) by MESFUNC9:def 7

 .= ((( ProjMap2 (( Partial_Sums f),n1)) . i1) + (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i1)) by MESFUNC9:def 7;

 (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i1) <> -infty by SUPINF_2: 51;

 then

 A30: ((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i1) - Gn1a) <> -infty by XXREAL_3: 19;

 then

 A31: (((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i1) - Gn1a) + Gn1a) <> -infty by XXREAL_3: 17;

 (( ProjMap2 (( Partial_Sums f),n1)) . i1) >= 0 by SUPINF_2: 51;

 then

 A32: ((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) <> -infty by XXREAL_3: 19;

 (( ProjMap2 (( Partial_Sums f),n1)) . i) = (((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) + Gna) & (( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) = (((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a) + Gn1a) by XXREAL_3: 22;



 then (( ProjMap2 (( Partial_Sums f),k)) . i) = (((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) + ((((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a) + Gn1a) + Gna)) by A29, A31, A32, XXREAL_3: 29

 .= (((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) + (((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a) + (Gn1a + Gna))) by XXREAL_3: 29

 .= ((((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) + ((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a)) + (Gna + Gn1a)) by A30, A32, XXREAL_3: 29;

 then ((( ProjMap2 (( Partial_Sums f),k)) . i) - G) = (((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna) + ((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a)) by XXREAL_3: 22;

 then |.((( ProjMap2 (( Partial_Sums f),k)) . i) - G).| <= ( |.((( ProjMap2 (( Partial_Sums f),n1)) . i) - Gna).| + |.((( ProjMap2 (( Partial_Sums_in_cod1 f),k)) . i) - Gn1a).|) by EXTREAL1: 24;

 hence |.((( ProjMap2 (( Partial_Sums f),k)) . i) - G).| < p by A28, XXREAL_0: 2;

 end;

 hence thesis;

 end;

 hence (( Partial_Sums ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) . k) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)),k))) by A18, A19, A20, A21, A22, MESFUNC5:def 12;

 end;

 end;

 hence P[(n + 1)];

 end;

 for n be Nat holds P[n] from NAT_1:sch 2( A13, A14);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:81

 for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( Partial_Sums f) is convergent_in_cod2_to_finite iff ( Partial_Sums_in_cod2 f) is convergent_in_cod2_to_finite

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;

 hereby

 assume ( Partial_Sums f) is convergent_in_cod2_to_finite;

 then ( ~ ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f))) is convergent_in_cod1_to_finite by Th36;

 then ( Partial_Sums_in_cod1 ( ~ ( Partial_Sums_in_cod1 f))) is convergent_in_cod1_to_finite by Th40;

 then ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( ~ f))) is convergent_in_cod1_to_finite by Th40;

 then ( Partial_Sums ( ~ f)) is convergent_in_cod1_to_finite by Lm8;

 then ( Partial_Sums_in_cod1 ( ~ f)) is convergent_in_cod1_to_finite by Th79;

 then ( ~ ( Partial_Sums_in_cod2 f)) is convergent_in_cod1_to_finite by Th40;

 hence ( Partial_Sums_in_cod2 f) is convergent_in_cod2_to_finite by Th36;

 end;

 assume ( Partial_Sums_in_cod2 f) is convergent_in_cod2_to_finite;

 then ( ~ ( Partial_Sums_in_cod2 f)) is convergent_in_cod1_to_finite by Th36;

 then ( Partial_Sums_in_cod1 ( ~ f)) is convergent_in_cod1_to_finite by Th40;

 then ( Partial_Sums ( ~ f)) is convergent_in_cod1_to_finite by Th79;

 then ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( ~ f))) is convergent_in_cod1_to_finite by Lm8;

 then ( Partial_Sums_in_cod1 ( ~ ( Partial_Sums_in_cod1 f))) is convergent_in_cod1_to_finite by Th40;

 then ( ~ ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f))) is convergent_in_cod1_to_finite by Th40;

 hence ( Partial_Sums f) is convergent_in_cod2_to_finite by Th36;

 end;

 theorem :: DBLSEQ_3:82

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL st ( Partial_Sums f) is convergent_in_cod2_to_finite holds for m be Element of NAT holds (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 assume ( Partial_Sums f) is convergent_in_cod2_to_finite;

 then ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f)) is convergent_in_cod2_to_finite by Lm8;

 then ( ~ ( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 f))) is convergent_in_cod1_to_finite by Th36;

 then ( Partial_Sums_in_cod2 ( ~ ( Partial_Sums_in_cod2 f))) is convergent_in_cod1_to_finite by Th40;

 then

 A1: ( Partial_Sums ( ~ f)) is convergent_in_cod1_to_finite by Th40;

 hereby

 let m be Element of NAT ;

 ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) = ( lim_in_cod1 ( ~ ( Partial_Sums_in_cod2 f))) by Th38

 .= ( lim_in_cod1 ( Partial_Sums_in_cod1 ( ~ f))) by Th40;



 then (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( Partial_Sums_in_cod2 ( ~ f))),m))) by A1, Th80

 .= ( lim ( ProjMap2 (( Partial_Sums_in_cod1 ( ~ ( Partial_Sums_in_cod1 f))),m))) by Th40

 .= ( lim ( ProjMap2 (( ~ ( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f))),m))) by Th40;

 hence (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . m) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 f)),m))) by Th32;

 end;

 end;

 theorem :: DBLSEQ_3:83



 Th83: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence st (for m be Element of NAT holds (seq . m) = ( lim_inf ( ProjMap2 (f,m)))) holds ( Sum seq) <= ( lim_inf ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence;

 assume

 A1: for m be Element of NAT holds (seq . m) = ( lim_inf ( ProjMap2 (f,m)));



 A2: for m be Element of NAT holds for N,n be Element of NAT st n >= N holds (( inferior_realsequence ( ProjMap2 (f,m))) . N) <= (f . (n,m))

 proof

 let m be Element of NAT ;

 let N,n be Element of NAT ;

 assume n >= N;

 then

 A4: (( inferior_realsequence ( ProjMap2 (f,m))) . N) <= (( inferior_realsequence ( ProjMap2 (f,m))) . n) by RINFSUP2: 7;

 (( inferior_realsequence ( ProjMap2 (f,m))) . n) <= (( ProjMap2 (f,m)) . n) by RINFSUP2: 8;

 then (( inferior_realsequence ( ProjMap2 (f,m))) . n) <= (f . (n,m)) by MESFUNC9:def 7;

 hence (( inferior_realsequence ( ProjMap2 (f,m))) . N) <= (f . (n,m)) by A4, XXREAL_0: 2;

 end;

 deffunc F( Element of NAT ) = ( inferior_realsequence ( ProjMap2 (f,$1)));

 deffunc G( Element of NAT , Element of NAT ) = (( inferior_realsequence ( ProjMap2 (f,$2))) . $1);

 consider g be Function of [: NAT , NAT :], ExtREAL such that

 A5: for n be Element of NAT holds for m be Element of NAT holds (g . (n,m)) = G(n,m) from BINOP_1:sch 4;

 now

 let z be object;

 per cases ;

 suppose z in ( dom g);

 then

 consider n,m be object such that

 D1: n in NAT & m in NAT & z = [n, m] by ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by D1;

 (g . (n,m)) = (( inferior_realsequence ( ProjMap2 (f,m))) . n) by A5;

 then

 consider Y be non empty Subset of ExtREAL such that

 D2: Y = { (( ProjMap2 (f,m)) . k) where k be Nat : n <= k } & (g . z) = ( inf Y) by D1, RINFSUP2:def 6;

 for x be ExtReal st x in Y holds 0 <= x

 proof

 let x be ExtReal;

 assume x in Y;

 then ex k be Nat st x = (( ProjMap2 (f,m)) . k) & n <= k by D2;

 hence 0 <= x by SUPINF_2: 51;

 end;

 then 0 is LowerBound of Y by XXREAL_2:def 2;

 hence 0 <= (g . z) by D2, XXREAL_2:def 4;

 end;

 suppose not z in ( dom g);

 hence 0 <= (g . z) by FUNCT_1:def 2;

 end;

 end;

 then g is nonnegative by SUPINF_2: 51;

 then

 reconsider g as nonnegative without-infty Function of [: NAT , NAT :], ExtREAL ;



 A6: for m be Element of NAT holds for N,n be Element of NAT st n >= N holds (( Partial_Sums_in_cod2 g) . (N,m)) <= (( Partial_Sums_in_cod2 f) . (n,m))

 proof

 let m be Element of NAT ;

 let N,n be Element of NAT ;

 assume

 A7: n >= N;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 g) . (N,$1)) <= (( Partial_Sums_in_cod2 f) . (n,$1));



 A8: (( Partial_Sums_in_cod2 g) . (N, 0 )) = (g . (N, 0 )) by DefCSM

 .= (( inferior_realsequence ( ProjMap2 (f, 0 ))) . N) by A5;

 (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 then

 A9: P[ 0 ] by A2, A7, A8;



 A10: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

 assume

 A11: P[k];

 (g . (N,(k1 + 1))) = (( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) . N) by A5;

 then

 A12: (g . (N,(k1 + 1))) <= (f . (n,(k1 + 1))) by A2, A7;

 (( Partial_Sums_in_cod2 g) . (N,(k + 1))) = ((( Partial_Sums_in_cod2 g) . (N,k)) + (g . (N,(k1 + 1)))) & (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k1 + 1)))) by DefCSM;

 hence P[(k + 1)] by A11, A12, XXREAL_3: 36;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A9, A10);

 hence thesis;

 end;



 A13: for m be Element of NAT holds for N,n be Element of NAT st n >= N holds (( Partial_Sums_in_cod2 g) . (N,m)) <= (( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . n)

 proof

 let m be Element of NAT ;

 let N,n be Element of NAT ;

 assume

 A14: n >= N;

 consider Y be non empty Subset of ExtREAL such that

 A15: Y = { (( ProjMap2 (( Partial_Sums_in_cod2 f),m)) . k) where k be Nat : n <= k } & (( inferior_realsequence ( ProjMap2 (( Partial_Sums_in_cod2 f),m))) . n) = ( inf Y) by RINFSUP2:def 6;

 for x be ExtReal st x in Y holds (( Partial_Sums_in_cod2 g) . (N,m)) <= x

 proof

 let x be ExtReal;

 assume x in Y;

 then

 consider k be Nat such that

 A17: x = (( ProjMap2 (( Partial_Sums_in_cod2 f),m)) . k) & n <= k by A15;

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;



 A18: N <= k1 by A14, A17, XXREAL_0: 2;

 x = (( Partial_Sums_in_cod2 f) . (k1,m)) by A17, MESFUNC9:def 7;

 hence (( Partial_Sums_in_cod2 g) . (N,m)) <= x by A6, A18;

 end;

 then (( Partial_Sums_in_cod2 g) . (N,m)) is LowerBound of Y by XXREAL_2:def 2;

 then

 A19: (( Partial_Sums_in_cod2 g) . (N,m)) <= (( inferior_realsequence ( ProjMap2 (( Partial_Sums_in_cod2 f),m))) . n) by A15, XXREAL_2:def 4;

 consider Z be non empty Subset of ExtREAL such that

 A20: Z = { (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . k) where k be Nat : n <= k } & (( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . n) = ( inf Z) by RINFSUP2:def 6;

 for z be ExtReal st z in Z holds ex y be ExtReal st y in Y & y <= z

 proof

 let z be ExtReal;

 assume z in Z;

 then

 consider j be Nat such that

 A21: z = (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . j) & n <= j by A20;

 reconsider j1 = j as Element of NAT by ORDINAL1:def 12;

 z = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),j1))) by A21, D1DEF6;

 then

 A23: z = ( sup ( ProjMap1 (( Partial_Sums_in_cod2 f),j1))) by RINFSUP2: 37;

 set y = (( ProjMap2 (( Partial_Sums_in_cod2 f),m)) . j1);

 take y;

 y = (( Partial_Sums_in_cod2 f) . (j1,m)) by MESFUNC9:def 7

 .= (( ProjMap1 (( Partial_Sums_in_cod2 f),j1)) . m) by MESFUNC9:def 6;

 hence y in Y & y <= z by A15, A21, A23, RINFSUP2: 23;

 end;

 then ( inf Y) <= ( inf Z) by XXREAL_2: 64;

 hence thesis by A15, A20, A19, XXREAL_0: 2;

 end;

 defpred Q[ Nat] means for m be Element of NAT st m = $1 holds (( Partial_Sums seq) . m) = ( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),m)));

 now

 let m be Element of NAT ;

 assume

 A24: m = 0 ;



 then (( Partial_Sums seq) . m) = (seq . 0 ) by MESFUNC9:def 1

 .= ( lim_inf ( ProjMap2 (f, 0 ))) by A1

 .= ( sup ( inferior_realsequence ( ProjMap2 (f, 0 )))) by RINFSUP2:def 9;

 then

 A26: (( Partial_Sums seq) . m) = ( lim ( inferior_realsequence ( ProjMap2 (f, 0 )))) by RINFSUP2: 37;

 now

 let n be Element of NAT ;

 (( ProjMap2 (( Partial_Sums_in_cod2 g), 0 )) . n) = (( Partial_Sums_in_cod2 g) . (n, 0 )) by MESFUNC9:def 7

 .= (g . (n, 0 )) by DefCSM

 .= (( inferior_realsequence ( ProjMap2 (f, 0 ))) . n) by A5;

 hence (( inferior_realsequence ( ProjMap2 (f, 0 ))) . n) = (( ProjMap2 (( Partial_Sums_in_cod2 g), 0 )) . n);

 end;

 hence (( Partial_Sums seq) . m) = ( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),m))) by A24, A26, FUNCT_2: 63;

 end;

 then

 A28: Q[ 0 ];



 P1: for m be Element of NAT holds ( ProjMap2 (( Partial_Sums_in_cod2 g),m)) is convergent

 proof

 let m be Element of NAT ;

 for j,i be Nat st i <= j holds (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . i) <= (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . j)

 proof

 let j,i be Nat;

 reconsider i1 = i, j1 = j as Element of NAT by ORDINAL1:def 12;

 assume

 B2: i <= j;



 B3: (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . i1) = (( Partial_Sums_in_cod2 g) . (i,m)) & (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . j1) = (( Partial_Sums_in_cod2 g) . (j,m)) by MESFUNC9:def 7;

 defpred R[ Nat] means (( Partial_Sums_in_cod2 g) . (i,$1)) <= (( Partial_Sums_in_cod2 g) . (j,$1));



 B4: (( Partial_Sums_in_cod2 g) . (i, 0 )) = (g . (i, 0 )) by DefCSM

 .= (( inferior_realsequence ( ProjMap2 (f, 0 ))) . i1) by A5;

 (( Partial_Sums_in_cod2 g) . (j, 0 )) = (g . (j, 0 )) by DefCSM

 .= (( inferior_realsequence ( ProjMap2 (f, 0 ))) . j1) by A5;

 then

 B5: R[ 0 ] by B2, B4, RINFSUP2: 7;



 B6: for l be Nat st R[l] holds R[(l + 1)]

 proof

 let l be Nat;

 reconsider l1 = l as Element of NAT by ORDINAL1:def 12;

 assume

 B7: R[l];

 (g . (i,(l + 1))) = (( inferior_realsequence ( ProjMap2 (f,(l1 + 1)))) . i1) & (g . (j,(l + 1))) = (( inferior_realsequence ( ProjMap2 (f,(l1 + 1)))) . j1) by A5;

 then

 B8: (g . (i,(l + 1))) <= (g . (j,(l + 1))) by B2, RINFSUP2: 7;

 (( Partial_Sums_in_cod2 g) . (i,(l + 1))) = ((( Partial_Sums_in_cod2 g) . (i,l)) + (g . (i,(l + 1)))) & (( Partial_Sums_in_cod2 g) . (j,(l + 1))) = ((( Partial_Sums_in_cod2 g) . (j,l)) + (g . (j,(l + 1)))) by DefCSM;

 hence R[(l + 1)] by B7, B8, XXREAL_3: 36;

 end;

 for l be Nat holds R[l] from NAT_1:sch 2( B5, B6);

 hence thesis by B3;

 end;

 then ( ProjMap2 (( Partial_Sums_in_cod2 g),m)) is non-decreasing by RINFSUP2: 7;

 hence ( ProjMap2 (( Partial_Sums_in_cod2 g),m)) is convergent by RINFSUP2: 37;

 end;



 A29: for k be Nat st Q[k] holds Q[(k + 1)]

 proof

 let k be Nat;

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

 assume

 A30: Q[k];

 now

 let m be Element of NAT ;

 assume

 B00: m = (k + 1);



 then

 B0: (( Partial_Sums seq) . m) = ((( Partial_Sums seq) . k) + (seq . (k + 1))) by MESFUNC9:def 1

 .= (( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),k1))) + (seq . (k + 1))) by A30

 .= (( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),k1))) + ( lim_inf ( ProjMap2 (f,(k + 1))))) by A1;



 B1: ( lim_inf ( ProjMap2 (f,(k + 1)))) = ( sup ( inferior_realsequence ( ProjMap2 (f,(k1 + 1))))) by RINFSUP2:def 9

 .= ( lim ( inferior_realsequence ( ProjMap2 (f,(k1 + 1))))) by RINFSUP2: 37;



 B9: ( ProjMap2 (( Partial_Sums_in_cod2 g),k1)) is convergent by P1;



 B10: ( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) is convergent by RINFSUP2: 37;

 for n be object st n in ( dom ( inferior_realsequence ( ProjMap2 (f,(k1 + 1))))) holds 0. <= (( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) . n)

 proof

 let n be object;

 assume n in ( dom ( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))));

 then (g . (n,(k1 + 1))) = (( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) . n) by A5;

 hence thesis by SUPINF_2: 51;

 end;

 then

 C2: ( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) is nonnegative by SUPINF_2: 52;

 for i be Nat holds (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . i) = ((( ProjMap2 (( Partial_Sums_in_cod2 g),k1)) . i) + (( inferior_realsequence ( ProjMap2 (f,(k1 + 1)))) . i))

 proof

 let i be Nat;

 reconsider i1 = i as Element of NAT by ORDINAL1:def 12;

 (( ProjMap2 (( Partial_Sums_in_cod2 g),m)) . i1) = (( Partial_Sums_in_cod2 g) . (i,m)) by MESFUNC9:def 7

 .= ((( Partial_Sums_in_cod2 g) . (i,k)) + (g . (i,(k + 1)))) by B00, DefCSM

 .= ((( ProjMap2 (( Partial_Sums_in_cod2 g),k1)) . i1) + (g . (i,(k + 1)))) by MESFUNC9:def 7;

 hence thesis by A5;

 end;

 hence (( Partial_Sums seq) . m) = ( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),m))) by B0, B1, B9, B10, C2, MESFUNC9: 11;

 end;

 hence Q[(k + 1)];

 end;



 A30: for k be Nat holds Q[k] from NAT_1:sch 2( A28, A29);



 A31: for m be Nat holds (( Partial_Sums seq) . m) <= ( lim_inf ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))

 proof

 let m be Nat;

 reconsider m1 = m as Element of NAT by ORDINAL1:def 12;



 A32: for n be Nat holds (( ProjMap2 (( Partial_Sums_in_cod2 g),m1)) . n) <= (( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . n)

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;

 (( ProjMap2 (( Partial_Sums_in_cod2 g),m1)) . n1) = (( Partial_Sums_in_cod2 g) . (n,m)) by MESFUNC9:def 7;

 hence (( ProjMap2 (( Partial_Sums_in_cod2 g),m1)) . n) <= (( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) . n) by A13;

 end;



 A33: ( ProjMap2 (( Partial_Sums_in_cod2 g),m1)) is convergent by P1;

 ( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) is convergent by RINFSUP2: 37;

 then ( lim ( ProjMap2 (( Partial_Sums_in_cod2 g),m1))) <= ( lim ( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))) by A32, A33, RINFSUP2: 38;

 then (( Partial_Sums seq) . m) <= ( lim ( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))) by A30;

 then (( Partial_Sums seq) . m) <= ( sup ( inferior_realsequence ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))) by RINFSUP2: 37;

 hence thesis by RINFSUP2:def 9;

 end;

 for m be object st m in ( dom seq) holds 0 <= (seq . m)

 proof

 let m be object;

 assume m in ( dom seq);

 then

 reconsider m1 = m as Element of NAT ;



 E1: (seq . m) = ( lim_inf ( ProjMap2 (f,m1))) by A1

 .= ( sup ( inferior_realsequence ( ProjMap2 (f,m1)))) by RINFSUP2:def 9;

 for n be object st n in ( dom ( inferior_realsequence ( ProjMap2 (f,m1)))) holds 0. <= (( inferior_realsequence ( ProjMap2 (f,m1))) . n)

 proof

 let n be object;

 assume n in ( dom ( inferior_realsequence ( ProjMap2 (f,m1))));

 then (g . (n,m1)) = (( inferior_realsequence ( ProjMap2 (f,m1))) . n) by A5;

 hence thesis by SUPINF_2: 51;

 end;

 then (( inferior_realsequence ( ProjMap2 (f,m1))) . 0 ) >= 0 by SUPINF_2: 51, SUPINF_2: 52;

 hence thesis by E1, RINFSUP2: 23;

 end;

 then seq is nonnegative by SUPINF_2: 52;

 then ( Partial_Sums seq) is non-decreasing by MESFUNC9: 16;

 then ( lim ( Partial_Sums seq)) <= ( lim_inf ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) by A31, MESFUNC9: 9, RINFSUP2: 37;

 hence thesis by MESFUNC9:def 3;

 end;

 theorem :: DBLSEQ_3:84

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence st (for m be Element of NAT holds (seq . m) = ( lim_inf ( ProjMap1 (f,m)))) holds ( Sum seq) <= ( lim_inf ( lim_in_cod1 ( Partial_Sums_in_cod1 f)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence;

 assume

 A1: for m be Element of NAT holds (seq . m) = ( lim_inf ( ProjMap1 (f,m)));

 now

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) = ( ProjMap2 (( ~ f),m)) by Th32;

 hence (seq . m) = ( lim_inf ( ProjMap2 (( ~ f),m))) by A1;

 end;

 then

 A2: ( Sum seq) <= ( lim_inf ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f)))) by Th83;

 ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f))) = ( lim_in_cod1 ( ~ ( Partial_Sums_in_cod2 ( ~ f)))) by Th38

 .= ( lim_in_cod1 ( Partial_Sums_in_cod1 ( ~ ( ~ f)))) by Th40;

 hence ( Sum seq) <= ( lim_inf ( lim_in_cod1 ( Partial_Sums_in_cod1 f))) by A2, DBLSEQ_2: 7;

 end;

 theorem :: DBLSEQ_3:85



 Th85: for f be Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence, n,m be Nat holds ((for i,j be Nat holds (f . (i,j)) <= (seq . i)) implies (( Partial_Sums_in_cod1 f) . (n,m)) <= (( Partial_Sums seq) . n)) & ((for i,j be Nat holds (f . (i,j)) <= (seq . j)) implies (( Partial_Sums_in_cod2 f) . (n,m)) <= (( Partial_Sums seq) . m))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence, n,m be Nat;

 hereby

 assume

 A1: for i,j be Nat holds (f . (i,j)) <= (seq . i);

 defpred P[ Nat] means (( Partial_Sums_in_cod1 f) . ($1,m)) <= (( Partial_Sums seq) . $1);



 A2: (( Partial_Sums_in_cod1 f) . ( 0 ,m)) = (f . ( 0 ,m)) by DefRSM;

 (( Partial_Sums seq) . 0 ) = (seq . 0 ) by MESFUNC9:def 1;

 then

 A3: P[ 0 ] by A1, A2;



 A4: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A5: P[k];



 A6: (f . ((k + 1),m)) <= (seq . (k + 1)) by A1;



 A7: (( Partial_Sums_in_cod1 f) . ((k + 1),m)) = ((( Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m))) by DefRSM;

 (( Partial_Sums seq) . (k + 1)) = ((( Partial_Sums seq) . k) + (seq . (k + 1))) by MESFUNC9:def 1;

 hence thesis by A5, A6, A7, XXREAL_3: 36;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A3, A4);

 hence (( Partial_Sums_in_cod1 f) . (n,m)) <= (( Partial_Sums seq) . n);

 end;

 assume

 A1: for i,j be Nat holds (f . (i,j)) <= (seq . j);

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (n,$1)) <= (( Partial_Sums seq) . $1);



 A2: (( Partial_Sums_in_cod2 f) . (n, 0 )) = (f . (n, 0 )) by DefCSM;

 (( Partial_Sums seq) . 0 ) = (seq . 0 ) by MESFUNC9:def 1;

 then

 A3: P[ 0 ] by A1, A2;



 A4: for k be Nat st P[k] holds P[(k + 1)]

 proof

 let k be Nat;

 assume

 A5: P[k];



 A6: (f . (n,(k + 1))) <= (seq . (k + 1)) by A1;



 A7: (( Partial_Sums_in_cod2 f) . (n,(k + 1))) = ((( Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1)))) by DefCSM;

 (( Partial_Sums seq) . (k + 1)) = ((( Partial_Sums seq) . k) + (seq . (k + 1))) by MESFUNC9:def 1;

 hence thesis by A5, A6, A7, XXREAL_3: 36;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A3, A4);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:86



 Th86: for seq be ExtREAL_sequence, r be R_eal st (for n be Nat holds (seq . n) <= r) holds ( lim_sup seq) <= r

 proof

 let seq be ExtREAL_sequence, r be R_eal;

 assume

 A1: for n be Nat holds (seq . n) <= r;

 deffunc F( Element of NAT ) = r;

 consider f be Function of NAT , ExtREAL such that

 A2: for n be Element of NAT holds (f . n) = F(n) from FUNCT_2:sch 4;



 A4: for n be Nat holds (f . n) = r

 proof

 let n be Nat;

 n is Element of NAT by ORDINAL1:def 12;

 hence (f . n) = r by A2;

 end;

 then

 A5: f is convergent & ( lim f) = r by MESFUNC5: 60;

 for n be Nat holds (seq . n) <= (f . n)

 proof

 let n be Nat;

 (f . n) = r by A4;

 hence (seq . n) <= (f . n) by A1;

 end;

 then ( lim_sup seq) <= ( lim_sup f) by MESFUN10: 3;

 hence ( lim_sup seq) <= r by A5, RINFSUP2: 41;

 end;

 theorem :: DBLSEQ_3:87



 Th87: for seq be ExtREAL_sequence, r be R_eal st (for n be Nat holds r <= (seq . n)) holds r <= ( lim_inf seq)

 proof

 let seq be ExtREAL_sequence, r be R_eal;

 assume

 A1: for n be Nat holds r <= (seq . n);

 deffunc F( Element of NAT ) = r;

 consider f be Function of NAT , ExtREAL such that

 A2: for n be Element of NAT holds (f . n) = F(n) from FUNCT_2:sch 4;



 A4: for n be Nat holds (f . n) = r

 proof

 let n be Nat;

 n is Element of NAT by ORDINAL1:def 12;

 hence (f . n) = r by A2;

 end;

 then

 A5: f is convergent & ( lim f) = r by MESFUNC5: 60;

 for n be Nat holds (f . n) <= (seq . n)

 proof

 let n be Nat;

 (f . n) = r by A4;

 hence (f . n) <= (seq . n) by A1;

 end;

 then ( lim_inf f) <= ( lim_inf seq) by MESFUN10: 3;

 hence r <= ( lim_inf seq) by A5, RINFSUP2: 41;

 end;

 theorem :: DBLSEQ_3:88

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL holds (for i1,i2,j be Nat st i1 <= i2 holds (( Partial_Sums_in_cod1 f) . (i1,j)) <= (( Partial_Sums_in_cod1 f) . (i2,j))) & (for i,j1,j2 be Nat st j1 <= j2 holds (( Partial_Sums_in_cod2 f) . (i,j1)) <= (( Partial_Sums_in_cod2 f) . (i,j2)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL ;

 A2:

 now

 let i1,i2,j be natural number;

 assume i1 <= i2;

 then

 consider k be Nat such that

 A3: i2 = (i1 + k) by NAT_1: 10;

 defpred P[ Nat] means $1 <= k implies (( Partial_Sums_in_cod1 f) . (i1,j)) <= (( Partial_Sums_in_cod1 f) . ((i1 + $1),j));



 A4: P[ 0 ];



 A5: for l be Nat st P[l] holds P[(l + 1)]

 proof

 let l be Nat;

 assume

 A6: P[l];

 now

 assume

 A7: (l + 1) <= k;

 (( Partial_Sums_in_cod1 f) . (((i1 + l) + 1),j)) = ((( Partial_Sums_in_cod1 f) . ((i1 + l),j)) + (f . (((i1 + l) + 1),j))) by DefRSM;

 then (( Partial_Sums_in_cod1 f) . ((i1 + l),j)) <= (( Partial_Sums_in_cod1 f) . (((i1 + l) + 1),j)) by SUPINF_2: 51, XXREAL_3: 39;

 hence (( Partial_Sums_in_cod1 f) . (i1,j)) <= (( Partial_Sums_in_cod1 f) . ((i1 + (l + 1)),j)) by A6, A7, NAT_1: 13, XXREAL_0: 2;

 end;

 hence P[(l + 1)];

 end;

 for l be Nat holds P[l] from NAT_1:sch 2( A4, A5);

 hence (( Partial_Sums_in_cod1 f) . (i1,j)) <= (( Partial_Sums_in_cod1 f) . (i2,j)) by A3;

 end;

 now

 let i,j1,j2 be natural number;

 assume j1 <= j2;

 then

 consider k be Nat such that

 B3: j2 = (j1 + k) by NAT_1: 10;

 defpred P[ Nat] means $1 <= k implies (( Partial_Sums_in_cod2 f) . (i,j1)) <= (( Partial_Sums_in_cod2 f) . (i,(j1 + $1)));



 B4: P[ 0 ];



 B5: for l be Nat st P[l] holds P[(l + 1)]

 proof

 let l be Nat;

 assume

 B6: P[l];

 now

 assume

 B7: (l + 1) <= k;

 (( Partial_Sums_in_cod2 f) . (i,((j1 + l) + 1))) = ((( Partial_Sums_in_cod2 f) . (i,(j1 + l))) + (f . (i,((j1 + l) + 1)))) by DefCSM;

 then (( Partial_Sums_in_cod2 f) . (i,(j1 + l))) <= (( Partial_Sums_in_cod2 f) . (i,((j1 + l) + 1))) by SUPINF_2: 51, XXREAL_3: 39;

 hence (( Partial_Sums_in_cod2 f) . (i,j1)) <= (( Partial_Sums_in_cod2 f) . (i,(j1 + (l + 1)))) by B6, B7, NAT_1: 13, XXREAL_0: 2;

 end;

 hence P[(l + 1)];

 end;

 for l be Nat holds P[l] from NAT_1:sch 2( B4, B5);

 hence (( Partial_Sums_in_cod2 f) . (i,j1)) <= (( Partial_Sums_in_cod2 f) . (i,j2)) by B3;

 end;

 hence thesis by A2;

 end;

 theorem :: DBLSEQ_3:89



 Th89: for f be Function of [: NAT , NAT :], ExtREAL , i,j,k be Nat st (for m be Element of NAT holds ( ProjMap2 (f,m)) is non-decreasing) & i <= j holds (( Partial_Sums_in_cod2 f) . (i,k)) <= (( Partial_Sums_in_cod2 f) . (j,k))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , i,j,k be Nat;

 assume that

 A1: for m be Element of NAT holds ( ProjMap2 (f,m)) is non-decreasing and

 A2: i <= j;

 reconsider i1 = i, j1 = j as Element of NAT by ORDINAL1:def 12;

 defpred P[ Nat] means (( Partial_Sums_in_cod2 f) . (i,$1)) <= (( Partial_Sums_in_cod2 f) . (j,$1));

 (( ProjMap2 (f, 0 )) . i1) = (f . (i, 0 )) & (( ProjMap2 (f, 0 )) . j1) = (f . (j, 0 )) by MESFUNC9:def 7;

 then (( ProjMap2 (f, 0 )) . i1) = (( Partial_Sums_in_cod2 f) . (i, 0 )) & (( ProjMap2 (f, 0 )) . j1) = (( Partial_Sums_in_cod2 f) . (j, 0 )) by DefCSM;

 then

 A4: P[ 0 ] by A1, A2, RINFSUP2: 7;



 A5: for n be Nat st P[n] holds P[(n + 1)]

 proof

 let n be Nat;

 assume

 A6: P[n];



 A7: (( Partial_Sums_in_cod2 f) . (i,(n + 1))) = ((( Partial_Sums_in_cod2 f) . (i,n)) + (f . (i,(n + 1)))) & (( Partial_Sums_in_cod2 f) . (j,(n + 1))) = ((( Partial_Sums_in_cod2 f) . (j,n)) + (f . (j,(n + 1)))) by DefCSM;



 A8: (( ProjMap2 (f,(n + 1))) . i) <= (( ProjMap2 (f,(n + 1))) . j) by A1, A2, RINFSUP2: 7;

 (( ProjMap2 (f,(n + 1))) . i1) = (f . (i,(n + 1))) & (( ProjMap2 (f,(n + 1))) . j1) = (f . (j,(n + 1))) by MESFUNC9:def 7;

 hence P[(n + 1)] by A6, A7, A8, XXREAL_3: 36;

 end;

 for n be Nat holds P[n] from NAT_1:sch 2( A4, A5);

 hence thesis;

 end;

 theorem :: DBLSEQ_3:90

 for f be Function of [: NAT , NAT :], ExtREAL , i,j,k be Nat st (for n be Element of NAT holds ( ProjMap1 (f,n)) is non-decreasing) & i <= j holds (( Partial_Sums_in_cod1 f) . (k,i)) <= (( Partial_Sums_in_cod1 f) . (k,j))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , i,j,k be Nat;

 assume that

 A1: for n be Element of NAT holds ( ProjMap1 (f,n)) is non-decreasing and

 A2: i <= j;

 for n be Element of NAT holds ( ProjMap2 (( ~ f),n)) is non-decreasing

 proof

 let n be Element of NAT ;

 ( ProjMap1 (f,n)) = ( ProjMap2 (( ~ f),n)) by Th32;

 hence thesis by A1;

 end;

 then (( Partial_Sums_in_cod2 ( ~ f)) . (i,k)) <= (( Partial_Sums_in_cod2 ( ~ f)) . (j,k)) by A2, Th89;

 then (( Partial_Sums_in_cod1 f) . (k,i)) <= (( Partial_Sums_in_cod2 ( ~ f)) . (j,k)) by Th39;

 hence thesis by Th39;

 end;

 theorem :: DBLSEQ_3:91



 Th91: for f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence st (for m be Element of NAT holds ( ProjMap2 (f,m)) is non-decreasing & (seq . m) = ( lim ( ProjMap2 (f,m)))) holds ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) is non-decreasing & ( Sum seq) = ( lim ( lim_in_cod2 ( Partial_Sums_in_cod2 f)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence;

 assume

 A1: for m be Element of NAT holds ( ProjMap2 (f,m)) is non-decreasing & (seq . m) = ( lim ( ProjMap2 (f,m)));

 now

 let m be Element of NAT ;

 ( ProjMap2 (f,m)) is non-decreasing by A1;

 then ( ProjMap2 (f,m)) is convergent by RINFSUP2: 37;

 then ( lim_inf ( ProjMap2 (f,m))) = ( lim ( ProjMap2 (f,m))) by RINFSUP2: 41;

 hence (seq . m) = ( lim_inf ( ProjMap2 (f,m))) by A1;

 end;

 then

 A2: ( Sum seq) <= ( lim_inf ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) by Th83;



 A3: for n,m be Nat holds (f . (n,m)) <= (seq . m)

 proof

 let n,m be Nat;

 reconsider m1 = m as Element of NAT by ORDINAL1:def 12;

 ( ProjMap2 (f,m1)) is non-decreasing by A1;

 then ( lim ( ProjMap2 (f,m1))) = ( sup ( ProjMap2 (f,m1))) by RINFSUP2: 37;

 then

 A4: (seq . m) = ( sup ( ProjMap2 (f,m1))) by A1;



 A5: n is Element of NAT by ORDINAL1:def 12;

 ( dom ( ProjMap2 (f,m1))) = NAT by FUNCT_2:def 1;

 then (( ProjMap2 (f,m1)) . n) <= ( sup ( rng ( ProjMap2 (f,m1)))) by A5, FUNCT_1: 3, XXREAL_2: 4;

 then (( ProjMap2 (f,m1)) . n) <= ( sup ( ProjMap2 (f,m1))) by RINFSUP2:def 1;

 hence thesis by A4, A5, MESFUNC9:def 7;

 end;

 for n,m be Nat st m <= n holds (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . m) <= (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . n)

 proof

 let n,m be Nat;

 assume

 C1: m <= n;

 reconsider m1 = m, n1 = n as Element of NAT by ORDINAL1:def 12;



 C2: (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . m) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),m1))) & (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . n) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),n1))) by D1DEF6;



 C3: ( ProjMap1 (( Partial_Sums_in_cod2 f),m1)) is convergent & ( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) is convergent by RINFSUP2: 37;

 for k be Nat holds (( ProjMap1 (( Partial_Sums_in_cod2 f),m1)) . k) <= (( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) . k)

 proof

 let k be Nat;

 reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

 (( ProjMap1 (( Partial_Sums_in_cod2 f),m1)) . k1) = (( Partial_Sums_in_cod2 f) . (m1,k1)) & (( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) . k1) = (( Partial_Sums_in_cod2 f) . (n1,k1)) by MESFUNC9:def 6;

 hence thesis by A1, C1, Th89;

 end;

 hence thesis by C2, C3, RINFSUP2: 38;

 end;

 hence ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) is non-decreasing by RINFSUP2: 7;

 then

 B1: ( lim_in_cod2 ( Partial_Sums_in_cod2 f)) is convergent by RINFSUP2: 37;

 then

 B3: ( Sum seq) <= ( lim ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) by A2, RINFSUP2: 41;

 for n be Nat holds (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . n) <= ( Sum seq)

 proof

 let n be Nat;

 reconsider n1 = n as Element of NAT by ORDINAL1:def 12;



 A6: (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . n) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),n1))) by D1DEF6;



 A7: ( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) is convergent by RINFSUP2: 37;

 for m be Element of NAT holds 0 <= (seq . m)

 proof

 let m be Element of NAT ;

 for n be Nat holds 0. <= (( ProjMap2 (f,m)) . n)

 proof

 let n be Nat;

 n is Element of NAT by ORDINAL1:def 12;

 hence 0. <= (( ProjMap2 (f,m)) . n) by SUPINF_2: 39;

 end;

 then

 B2: 0. <= ( lim_inf ( ProjMap2 (f,m))) by Th87;

 ( ProjMap2 (f,m)) is non-decreasing by A1;

 then ( ProjMap2 (f,m)) is convergent by RINFSUP2: 37;

 then ( lim_inf ( ProjMap2 (f,m))) = ( lim ( ProjMap2 (f,m))) by RINFSUP2: 41;

 hence 0 <= (seq . m) by B2, A1;

 end;

 then seq is nonnegative by SUPINF_2: 39;

 then ( Partial_Sums seq) is non-decreasing by MESFUNC9: 16;

 then

 A8: ( Partial_Sums seq) is convergent by RINFSUP2: 37;

 for m be Nat holds (( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) . m) <= (( Partial_Sums seq) . m)

 proof

 let m be Nat;

 m is Element of NAT by ORDINAL1:def 12;

 then (( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) . m) = (( Partial_Sums_in_cod2 f) . (n,m)) by MESFUNC9:def 6;

 hence (( ProjMap1 (( Partial_Sums_in_cod2 f),n1)) . m) <= (( Partial_Sums seq) . m) by A3, Th85;

 end;

 then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 f),n1))) <= ( lim ( Partial_Sums seq)) by A7, A8, RINFSUP2: 38;

 hence (( lim_in_cod2 ( Partial_Sums_in_cod2 f)) . n) <= ( Sum seq) by A6, MESFUNC9:def 3;

 end;

 then ( lim_sup ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) <= ( Sum seq) by Th86;

 then ( lim ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) <= ( Sum seq) by B1, RINFSUP2: 41;

 hence ( Sum seq) = ( lim ( lim_in_cod2 ( Partial_Sums_in_cod2 f))) by B3, XXREAL_0: 1;

 end;

 theorem :: DBLSEQ_3:92

 for f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence st (for m be Element of NAT holds ( ProjMap1 (f,m)) is non-decreasing & (seq . m) = ( lim ( ProjMap1 (f,m)))) holds ( lim_in_cod1 ( Partial_Sums_in_cod1 f)) is non-decreasing & ( Sum seq) = ( lim ( lim_in_cod1 ( Partial_Sums_in_cod1 f)))

 proof

 let f be nonnegative Function of [: NAT , NAT :], ExtREAL , seq be ExtREAL_sequence;

 assume

 A1: for m be Element of NAT holds ( ProjMap1 (f,m)) is non-decreasing & (seq . m) = ( lim ( ProjMap1 (f,m)));

 for m be Element of NAT holds ( ProjMap2 (( ~ f),m)) is non-decreasing & (seq . m) = ( lim ( ProjMap2 (( ~ f),m)))

 proof

 let m be Element of NAT ;

 ( ProjMap1 (f,m)) is non-decreasing by A1;

 hence ( ProjMap2 (( ~ f),m)) is non-decreasing by Th32;

 (seq . m) = ( lim ( ProjMap1 (f,m))) by A1;

 hence (seq . m) = ( lim ( ProjMap2 (( ~ f),m))) by Th32;

 end;

 then

 A2: ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f))) is non-decreasing & ( Sum seq) = ( lim ( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f)))) by Th91;

 for n be Element of NAT holds (( lim_in_cod2 ( Partial_Sums_in_cod2 ( ~ f))) . n) = (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . n)

 proof

 let n be Element of NAT ;

 (( lim_in_cod1 ( Partial_Sums_in_cod1 f)) . n) = ( lim ( ProjMap2 (( Partial_Sums_in_cod1 f),n))) by D1DEF5

 .= ( lim ( ProjMap1 (( ~ ( Partial_Sums_in_cod1 f)),n))) by Th33

 .= ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( ~ f)),n))) by Th40;

 hence thesis by D1DEF6;

 end;

 hence thesis by A2, FUNCT_2:def 8;

 end;

 begin

 theorem :: DBLSEQ_3:93



 Th93: for f be Function of [: NAT , NAT :], ExtREAL st f is P-convergent_to_+infty holds not f is P-convergent_to_-infty & not f is P-convergent_to_finite_number

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: f is P-convergent_to_+infty;

 hereby

 assume f is P-convergent_to_-infty;

 then

 consider N1 be Nat such that

 A3: for n,m be Nat st n >= N1 & m >= N1 holds (f . (n,m)) <= ( - 1);

 consider N2 be Nat such that

 A4: for n,m be Nat st n >= N2 & m >= N2 holds 1 <= (f . (n,m)) by A1;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A5: N >= N1 & N >= N2 by XXREAL_0: 25;

 then (f . (N,N)) <= ( - 1) by A3;

 hence contradiction by A4, A5;

 end;

 assume f is P-convergent_to_finite_number;

 then

 consider p be Real such that

 A6: for e be Real st 0 < e holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds |.((f . (n,m)) - p) qua ExtReal.| < e;

 reconsider p1 = p as ExtReal;

 per cases ;

 suppose

 A9: p > 0 ;

 then

 consider N1 be Nat such that

 A7: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p1).| = N1 & m >= N1;

 then |.((f . (n,m)) - p) qua ExtReal.| < p by A7;

 then ((f . (n,m)) - p1) < p by EXTREAL1: 21;

 then (f . (n,m)) < (p1 + p1) by XXREAL_3: 54;

 then (f . (n,m)) < (2 * p1) by XXREAL_3: 94;

 hence (f . (n,m)) < (2 * p) by XXREAL_3:def 5;

 end;

 consider N2 be Nat such that

 A10: for n,m be Nat st n >= N2 & m >= N2 holds (2 * p) <= (f . (n,m)) by A1, A9;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A11: N >= N1 & N >= N2 by XXREAL_0: 25;

 then (f . (N,N)) < (2 * p) by A8;

 hence contradiction by A11, A10;

 end;

 suppose

 A12: p = 0 ;

 consider N1 be Nat such that

 A13: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p).| < 1 by A6;

 consider N2 be Nat such that

 A14: for n,m be Nat st n >= N2 & m >= N2 holds 1 <= (f . (n,m)) by A1;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 reconsider jj = 1 as ExtReal;

 set N = ( max (N1,N2));



 A15: N >= N1 & N >= N2 by XXREAL_0: 25;

 then |.((f . (N,N)) - p1).| < jj by A13;

 then ((f . (N,N)) - p1) < jj by EXTREAL1: 21;

 then (f . (N,N)) < (jj + p1) by XXREAL_3: 54;

 then (f . (N,N)) < (1 + 0 ) by A12, XXREAL_3:def 2;

 hence contradiction by A14, A15;

 end;

 suppose p < 0 ;

 then

 consider N1 be Nat such that

 A17: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p).| < ( - p) by A6;

 A18:

 now

 let n,m be Nat;

 assume n >= N1 & m >= N1;

 then |.((f . (n,m)) - p).| < ( - p) by A17;

 then ((f . (n,m)) - p1) < ( - p) by EXTREAL1: 21;

 then (f . (n,m)) < (p1 + ( - p)) by XXREAL_3: 54;

 then (f . (n,m)) < (p + ( - p)) by XXREAL_3:def 2;

 hence (f . (n,m)) < 0 ;

 end;

 consider N2 be Nat such that

 A19: for n,m be Nat st n >= N2 & m >= N2 holds 1 <= (f . (n,m)) by A1;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A20: N >= N1 & N >= N2 by XXREAL_0: 25;

 then (f . (N,N)) < 0 by A18;

 hence contradiction by A19, A20;

 end;

 end;

 theorem :: DBLSEQ_3:94



 Th94: for f be Function of [: NAT , NAT :], ExtREAL st f is P-convergent_to_-infty holds not f is P-convergent_to_+infty & not f is P-convergent_to_finite_number

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: f is P-convergent_to_-infty;

 hereby

 assume f is P-convergent_to_+infty;

 then

 consider N1 be Nat such that

 A3: for n,m be Nat st n >= N1 & m >= N1 holds (f . (n,m)) >= 1;

 consider N2 be Nat such that

 A4: for n,m be Nat st n >= N2 & m >= N2 holds ( - 1) >= (f . (n,m)) by A1;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A5: N >= N1 & N >= N2 by XXREAL_0: 25;

 then (f . (N,N)) >= 1 by A3;

 hence contradiction by A4, A5;

 end;

 assume f is P-convergent_to_finite_number;

 then

 consider p be Real such that

 A6: for e be Real st 0 < e holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds |.((f . (n,m)) - p) qua ExtReal.| < e;

 reconsider p1 = p as ExtReal;

 per cases ;

 suppose

 A9: p > 0 ;

 then

 consider N1 be Nat such that

 A7: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p1).| = N1 & m >= N1;

 then |.((f . (n,m)) - p) qua ExtReal.| < p by A7;

 then ( - p1) < ((f . (n,m)) - p) by EXTREAL1: 21;

 then (( - p1) + p) < (f . (n,m)) by XXREAL_3: 53;

 hence 0 < (f . (n,m)) by XXREAL_3: 7;

 end;

 consider N2 be Nat such that

 A10: for n,m be Nat st n >= N2 & m >= N2 holds ( - (2 * p)) >= (f . (n,m)) by A1, A9;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A11: N >= N1 & N >= N2 by XXREAL_0: 25;

 then 0 < (f . (N,N)) by A8;

 hence contradiction by A9, A11, A10;

 end;

 suppose

 A12: p = 0 ;

 consider N1 be Nat such that

 A13: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p).| < 1 by A6;

 consider N2 be Nat such that

 A14: for n,m be Nat st n >= N2 & m >= N2 holds ( - 1) >= (f . (n,m)) by A1;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 reconsider jj = 1 as ExtReal;

 set N = ( max (N1,N2));



 A15: N >= N1 & N >= N2 by XXREAL_0: 25;

 then |.((f . (N,N)) - p1).| < jj by A13;

 then ( - jj) < ((f . (N,N)) - p1) by EXTREAL1: 21;

 then (( - jj) + p) < (f . (N,N)) by XXREAL_3: 53;

 then ( - jj) < (f . (N,N)) by A12, XXREAL_3: 4;

 then ( - 1) < (f . (N,N)) by XXREAL_3:def 3;

 hence contradiction by A14, A15;

 end;

 suppose

 A16: p < 0 ;

 then

 consider N1 be Nat such that

 A17: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - p).| < ( - p) by A6;

 A18:

 now

 let n,m be Nat;

 assume n >= N1 & m >= N1;

 then |.((f . (n,m)) - p) qua ExtReal.| < ( - p) by A17;

 then ( - ( - p1)) < ((f . (n,m)) - p1) by EXTREAL1: 21;

 then (p1 + p1) < (f . (n,m)) by XXREAL_3: 53;

 then (2 * p1) < (f . (n,m)) by XXREAL_3: 94;

 hence (2 * p) < (f . (n,m)) by XXREAL_3:def 5;

 end;

 consider N2 be Nat such that

 A19: for n,m be Nat st n >= N2 & m >= N2 holds (f . (n,m)) <= (2 * p) by A1, A16;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 A20: N >= N1 & N >= N2 by XXREAL_0: 25;

 then (2 * p) < (f . (N,N)) by A18;

 hence contradiction by A19, A20;

 end;

 end;

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 :: DBLSEQ_3:def17

 attr f is P-convergent means f is P-convergent_to_finite_number or f is P-convergent_to_+infty or f is P-convergent_to_-infty;

 end

 definition

 let f be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: f is P-convergent;

 :: DBLSEQ_3:def18

 func P-lim f -> ExtReal means

 : Def5: (ex p be Real st it = p & (for e be Real st 0 < e holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds |.((f . (n,m)) - it ).| < e) & f is P-convergent_to_finite_number) or (it = +infty & f is P-convergent_to_+infty) or (it = -infty & f is P-convergent_to_-infty);

 existence

 proof

 per cases by A1;

 suppose

 A2: f is P-convergent_to_finite_number;

 then

 consider p be Real such that

 A3: for e be Real st 0 < e holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds |.((f . (n,m)) - p).| < e;

 reconsider p as R_eal by XXREAL_0:def 1;

 take p;

 thus thesis by A2, A3;

 end;

 suppose f is P-convergent_to_+infty or f is P-convergent_to_-infty;

 hence thesis;

 end;

 end;

 uniqueness

 proof

 defpred P[ ExtReal] means (ex g be Real st $1 = g & (for p be Real st 0 = N & m >= N holds |.((f . (n,m)) - $1).| < p) & f is P-convergent_to_finite_number) or ($1 = +infty & f is P-convergent_to_+infty) or ($1 = -infty & f is P-convergent_to_-infty);

 given g1,g2 be ExtReal such that

 A4: P[g1] and

 A5: P[g2] and

 A6: g1 <> g2;

 per cases by A1;

 suppose

 A7: f is P-convergent_to_finite_number;

 then

 consider g be Real such that

 A8: g1 = g and

 A9: for p be Real st 0 = N & m >= N holds |.((f . (n,m)) - g1).| < p and f is P-convergent_to_finite_number by A4, Th93, Th94;

 consider h be Real such that

 A10: g2 = h and

 A11: for p be Real st 0 = N & m >= N holds |.((f . (n,m)) - g2).| 0 by A6, A8, A10;

 then

 consider N1 be Nat such that

 A13: for n,m be Nat st n >= N1 & m >= N1 holds |.((f . (n,m)) - g1).| < ( |.(g - h).| / 2) by A9;

 consider N2 be Nat such that

 A14: for n,m be Nat st n >= N2 & m >= N2 holds |.((f . (n,m)) - g2).| < ( |.(g - h).| / 2) by A11, A12;

 reconsider N1, N2 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (N1,N2));



 B1: N >= N1 & N >= N2 by XXREAL_0: 25;

 then

 A15: |.((f . (N,N)) - g1).| < ( |.(g - h).| / 2) by A13;



 A16: |.((f . (N,N)) - g2).| < ( |.(g - h).| / 2) by A14, B1;

 reconsider g, h as Complex;



 A17: ((f . (N,N)) - g2) < ( |.(g - h).| / 2) by A16, EXTREAL1: 21;



 A18: ( - ( |.(g - h).| / 2) qua ExtReal) < ((f . (N,N)) - g2) by A16, EXTREAL1: 21;

 then

 reconsider w = ((f . (N,N)) - g2) as Element of REAL by A17, XXREAL_0: 48;



 A19: ((f . (N,N)) - g2) in REAL by A18, A17, XXREAL_0: 48;

 then

 A20: (f . (N,N)) <> +infty by A10;



 A21: (( - (f . (N,N))) + g1) = ( - ((f . (N,N)) - g1)) by XXREAL_3: 26;

 then

 A22: |.(( - (f . (N,N))) + g1).| < ( |.(g - h).| / 2) by A15, EXTREAL1: 29;

 then

 A23: (( - (f . (N,N))) + g1) < ( |.(g - h).| / 2) by EXTREAL1: 21;

 ( - ( |.(g - h).| / 2) qua ExtReal) < (( - (f . (N,N))) + g1) by A22, EXTREAL1: 21;

 then

 A24: (( - (f . (N,N))) + g1) in REAL by A23, XXREAL_0: 48;



 A25: (f . (N,N)) <> -infty by A10, A19;

 |.(g1 - g2).| = |.((g1 + 0. ) - g2).| by XXREAL_3: 4

 .= |.((g1 + ((f . (N,N)) + ( - (f . (N,N))))) - g2).| by XXREAL_3: 7

 .= |.(((( - (f . (N,N))) + g1) + (f . (N,N))) - g2).| by A8, A20, A25, XXREAL_3: 29

 .= |.((( - (f . (N,N))) + g1) + ((f . (N,N)) - g2)).| by A10, A24, XXREAL_3: 30;

 then |.(g1 - g2).| <= ( |.(( - (f . (N,N))) + g1).| + |.((f . (N,N)) - g2).|) by EXTREAL1: 24;

 then

 A26: |.(g1 - g2).| <= ( |.((f . (N,N)) - g1).| + |.((f . (N,N)) - g2).|) by A21, EXTREAL1: 29;

 |.((f . (N,N)) - g2).| < +infty & |.((f . (N,N)) - g2).| >= 0 by A19, EXTREAL1: 41, EXTREAL1: 14;

 then |.((f . (N,N)) - g2).| in REAL by XXREAL_0: 14;

 then

 A27: ( |.((f . (N,N)) - g1).| + |.((f . (N,N)) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + |.((f . (N,N)) - g2).|) by A15, XXREAL_3: 43;

 ( |.(g - h).| / 2) in REAL by XREAL_0:def 1;

 then (( |.(g - h).| / 2) qua ExtReal + |.((f . (N,N)) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + ( |.(g - h).| / 2)) by A16, XXREAL_3: 43;

 then

 A28: ( |.((f . (N,N)) - g1).| + |.((f . (N,N)) - g2).|) < (( |.(g - h).| / 2) qua ExtReal + ( |.(g - h).| / 2)) by A27, XXREAL_0: 2;

 (g - h) = (g1 - g2) by A8, A10, SUPINF_2: 3;

 hence contradiction by A28, A26, EXTREAL1: 12;

 end;

 suppose f is P-convergent_to_+infty or f is P-convergent_to_-infty;

 hence contradiction by A4, A5, A6, Th93, Th94;

 end;

 end;

 end

 theorem :: DBLSEQ_3:95

 for f be Function of [: NAT , NAT :], ExtREAL , r be Real st (for n,m be Nat holds (f . (n,m)) = r) holds f is P-convergent_to_finite_number & ( P-lim f) = r

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , r be Real;

 assume

 A1: for n,m be Nat holds (f . (n,m)) = r;

 A2:

 now

 reconsider N = 1 as Nat;

 let p be Real;

 assume

 A3: 0 < p;

 take N;

 let n,m be Nat such that n >= N & m >= N;

 (f . (n,m)) = r by A1;

 hence |.((f . (n,m)) - r).| < p by A3, XXREAL_3: 7, EXTREAL1: 16;

 end;

 hence

 A4: f is P-convergent_to_finite_number;

 then f is P-convergent;

 hence thesis by A2, A4, Def5;

 end;

 theorem :: DBLSEQ_3:96



 Th96: for f be Function of [: NAT , NAT :], ExtREAL st (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f . (n1,m1)) <= (f . (n2,m2))) holds f is P-convergent & ( P-lim f) = ( sup ( rng f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f . (n1,m1)) <= (f . (n2,m2));

 A2:

 now

 let n,m be Nat;

 reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def 12;

 [n1, m1] in [: NAT , NAT :] & ( dom f) = [: NAT , NAT :] by ZFMISC_1:def 2, FUNCT_2:def 1;

 then

 A3: (f . (n1,m1)) in ( rng f) by FUNCT_1:def 3;

 ( sup ( rng f)) is UpperBound of ( rng f) by XXREAL_2:def 3;

 hence (f . (n,m)) <= ( sup ( rng f)) by A3, XXREAL_2:def 1;

 end;

 per cases ;

 suppose

 A4: not ex n0,m0 be Nat st -infty < (f . (n0,m0));

 now

 let x be ExtReal;

 assume x in ( rng f);

 then

 consider z be object such that

 B1: z in ( dom f) & x = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 B2: n in NAT & m in NAT & z = [n, m] by B1, ZFMISC_1:def 2;

 reconsider n, m as Nat by B2;

 not ( -infty < (f . (n,m))) by A4;

 hence x <= -infty by B1, B2;

 end;

 then

 A5: -infty is UpperBound of ( rng f) by XXREAL_2:def 1;

 for y be UpperBound of ( rng f) holds -infty <= y by XXREAL_0: 5;

 then

 A6: -infty = ( sup ( rng f)) by A5, XXREAL_2:def 3;

 now

 reconsider N0 = 0 as Nat;

 let K be Real such that K < 0 ;

 take N0;

 let n,m be Nat such that N0 <= n & N0 <= m;

 (f . (n,m)) = -infty by A4, XXREAL_0: 6;

 hence (f . (n,m)) <= K by XXREAL_0: 5;

 end;

 then

 A7: f is P-convergent_to_-infty;

 then f is P-convergent;

 hence thesis by A7, A6, Def5;

 end;

 suppose ex n0,m0 be Nat st -infty < (f . (n0,m0));

 then

 consider n0,m0 be Nat such that

 A8: -infty < (f . (n0,m0));

 reconsider n0, m0 as Element of NAT by ORDINAL1:def 12;

 per cases ;

 suppose ex K be Real st for n,m be Nat holds (f . (n,m)) < K;

 then

 consider K be Real such that

 A9: for n,m be Nat holds (f . (n,m)) < K;

 now

 let x be ExtReal;

 assume x in ( rng f);

 then

 consider z be object such that

 C1: z in ( dom f) & x = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 C2: n in NAT & m in NAT & z = [n, m] by C1, ZFMISC_1:def 2;

 reconsider n, m as Nat by C2;

 (f . (n,m)) < K by A9;

 hence x <= K by C1, C2;

 end;

 then K is UpperBound of ( rng f) by XXREAL_2:def 1;

 then ( sup ( rng f)) <= K by XXREAL_2:def 3;

 then

 A11: ( sup ( rng f)) <> +infty by XXREAL_0: 9, XREAL_0:def 1;



 A12: ( sup ( rng f)) <> -infty by A2, A8;

 then

 reconsider h = ( sup ( rng f)) as Element of REAL by A11, XXREAL_0: 14;



 A13: for p be Real st 0 = N & m >= N holds |.((f . (n,m)) - ( sup ( rng f))).| < p

 proof

 let p be Real;

 assume

 A14: 0 < p;

 reconsider e = p as R_eal by XXREAL_0:def 1;

 ( sup ( rng f)) in REAL by A12, A11, XXREAL_0: 14;

 then

 consider y be ExtReal such that

 A15: y in ( rng f) and

 A16: (( sup ( rng f)) - e) < y by A14, MEASURE6: 6;

 consider x be object such that

 A17: x in ( dom f) and

 A18: y = (f . x) by A15, FUNCT_1:def 3;

 consider i,j be object such that

 B1: i in NAT & j in NAT & x = [i, j] by A17, ZFMISC_1:def 2;

 reconsider i, j as Nat by B1;

 reconsider Ni = i, Nj = j as Element of NAT by B1;

 set N0 = ( max (Ni,n0)), M0 = ( max (Nj,m0)), N = ( max (N0,M0));

 take N;

 hereby

 let n,m be Nat;

 Ni <= N0 & n0 <= N0 & Nj <= M0 & m0 <= M0 & N0 <= N & M0 <= N by XXREAL_0: 25;

 then

 B2: Ni <= N & Nj <= N by XXREAL_0: 2;

 assume N <= n & N <= m;

 then Ni <= n & Nj <= m by B2, XXREAL_0: 2;

 then (f . (Ni,Nj)) <= (f . (n,m)) by A1;

 then (( sup ( rng f)) - e) < (f . (n,m)) by A16, A18, B1, XXREAL_0: 2;

 then ( sup ( rng f)) < ((f . (n,m)) + e) by XXREAL_3: 54;

 then (( sup ( rng f)) - (f . (n,m))) < e by XXREAL_3: 55;

 then ( - e) < ( - (( sup ( rng f)) - (f . (n,m)))) by XXREAL_3: 38;

 then

 A20: ( - e) < ((f . (n,m)) - ( sup ( rng f))) by XXREAL_3: 26;



 A21: (f . (n,m)) <= ( sup ( rng f)) by A2;

 A22:

 now

 assume

 A23: ( sup ( rng f)) = (( sup ( rng f)) + e);

 ((e + ( sup ( rng f))) + ( - ( sup ( rng f)))) = (e + (( sup ( rng f)) + ( - ( sup ( rng f))))) 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 f)) + 0 qua ExtReal) <= (( sup ( rng f)) + e) by A14, XXREAL_3: 36;

 then ( sup ( rng f)) <= (( sup ( rng f)) + e) by XXREAL_3: 4;

 then ( sup ( rng f)) < (( sup ( rng f)) + e) by A22, XXREAL_0: 1;

 then (f . (n,m)) < (( sup ( rng f)) + e) by A21, XXREAL_0: 2;

 then ((f . (n,m)) - ( sup ( rng f))) < e by XXREAL_3: 55;

 hence |.((f . (n,m)) - ( sup ( rng f))).| < p by A20, EXTREAL1: 22;

 end;

 end;



 A24: h = ( sup ( rng f));

 then

 A25: f is P-convergent_to_finite_number by A13;

 hence f is P-convergent;

 hence thesis by A13, A24, A25, Def5;

 end;

 suppose

 A26: not (ex K be Real st 0 < K & for n,m be Nat holds (f . (n,m)) < K);

 now

 let K be Real;

 assume 0 < K;

 then

 consider N0,N1 be Nat such that

 A27: K <= (f . (N0,N1)) by A26;

 reconsider n0 = N0, n1 = N1 as Element of NAT by ORDINAL1:def 12;

 set N = ( max (n0,n1));



 B3: N >= N0 & N >= N1 by XXREAL_0: 25;

 reconsider N as Nat;

 now

 let n,m be Nat;

 assume N <= n & N <= m;

 then N0 <= n & N1 <= m by B3, XXREAL_0: 2;

 then (f . (N0,N1)) <= (f . (n,m)) by A1;

 hence K <= (f . (n,m)) by A27, XXREAL_0: 2;

 end;

 hence ex N be Nat st for n,m be Nat st N <= n & N <= m holds K <= (f . (n,m));

 end;

 then

 A28: f is P-convergent_to_+infty;

 hence

 A29: f is P-convergent;

 now

 assume

 A30: ( sup ( rng f)) <> +infty ;

 (f . (n0,m0)) <= ( sup ( rng f)) by A2;

 then

 reconsider h = ( sup ( rng f)) as Element of REAL by A8, A30, XXREAL_0: 14;

 set K = ( max ( 0 ,h));

 0 <= K by XXREAL_0: 25;

 then

 consider N0,M0 be Nat such that

 A31: (K + 1) <= (f . (N0,M0)) by A26;

 (h + 0 ) < (K + 1) by XREAL_1: 8, XXREAL_0: 25;

 then ( sup ( rng f)) < (f . (N0,M0)) by A31, XXREAL_0: 2;

 hence contradiction by A2;

 end;

 hence thesis by A28, A29, Def5;

 end;

 end;

 end;

 theorem :: DBLSEQ_3:97

 for f1,f2 be Function of [: NAT , NAT :], ExtREAL st (for n,m be Nat holds (f1 . (n,m)) <= (f2 . (n,m))) holds ( sup ( rng f1)) <= ( sup ( rng f2))

 proof

 let f1,f2 be Function of [: NAT , NAT :], ExtREAL ;

 assume

 A1: for n,m be Nat holds (f1 . (n,m)) <= (f2 . (n,m));

 A2:

 now

 let n,m be Element of NAT ;

 ( dom f2) = [: NAT , NAT :] by FUNCT_2:def 1;

 then [n, m] in ( dom f2) by ZFMISC_1: 87;

 then

 A3: (f2 . (n,m)) in ( rng f2) by FUNCT_1:def 3;



 A4: (f1 . (n,m)) <= (f2 . (n,m)) by A1;

 ( sup ( rng f2)) is UpperBound of ( rng f2) by XXREAL_2:def 3;

 then (f2 . (n,m)) <= ( sup ( rng f2)) by A3, XXREAL_2:def 1;

 hence (f1 . (n,m)) <= ( sup ( rng f2)) by A4, XXREAL_0: 2;

 end;

 now

 let x be ExtReal;

 assume x in ( rng f1);

 then

 consider z be object such that

 A5: z in ( dom f1) & x = (f1 . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A6: n in NAT & m in NAT & z = [n, m] by A5, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A6;

 x = (f1 . (n,m)) by A5, A6;

 hence x <= ( sup ( rng f2)) by A2;

 end;

 then ( sup ( rng f2)) is UpperBound of ( rng f1) by XXREAL_2:def 1;

 hence thesis by XXREAL_2:def 3;

 end;

 theorem :: DBLSEQ_3:98

 for f be Function of [: NAT , NAT :], ExtREAL holds for n,m be Nat holds (f . (n,m)) <= ( sup ( rng f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL ;

 hereby

 let n,m be Nat;



 A0: n in NAT & m in NAT by ORDINAL1:def 12;

 ( dom f) = [: NAT , NAT :] by FUNCT_2:def 1;

 then [n, m] in ( dom f) by A0, ZFMISC_1: 87;

 then

 A1: (f . (n,m)) in ( rng f) by FUNCT_1:def 3;

 ( sup ( rng f)) is UpperBound of ( rng f) by XXREAL_2:def 3;

 hence (f . (n,m)) <= ( sup ( rng f)) by A1, XXREAL_2:def 1;

 end;

 end;

 theorem :: DBLSEQ_3:99

 for f be Function of [: NAT , NAT :], ExtREAL , K be R_eal st (for n,m be Nat holds (f . (n,m)) <= K) holds ( sup ( rng f)) <= K

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , K be R_eal;

 assume

 A1: for n,m be Nat holds (f . (n,m)) <= K;

 now

 let x be ExtReal;

 assume x in ( rng f);

 then

 consider z be object such that

 A2: z in ( dom f) & x = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A3: n in NAT & m in NAT & z = [n, m] by A2, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A3;

 x = (f . (n,m)) by A2, A3;

 hence x <= K by A1;

 end;

 then K is UpperBound of ( rng f) by XXREAL_2:def 1;

 hence thesis by XXREAL_2:def 3;

 end;

 theorem :: DBLSEQ_3:100

 for f be Function of [: NAT , NAT :], ExtREAL , K be R_eal st K <> +infty & (for n,m be Nat holds (f . (n,m)) <= K) holds ( sup ( rng f)) < +infty

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , K be R_eal;

 assume

 A1: K <> +infty & (for n,m be Nat holds (f . (n,m)) <= K);

 now

 let x be ExtReal;

 assume x in ( rng f);

 then

 consider z be object such that

 A2: z in ( dom f) & x = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A3: n in NAT & m in NAT & z = [n, m] by A2, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A3;

 x = (f . (n,m)) by A2, A3;

 hence x <= K by A1;

 end;

 then K is UpperBound of ( rng f) by XXREAL_2:def 1;

 then ( sup ( rng f)) <= K by XXREAL_2:def 3;

 hence thesis by A1, XXREAL_0: 2, XXREAL_0: 4;

 end;

 theorem :: DBLSEQ_3:101



 Th101: for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( sup ( rng f)) <> +infty iff ex K be Real st 0 < K & for n,m be Nat holds (f . (n,m)) <= K

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;



 A1: -infty < (f . (1,1)) by MESFUNC5:def 5;



 A2: ( dom f) = [: NAT , NAT :] by FUNCT_2:def 1;

 then [1, 1] in ( dom f) by ZFMISC_1: 87;

 then

 A3: (f . (1,1)) <= ( sup ( rng f)) by FUNCT_1: 3, XXREAL_2: 4;

 A4:

 now

 assume ( sup ( rng f)) <> +infty ;

 then not ( sup ( rng f)) in { -infty , +infty } by A1, A3, TARSKI:def 2;

 then ( sup ( rng f)) in REAL by XBOOLE_0:def 3, XXREAL_0:def 4;

 then

 reconsider S = ( sup ( rng f)) as Real;

 take K = ( max (S,1));

 thus 0 < K by XXREAL_0: 25;

 let n,m be Nat;

 n in NAT & m in NAT by ORDINAL1:def 12;

 then [n, m] in [: NAT , NAT :] by ZFMISC_1: 87;

 then

 A5: (f . (n,m)) <= ( sup ( rng f)) by A2, FUNCT_1: 3, XXREAL_2: 4;

 S <= K by XXREAL_0: 25;

 hence (f . (n,m)) <= K by A5, XXREAL_0: 2;

 end;

 now

 given K be Real such that 0 < K and

 A6: for n,m be Nat holds (f . (n,m)) <= K;

 now

 let w be ExtReal;

 assume w in ( rng f);

 then

 consider z be object such that

 A7: z in ( dom f) & w = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A8: n in NAT & m in NAT & z = [n, m] by A7, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A8;

 w = (f . (n,m)) by A7, A8;

 hence w <= K by A6;

 end;

 then K is UpperBound of ( rng f) by XXREAL_2:def 1;

 then ( sup ( rng f)) <= K by XXREAL_2:def 3;

 hence ( sup ( rng f)) <> +infty by XXREAL_0: 9, XREAL_0:def 1;

 end;

 hence thesis by A4;

 end;

 theorem :: DBLSEQ_3:102



 Th102: for f be Function of [: NAT , NAT :], ExtREAL , c be ExtReal st (for n,m be Nat holds (f . (n,m)) = c) holds f is P-convergent & ( P-lim f) = c & ( P-lim f) = ( sup ( rng f))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , c be ExtReal;

 reconsider cc = c as R_eal by XXREAL_0:def 1;



 A1: ( dom f) = [: NAT , 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,m be Nat holds (f . (n,m)) = c;

 then

 A4: (f . (1,1)) = c;

 now

 let v be ExtReal;

 assume v in ( rng f);

 then

 consider z be object such that

 A7: z in ( dom f) & v = (f . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A8: n in NAT & m in NAT & z = [n, m] by A7, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A8;

 v = (f . (n,m)) by A7, A8;

 hence v <= c by A3;

 end;

 then

 A5: c is UpperBound of ( rng f) 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 n1,m1 be Nat such that N <= n1 & N <= m1;

 ((f . (n1,m1)) - rc) = ((f . (n1,m1)) - (f . (n1,m1))) by A3;

 hence |.((f . (n1,m1)) - rc).| < p by A7, EXTREAL1: 16, XXREAL_3: 7;

 end;

 then

 A8: f is P-convergent_to_finite_number;

 hence f is P-convergent;

 hence

 A9: ( P-lim f) = c by A6, A8, Def5;

 [1, 1] in ( dom f) by A1, ZFMISC_1: 87;

 hence thesis by A9, A5, A4, FUNCT_1: 3, XXREAL_2: 55;

 end;

 suppose

 A10: c = -infty ;

 for p be Real st p < 0 holds ex N be Nat st for n,m be Nat st n >= N & m >= N holds (f . (n,m)) <= p

 proof

 let p be Real such that p < 0 ;

 take 0 ;

 now

 let n,m be Nat such that 0 <= n & 0 <= m;

 (f . (n,m)) = -infty by A3, A10;

 hence (f . (n,m)) <= p by XREAL_0:def 1, XXREAL_0: 12;

 end;

 hence thesis;

 end;

 then

 A12: f is P-convergent_to_-infty;

 hence f is P-convergent;

 hence

 A13: ( P-lim f) = c by A10, A12, Def5;

 [1, 1] in ( dom f) by A1, ZFMISC_1: 87;

 hence thesis by A5, A4, A13, FUNCT_1: 3, XXREAL_2: 55;

 end;

 suppose

 A14: c = +infty ;

 for p be Real st 0 = N & m >= N holds p <= (f . (n,m))

 proof

 let p be Real such that 0 < p;

 take 0 ;

 now

 let n,m be Nat such that n >= 0 & m >= 0 ;

 (f . (n,m)) = +infty by A3, A14;

 hence p <= (f . (n,m)) by XREAL_0:def 1, XXREAL_0: 9;

 end;

 hence thesis;

 end;

 then

 A16: f is P-convergent_to_+infty;

 hence f is P-convergent;

 hence

 A17: ( P-lim f) = c by A14, A16, Def5;

 [1, 1] in ( dom f) by A1, ZFMISC_1: 87;

 hence thesis by A5, A4, A17, FUNCT_1: 3, XXREAL_2: 55;

 end;

 end;



 Lm8: for f be without-infty Function of [: NAT , NAT :], ExtREAL holds ( sup ( rng f)) in REAL or ( sup ( rng f)) = +infty

 proof

 let f be without-infty Function of [: NAT , NAT :], ExtREAL ;



 A1: not -infty in ( rng f) by MESFUNC5:def 3;

 ( dom f) = [: NAT , NAT :] by FUNCT_2:def 1;

 then [1, 1] in ( dom f) by ZFMISC_1: 87;

 then (f . (1,1)) in ( rng f) by FUNCT_1: 3;

 then not -infty is UpperBound of ( rng f) by A1, XXREAL_0: 6, XXREAL_2:def 1;

 then ( sup ( rng f)) <> -infty by XXREAL_2:def 3;

 hence thesis by XXREAL_0: 14;

 end;

 theorem :: DBLSEQ_3:103

 for f be Function of [: NAT , NAT :], ExtREAL , f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL st (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f1 . (n1,m1)) <= (f1 . (n2,m2))) & (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f2 . (n1,m1)) <= (f2 . (n2,m2))) & (for n,m be Nat holds ((f1 . (n,m)) + (f2 . (n,m))) = (f . (n,m))) holds f is P-convergent & ( P-lim f) = ( sup ( rng f)) & ( P-lim f) = (( P-lim f1) + ( P-lim f2)) & ( sup ( rng f)) = (( sup ( rng f1)) + ( sup ( rng f2)))

 proof

 let f be Function of [: NAT , NAT :], ExtREAL , f1,f2 be without-infty Function of [: NAT , NAT :], ExtREAL ;

 assume that

 A1: for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f1 . (n1,m1)) <= (f1 . (n2,m2)) and

 A2: for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f2 . (n1,m1)) <= (f2 . (n2,m2)) and

 A5: for n,m be Nat holds ((f1 . (n,m)) + (f2 . (n,m))) = (f . (n,m));



 A6: ( dom f1) = [: NAT , NAT :] & ( dom f2) = [: NAT , NAT :] by FUNCT_2:def 1;



 B0: f1 is P-convergent & ( P-lim f1) = ( sup ( rng f1)) & f2 is P-convergent & ( P-lim f2) = ( sup ( rng f2)) by A1, A2, Th96;

 now

 let n1,m1,n2,m2 be Nat;

 assume n1 <= n2 & m1 <= m2;

 then (f1 . (n1,m1)) <= (f1 . (n2,m2)) & (f2 . (n1,m1)) <= (f2 . (n2,m2)) by A1, A2;

 then ((f1 . (n1,m1)) + (f2 . (n1,m1))) <= ((f1 . (n2,m2)) + (f2 . (n2,m2))) by XXREAL_3: 36;

 then (f . (n1,m1)) <= ((f1 . (n2,m2)) + (f2 . (n2,m2))) by A5;

 hence (f . (n1,m1)) <= (f . (n2,m2)) by A5;

 end;

 hence

 A7: f is P-convergent & ( P-lim f) = ( sup ( rng f)) by Th96;

 P1:

 now

 per cases by Lm8;

 suppose

 A9: ( sup ( rng f1)) in REAL ;

 set SE1 = ( sup ( rng f1));

 per cases by Lm8;

 suppose

 A10: ( sup ( rng f2)) in REAL ;

 set SE2 = ( sup ( rng f2));

 B1:

 now

 let p be Real;

 assume

 A11: 0 < p;

 then

 consider x1 be ExtReal such that

 A12: x1 in ( rng f1) & (( sup ( rng f1)) - (p / 2)) < x1 by A9, MEASURE6: 6;

 consider z1 be object such that

 A13: z1 in ( dom f1) & x1 = (f1 . z1) by A12, FUNCT_1:def 3;

 consider n1,m1 be object such that

 A14: n1 in NAT & m1 in NAT & z1 = [n1, m1] by A13, ZFMISC_1:def 2;

 reconsider n1, m1 as Element of NAT by A14;

 consider x2 be ExtReal such that

 A15: x2 in ( rng f2) & (( sup ( rng f2)) - (p / 2)) < x2 by A10, A11, MEASURE6: 6;

 consider z2 be object such that

 A16: z2 in ( dom f2) & x2 = (f2 . z2) by A15, FUNCT_1:def 3;

 consider n2,m2 be object such that

 A17: n2 in NAT & m2 in NAT & z2 = [n2, m2] by A16, ZFMISC_1:def 2;

 reconsider n2, m2 as Element of NAT by A17;

 reconsider N = ( max (( max (n1,m1)),( max (n2,m2)))) as Nat;

 take N;

 hereby

 let n,m be Nat;

 assume

 A18: n >= N & m >= N;

 N >= ( max (n1,m1)) & N >= ( max (n2,m2)) & ( max (n1,m1)) >= n1 & ( max (n1,m1)) >= m1 & ( max (n2,m2)) >= n2 & ( max (n2,m2)) >= m2 by XXREAL_0: 25;

 then N >= n1 & N >= m1 & N >= n2 & N >= m2 by XXREAL_0: 2;

 then n >= n1 & n >= n2 & m >= m1 & m >= m2 by A18, XXREAL_0: 2;

 then

 A22: (f1 . (n1,m1)) <= (f1 . (n,m)) & (f2 . (n2,m2)) <= (f2 . (n,m)) by A1, A2;

 then (SE1 - (f1 . (n,m))) <= (SE1 - x1) & (SE2 - (f2 . (n,m))) <= (SE2 - x2) by A13, A14, A16, A17, XXREAL_3: 37;

 then

 A19: ((SE1 - (f1 . (n,m))) + (SE2 - (f2 . (n,m)))) <= ((SE1 - x1) + (SE2 - x2)) by XXREAL_3: 36;



 A20: (p / 2) in REAL by XREAL_0:def 1;

 SE1 < ((p / 2) + x1) & SE2 < ((p / 2) + x2) by A12, A15, XXREAL_3: 54;

 then

 A21: (SE1 - x1) < (p / 2) & (SE2 - x2) < (p / 2) by XXREAL_3: 55;

 then

 A24: ((p / 2) + (SE2 - x2)) < ((p / 2) + (p / 2)) by A20, XXREAL_3: 43;

 n in NAT & m in NAT by ORDINAL1:def 12;

 then [n, m] in [: NAT , NAT :] by ZFMISC_1: 87;

 then

 B1: (f1 . (n,m)) in ( rng f1) & (f2 . (n,m)) in ( rng f2) & (f1 . (n,m)) <= SE1 & (f2 . (n,m)) <= SE2 by A6, FUNCT_1: 3, XXREAL_2: 4;

 then

 B2: (f1 . (n,m)) < +infty & (f2 . (n,m)) < +infty by A9, A10, XXREAL_0: 2, XXREAL_0: 9;



 B3: -infty <> (f1 . (n,m)) & -infty <> (f2 . (n,m)) by B1, MESFUNC5:def 3;

 -infty <> x1 & -infty <> x2 by A12, A15, MESFUNC5:def 3;

 then x1 in REAL & x2 in REAL by B2, A22, A13, A14, A16, A17, XXREAL_0: 14;

 then (SE2 - x2) in REAL by A10, XREAL_0:def 1;

 then ((SE1 - x1) + (SE2 - x2)) < ((p / 2) + (SE2 - x2)) by A21, XXREAL_3: 43;

 then ((SE1 - x1) + (SE2 - x2)) < ((p / 2) + (p / 2)) by A24, XXREAL_0: 2;

 then

 A26: ((SE1 - (f1 . (n,m))) + (SE2 - (f2 . (n,m)))) < p by A19, XXREAL_0: 2;



 B5: ((SE1 - (f1 . (n,m))) + (SE2 - (f2 . (n,m)))) = (((SE1 - (f1 . (n,m))) + SE2) - (f2 . (n,m))) by A10, B2, B3, XXREAL_3: 30

 .= (((SE2 + SE1) - (f1 . (n,m))) - (f2 . (n,m))) by A9, A10, XXREAL_3: 30

 .= ((SE1 + SE2) - ((f1 . (n,m)) + (f2 . (n,m)))) by A9, A10, B3, XXREAL_3: 31

 .= ((SE1 + SE2) - (f . (n,m))) by A5;

 (SE1 - (f1 . (n,m))) >= 0 & (SE2 - (f2 . (n,m))) >= 0 by B1, XXREAL_3: 40;

 then |.((SE1 + SE2) - (f . (n,m))).| < p by B5, A26, EXTREAL1:def 1;

 then |.( - ((f . (n,m)) - (SE1 + SE2))).| < p by XXREAL_3: 26;

 hence |.((f . (n,m)) - (SE1 + SE2)).| < p by EXTREAL1: 29;

 end;

 end;

 then f is P-convergent_to_finite_number by A9, A10;

 hence ( P-lim f) = (( P-lim f1) + ( P-lim f2)) by A9, A10, B0, B1, A7, Def5;

 end;

 suppose

 C1: ( sup ( rng f2)) = +infty ;

 then

 C2: (( P-lim f1) + ( P-lim f2)) = +infty by B0, A9, XXREAL_3:def 2;

 now

 let g be Real;

 assume 0 < g;

 then

 consider e1 be ExtReal such that

 C5: e1 in ( rng f1) & (SE1 - (g / 2)) < e1 by A9, MEASURE6: 6;

 consider z1 be object such that

 C6: z1 in ( dom f1) & e1 = (f1 . z1) by C5, FUNCT_1:def 3;

 consider n1,m1 be object such that

 C7: n1 in NAT & m1 in NAT & z1 = [n1, m1] by C6, ZFMISC_1:def 2;

 reconsider n1, m1 as Element of NAT by C7;

 (g - (SE1 - (g / 2))) in REAL & (SE1 - (g / 2)) in REAL by A9, XREAL_0:def 1;

 then

 consider e2 be Element of ExtREAL such that

 C8: e2 in ( rng f2) & (g - (SE1 - (g / 2))) < e2 by C1, XXREAL_0: 9, XXREAL_2: 94;

 consider z2 be object such that

 C9: z2 in ( dom f2) & e2 = (f2 . z2) by C8, FUNCT_1:def 3;

 consider n2,m2 be object such that

 C10: n2 in NAT & m2 in NAT & z2 = [n2, m2] by C9, ZFMISC_1:def 2;

 reconsider n2, m2 as Element of NAT by C10;

 reconsider N = ( max (( max (n2,m2)),( max (n1,m1)))) as Nat;

 take N;

 N >= ( max (n1,m1)) & N >= ( max (n2,m2)) & ( max (n1,m1)) >= n1 & ( max (n1,m1)) >= m1 & ( max (n2,m2)) >= n2 & ( max (n2,m2)) >= m2 by XXREAL_0: 25;

 then

 C13: N >= n1 & N >= m1 & N >= n2 & N >= m2 by XXREAL_0: 2;

 hereby

 let n,m be Nat;

 assume n >= N & m >= N;

 then n >= n1 & m >= m1 & n >= n2 & m >= m2 by C13, XXREAL_0: 2;

 then (f1 . (n,m)) >= (f1 . (n1,m1)) & (f2 . (n,m)) >= (f2 . (n2,m2)) by A1, A2;

 then

 C14: (SE1 - (g / 2)) < (f1 . (n,m)) & (g - (SE1 - (g / 2))) < (f2 . (n,m)) by C5, C6, C7, C8, C9, C10, XXREAL_0: 2;

 ((g - (SE1 - (g / 2))) + (SE1 - (g / 2))) = g by A9, XXREAL_3: 22;

 then g < ((f1 . (n,m)) + (f2 . (n,m))) by C14, XXREAL_3: 64;

 hence g <= (f . (n,m)) by A5;

 end;

 end;

 then f is P-convergent_to_+infty;

 hence ( P-lim f) = (( P-lim f1) + ( P-lim f2)) by C2, A7, Def5;

 end;

 end;

 suppose

 D1: ( sup ( rng f1)) = +infty ;

 per cases by Lm8;

 suppose

 D3: ( sup ( rng f2)) in REAL ;

 set SE2 = ( sup ( rng f2));



 D2: (( P-lim f1) + ( P-lim f2)) = +infty by B0, D1, D3, XXREAL_3:def 2;

 now

 let g be Real;

 assume 0 < g;

 then

 consider e2 be ExtReal such that

 D5: e2 in ( rng f2) & (SE2 - (g / 2)) < e2 by D3, MEASURE6: 6;

 consider z2 be object such that

 D6: z2 in ( dom f2) & e2 = (f2 . z2) by D5, FUNCT_1:def 3;

 consider n1,m1 be object such that

 D7: n1 in NAT & m1 in NAT & z2 = [n1, m1] by D6, ZFMISC_1:def 2;

 reconsider n1, m1 as Element of NAT by D7;

 (g - (SE2 - (g / 2))) in REAL & (SE2 - (g / 2)) in REAL by D3, XREAL_0:def 1;

 then

 consider e1 be Element of ExtREAL such that

 D8: e1 in ( rng f1) & (g - (SE2 - (g / 2))) < e1 by D1, XXREAL_0: 9, XXREAL_2: 94;

 consider z1 be object such that

 D9: z1 in ( dom f1) & e1 = (f1 . z1) by D8, FUNCT_1:def 3;

 consider n2,m2 be object such that

 D10: n2 in NAT & m2 in NAT & z1 = [n2, m2] by D9, ZFMISC_1:def 2;

 reconsider n2, m2 as Element of NAT by D10;

 reconsider N = ( max (( max (n2,m2)),( max (n1,m1)))) as Nat;

 take N;

 N >= ( max (n1,m1)) & N >= ( max (n2,m2)) & ( max (n1,m1)) >= n1 & ( max (n1,m1)) >= m1 & ( max (n2,m2)) >= n2 & ( max (n2,m2)) >= m2 by XXREAL_0: 25;

 then

 D13: N >= n1 & N >= m1 & N >= n2 & N >= m2 by XXREAL_0: 2;

 hereby

 let n,m be Nat;

 assume n >= N & m >= N;

 then n >= n1 & m >= m1 & n >= n2 & m >= m2 by D13, XXREAL_0: 2;

 then (f1 . (n,m)) >= (f1 . (n2,m2)) & (f2 . (n,m)) >= (f2 . (n1,m1)) by A1, A2;

 then

 D14: (SE2 - (g / 2)) < (f2 . (n,m)) & (g - (SE2 - (g / 2))) < (f1 . (n,m)) by D5, D6, D7, D8, D9, D10, XXREAL_0: 2;

 ((g - (SE2 - (g / 2))) + (SE2 - (g / 2))) = g by D3, XXREAL_3: 22;

 then g < ((f1 . (n,m)) + (f2 . (n,m))) by D14, XXREAL_3: 64;

 hence g <= (f . (n,m)) by A5;

 end;

 end;

 then f is P-convergent_to_+infty;

 hence ( P-lim f) = (( P-lim f1) + ( P-lim f2)) by D2, A7, Def5;

 end;

 suppose

 E1: ( sup ( rng f2)) = +infty ;

 now

 let p be Real;

 assume

 E2: 0 < p;

 then

 consider n1,m1 be Nat such that

 E3: (f1 . (n1,m1)) > (p / 2) by D1, Th101;

 consider n2,m2 be Nat such that

 E4: (f2 . (n2,m2)) > (p / 2) by E1, E2, Th101;

 reconsider n1, n2, m1, m2 as Element of NAT by ORDINAL1:def 12;

 reconsider N = ( max (( max (n2,m2)),( max (n1,m1)))) as Nat;

 take N;

 N >= ( max (n1,m1)) & N >= ( max (n2,m2)) & ( max (n1,m1)) >= n1 & ( max (n1,m1)) >= m1 & ( max (n2,m2)) >= n2 & ( max (n2,m2)) >= m2 by XXREAL_0: 25;

 then

 E5: N >= n1 & N >= m1 & N >= n2 & N >= m2 by XXREAL_0: 2;

 hereby

 let n,m be Nat;

 assume n >= N & m >= N;

 then n >= n1 & m >= m1 & n >= n2 & m >= m2 by E5, XXREAL_0: 2;

 then (f1 . (n,m)) >= (f1 . (n1,m1)) & (f2 . (n,m)) >= (f2 . (n2,m2)) by A1, A2;

 then (f1 . (n,m)) >= (p / 2) & (f2 . (n,m)) >= (p / 2) by E3, E4, XXREAL_0: 2;

 then ((p / 2) + (p / 2)) <= ((f1 . (n,m)) + (f2 . (n,m))) by XXREAL_3: 36;

 hence p <= (f . (n,m)) by A5;

 end;

 end;

 then f is P-convergent_to_+infty;

 then ( P-lim f) = +infty & ( P-lim f1) = +infty & ( P-lim f2) = +infty by A7, Def5, D1, E1, A1, A2, Th96;

 hence ( P-lim f) = (( P-lim f1) + ( P-lim f2)) by XXREAL_3:def 2;

 end;

 end;

 end;

 hence ( P-lim f) = (( P-lim f1) + ( P-lim f2));



 P2: ( P-lim f1) = ( sup ( rng f1)) & ( P-lim f2) = ( sup ( rng f2)) by A1, A2, Th96;

 now

 let n1,m1,n2,m2 be Nat;

 assume n1 <= n2 & m1 <= m2;

 then (f1 . (n1,m1)) <= (f1 . (n2,m2)) & (f2 . (n1,m1)) <= (f2 . (n2,m2)) by A1, A2;

 then ((f1 . (n1,m1)) + (f2 . (n1,m1))) <= ((f1 . (n2,m2)) + (f2 . (n2,m2))) by XXREAL_3: 36;

 then (f . (n1,m1)) <= ((f1 . (n2,m2)) + (f2 . (n2,m2))) by A5;

 hence (f . (n1,m1)) <= (f . (n2,m2)) by A5;

 end;

 hence thesis by P1, P2, Th96;

 end;

 theorem :: DBLSEQ_3:104



 Th104: for f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be Function of [: NAT , NAT :], ExtREAL , c be Real st 0 <= c & (for n,m be Nat holds (f2 . (n,m)) = (c * (f1 . (n,m)))) holds ( sup ( rng f2)) = (c * ( sup ( rng f1))) & f2 is without-infty

 proof

 let f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be Function of [: NAT , NAT :], ExtREAL , c be Real;

 assume that

 A1: 0 <= c and

 A3: for n,m be Nat holds (f2 . (n,m)) = (c * (f1 . (n,m)));



 A6: ( dom f1) = [: NAT , NAT :] & ( dom f2) = [: NAT , NAT :] by FUNCT_2:def 1;

 C6:

 now

 assume -infty in ( rng f2);

 then

 consider z be object such that

 C1: z in ( dom f2) & -infty = (f2 . z) by FUNCT_1:def 3;

 consider n,m be object such that

 C2: n in NAT & m in NAT & z = [n, m] by C1, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by C2;

 (f2 . (n,m)) = -infty by C1, C2;

 then

 C3: (c * (f1 . (n,m))) = -infty by A3;

 then

 C4: (f1 . (n,m)) = -infty or (f1 . (n,m)) = +infty by XXREAL_3: 70;

 z in [: NAT , NAT :] by C1;

 then [n, m] in ( dom f1) by C2, FUNCT_2:def 1;

 then -infty in ( rng f1) by A1, C3, C4, FUNCT_1: 3;

 hence contradiction by MESFUNC5:def 3;

 end;

 then

 C6a: f2 is without-infty by MESFUNC5:def 3;

 now

 per cases by Lm8;

 suppose

 A4: ( sup ( rng f1)) in REAL ;



 A5: for y be UpperBound of ( rng f2) holds (c * ( sup ( rng f1))) <= y

 proof

 let y be UpperBound of ( rng f2);

 reconsider y as R_eal by XXREAL_0:def 1;

 per cases ;

 suppose

 A8: c = 0 ;

 [1, 1] in [: NAT , NAT :] by ZFMISC_1: 87;

 then (f2 . (1,1)) <= y by A6, FUNCT_1: 3, XXREAL_2:def 1;

 then (c * (f1 . (1,1))) <= y by A3;

 hence thesis by A8;

 end;

 suppose

 A10: c <> 0 ;

 now

 let x be ExtReal;

 assume x in ( rng f1);

 then

 consider z be object such that

 A11: z in ( dom f1) & x = (f1 . z) by FUNCT_1:def 3;

 consider n,m be object such that

 A12: n in NAT & m in NAT & z = [n, m] by A11, ZFMISC_1:def 2;

 reconsider n, m as Element of NAT by A12;



 A14: (f2 . (n,m)) in ( rng f2) by A11, A12, A6, FUNCT_1: 3;

 (f2 . (n,m)) = (c * (f1 . (n,m))) by A3;

 then ((c * (f1 . (n,m))) / c) <= (y / c) by A1, A10, A14, XXREAL_2:def 1, XXREAL_3: 79;

 hence x <= (y / c) by A10, A11, A12, XXREAL_3: 88;

 end;

 then

 B14: (y / c) is UpperBound of ( rng f1) by XXREAL_2:def 1;

 A15:

 now

 assume

 A16: y = -infty ;



 A17: [1, 1] in [: NAT , NAT :] by ZFMISC_1: 87;

 then (f2 . (1,1)) in ( rng f2) by A6, FUNCT_1: 3;

 then (f2 . (1,1)) = -infty by A16, XXREAL_0: 6, XXREAL_2:def 1;

 then

 A19: (c * (f1 . (1,1))) = -infty by A3;

 (f1 . (1,1)) <= ( sup ( rng f1)) by A6, A17, FUNCT_1: 3, XXREAL_2: 4;

 then

 A20: (f1 . (1,1)) < +infty by A4, XXREAL_0: 2, XXREAL_0: 9;



 A21: not -infty in ( rng f1) by MESFUNC5:def 3;

 (f1 . (1,1)) in ( rng f1) by A6, A17, FUNCT_1: 3;

 hence contradiction by A19, A20, A21, 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 SE1 = ( sup ( rng f1)) as Real by A4;

 (y / c) = (ry / c);

 then (SE1 * c) <= ry by A1, A10, B14, XXREAL_2:def 3, XREAL_1: 83;

 hence thesis by XXREAL_3:def 5;

 end;

 end;

 end;

 now

 let x be ExtReal;

 assume x in ( rng f2);

 then

 consider z2 be object such that

 A22: z2 in ( dom f2) & x = (f2 . z2) by FUNCT_1:def 3;

 consider n2,m2 be object such that

 A23: n2 in NAT & m2 in NAT & z2 = [n2, m2] by A22, ZFMISC_1:def 2;

 reconsider n2, m2 as Element of NAT by A23;



 A24: ( sup ( rng f1)) is UpperBound of ( rng f1) by XXREAL_2:def 3;

 [n2, m2] in ( dom f1) by A6, ZFMISC_1: 87;

 then

 A25: (f1 . (n2,m2)) <= ( sup ( rng f1)) by A24, XXREAL_2:def 1, FUNCT_1: 3;

 x = (f2 . (n2,m2)) by A22, A23;

 then x = (c * (f1 . (n2,m2))) by A3;

 hence x <= (c * ( sup ( rng f1))) by A1, A25, XXREAL_3: 71;

 end;

 then (c * ( sup ( rng f1))) is UpperBound of ( rng f2) by XXREAL_2:def 1;

 hence ( sup ( rng f2)) = (c * ( sup ( rng f1))) by A5, XXREAL_2:def 3;

 end;

 suppose

 A30: ( sup ( rng f1)) = +infty ;

 per cases ;

 suppose

 A27: c = 0 ;

 A28:

 now

 let n,m be Nat;

 (f2 . (n,m)) = (c * (f1 . (n,m))) by A3;

 hence (f2 . (n,m)) = 0 by A27;

 end;

 then ( P-lim f2) = ( sup ( rng f2)) by Th102;

 hence ( sup ( rng f2)) = (c * ( sup ( rng f1))) by A27, A28, Th102;

 end;

 suppose

 A29: c <> 0 ;

 A34:

 now

 let k be Real;

 reconsider k1 = k as Real;

 assume

 C5: 0 < k;

 then

 consider n,m be Nat such that

 A31: (k / c) < (f1 . (n,m)) by A1, A29, A30, Th101;



 C10: (f2 . (n,m)) = (c * (f1 . (n,m))) by A3;

 n in NAT & m in NAT by ORDINAL1:def 12;

 then [n, m] in ( dom f2) by A6, ZFMISC_1: 87;

 then

 C3: (f2 . (n,m)) <> -infty by C6, FUNCT_1: 3;

 now

 per cases by C3, XXREAL_0: 14;

 suppose

 C7: (f2 . (n,m)) in REAL ;

 (f1 . (n,m)) > 0 & 0 >= -infty by A1, C5, A31;

 then

 C9: (f1 . (n,m)) in REAL or (f1 . (n,m)) = +infty by XXREAL_0: 14;

 now

 assume (f1 . (n,m)) = +infty ;

 then (c * (f1 . (n,m))) = +infty by A1, A29, XXREAL_3:def 5;

 hence contradiction by A3, C7;

 end;

 then

 reconsider ES1 = (f1 . (n,m)) as Real by C9;

 k < (c * ES1) by A1, A29, A31, XREAL_1: 77;

 hence k < (f2 . (n,m)) by C10, XXREAL_3:def 5;

 end;

 suppose (f2 . (n,m)) = +infty ;

 hence k < (f2 . (n,m)) by XREAL_0:def 1, XXREAL_0: 9;

 end;

 end;

 hence ex n,m be Nat st not (f2 . (n,m)) <= k;

 end;

 (c * ( sup ( rng f1))) = +infty by A1, A29, A30, XXREAL_3:def 5;

 hence ( sup ( rng f2)) = (c * ( sup ( rng f1))) by C6a, A34, Th101;

 end;

 end;

 end;

 hence ( sup ( rng f2)) = (c * ( sup ( rng f1)));

 thus f2 is without-infty by C6, MESFUNC5:def 3;

 end;

 theorem :: DBLSEQ_3:105

 for f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be Function of [: NAT , NAT :], ExtREAL , c be Real st 0 <= c & (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f1 . (n1,m1)) <= (f1 . (n2,m2))) & (for n,m be Nat holds (f2 . (n,m)) = (c * (f1 . (n,m)))) holds (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f2 . (n1,m1)) <= (f2 . (n2,m2))) & f2 is without-infty & f2 is P-convergent & ( P-lim f2) = ( sup ( rng f2)) & ( P-lim f2) = (c * ( P-lim f1))

 proof

 let f1 be without-infty Function of [: NAT , NAT :], ExtREAL , f2 be Function of [: NAT , NAT :], ExtREAL , c be Real;

 assume that

 A1: 0 <= c and

 A2: for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds (f1 . (n1,m1)) <= (f1 . (n2,m2)) and

 A3: for n,m be Nat holds (f2 . (n,m)) = (c * (f1 . (n,m)));



 A5: ( sup ( rng f1)) = ( P-lim f1) by A2, Th96;

 now

 let n1,m1,n2,m2 be Nat;

 assume n1 <= n2 & m1 <= m2;

 then (c * (f1 . (n1,m1))) <= (c * (f1 . (n2,m2))) by A1, A2, XXREAL_3: 71;

 then (f2 . (n1,m1)) <= (c * (f1 . (n2,m2))) by A3;

 hence (f2 . (n1,m1)) <= (f2 . (n2,m2)) by A3;

 end;

 thus f2 is without-infty by A1, A3, Th104;

 thus f2 is P-convergent & ( P-lim f2) = ( sup ( rng f2)) by A6, Th96;

 ( sup ( rng f2)) = ( P-lim f2) by A6, Th96;

 hence thesis by A1, A3, A5, Th104;

 end;