mesfun12.miz



    begin

    definition

      let A,X be set, er be ExtReal;

      :: MESFUN12:def1

      func chi (er,A,X) -> Function of X, ExtREAL means

      : Def1: for x be object st x in X holds (x in A implies (it . x) = er) & ( not x in A implies (it . x) = 0 );

      existence

      proof

        defpred P[ object, object] means ($1 in A implies $2 = er) & ( not $1 in A implies $2 = 0 );

        

         A1: for x be object st x in X holds ex y be object st P[x, y]

        proof

          let x be object;

          assume x in X;

           not x in A implies ex y be object st y = {} & (x in A implies y = er) & ( not x in A implies y = {} );

          hence thesis;

        end;

        

         A2: for x,y1,y2 be object st x in X & P[x, y1] & P[x, y2] holds y1 = y2;

        consider f be Function such that

         A3: ( dom f) = X & for x be object st x in X holds P[x, (f . x)] from FUNCT_1:sch 2( A2, A1);

        for x be object st x in X holds (f . x) in ExtREAL

        proof

          let x be object;

          assume

           A4: x in X;

          per cases ;

            suppose x in A;

            then (f . x) = er by A3, A4;

            hence (f . x) in ExtREAL by XXREAL_0:def 1;

          end;

            suppose not x in A;

            then (f . x) = 0. by A3, A4;

            hence (f . x) in ExtREAL ;

          end;

        end;

        then

        reconsider f as Function of X, ExtREAL by A3, FUNCT_2: 3;

        take f;

        thus thesis by A3;

      end;

      uniqueness

      proof

        let f1,f2 be Function of X, ExtREAL such that

         A4: for x be object st x in X holds (x in A implies (f1 . x) = er) & ( not x in A implies (f1 . x) = 0 ) and

         A6: for x be object st x in X holds (x in A implies (f2 . x) = er) & ( not x in A implies (f2 . x) = 0 );

        for x be object st x in X holds (f1 . x) = (f2 . x)

        proof

          let x be object;

          assume

           A7: x in X;

          then

           A8: not x in A implies (f1 . x) = 0 & (f2 . x) = 0 by A4, A6;

          x in A implies (f1 . x) = er & (f2 . x) = er by A4, A6, A7;

          hence thesis by A8;

        end;

        hence thesis by FUNCT_2: 12;

      end;

    end

    theorem :: MESFUN12:1

    

     Th1: for X be non empty set, A be set, r be Real holds (r (#) ( chi (A,X))) = ( chi (r,A,X))

    proof

      let X be non empty set, A be set, r be Real;

      for x be Element of X holds ((r (#) ( chi (A,X))) . x) = (( chi (r,A,X)) . x)

      proof

        let x be Element of X;

        x in X;

        then x in ( dom (r (#) ( chi (A,X)))) by FUNCT_2:def 1;

        then

         A2: ((r (#) ( chi (A,X))) . x) = (r * (( chi (A,X)) . x)) by MESFUNC1:def 6;

        per cases ;

          suppose x in A;

          then (( chi (A,X)) . x) = 1 & (( chi (r,A,X)) . x) = r by Def1, FUNCT_3:def 3;

          hence ((r (#) ( chi (A,X))) . x) = (( chi (r,A,X)) . x) by A2, XXREAL_3: 81;

        end;

          suppose not x in A;

          then (( chi (A,X)) . x) = 0 & (( chi (r,A,X)) . x) = 0 by Def1, FUNCT_3:def 3;

          hence ((r (#) ( chi (A,X))) . x) = (( chi (r,A,X)) . x) by A2;

        end;

      end;

      hence thesis by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:2

    

     Th2: for X be non empty set, A be set holds ( chi ( +infty ,A,X)) = ( Xchi (A,X)) & ( chi ( -infty ,A,X)) = ( - ( Xchi (A,X)))

    proof

      let X be non empty set, A be set;

      for x be Element of X holds (( chi ( +infty ,A,X)) . x) = (( Xchi (A,X)) . x)

      proof

        let x be Element of X;

        per cases ;

          suppose x in A;

          then (( chi ( +infty ,A,X)) . x) = +infty & (( Xchi (A,X)) . x) = +infty by Def1, MEASUR10:def 7;

          hence (( chi ( +infty ,A,X)) . x) = (( Xchi (A,X)) . x);

        end;

          suppose not x in A;

          then (( chi ( +infty ,A,X)) . x) = 0 & (( Xchi (A,X)) . x) = 0 by Def1, MEASUR10:def 7;

          hence (( chi ( +infty ,A,X)) . x) = (( Xchi (A,X)) . x);

        end;

      end;

      hence ( chi ( +infty ,A,X)) = ( Xchi (A,X)) by FUNCT_2:def 8;

      for x be Element of X holds (( chi ( -infty ,A,X)) . x) = (( - ( Xchi (A,X))) . x)

      proof

        let x be Element of X;

        x in X;

        then

         A1: x in ( dom ( - ( Xchi (A,X)))) by FUNCT_2:def 1;

        then

         A2: (( - ( Xchi (A,X))) . x) = ( - (( Xchi (A,X)) . x)) by MESFUNC1:def 7;

        per cases ;

          suppose x in A;

          then (( chi ( -infty ,A,X)) . x) = -infty & (( Xchi (A,X)) . x) = +infty by Def1, MEASUR10:def 7;

          hence (( chi ( -infty ,A,X)) . x) = (( - ( Xchi (A,X))) . x) by A1, XXREAL_3: 6, MESFUNC1:def 7;

        end;

          suppose not x in A;

          then (( chi ( -infty ,A,X)) . x) = 0 & (( Xchi (A,X)) . x) = 0 by Def1, MEASUR10:def 7;

          hence (( chi ( -infty ,A,X)) . x) = (( - ( Xchi (A,X))) . x) by A2;

        end;

      end;

      hence ( chi ( -infty ,A,X)) = ( - ( Xchi (A,X))) by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:3

    

     Th3: for X,A be set holds ( chi (A,X)) is without+infty without-infty

    proof

      let X,A be set;

      ( rng ( chi (A,X))) c= { 0 , 1} by FUNCT_3: 39;

      then not +infty in ( rng ( chi (A,X))) & not -infty in ( rng ( chi (A,X)));

      hence ( chi (A,X)) is without+infty without-infty by MESFUNC5:def 3, MESFUNC5:def 4;

    end;

    theorem :: MESFUN12:4

    

     Th4: for X be non empty set, A be set, r be Real holds ( rng ( chi (r,A,X))) c= { 0 , r} & ( chi (r,A,X)) is without+infty without-infty

    proof

      let X be non empty set, A be set, r be Real;

      now

        let y be object;

        assume y in ( rng ( chi (r,A,X)));

        then

        consider x be object such that

         A1: x in ( dom ( chi (r,A,X))) & y = (( chi (r,A,X)) . x) by FUNCT_1:def 3;

        per cases ;

          suppose x in A;

          then (( chi (r,A,X)) . x) = r by A1, Def1;

          hence y in { 0 , r} by A1, TARSKI:def 2;

        end;

          suppose not x in A;

          then (( chi (r,A,X)) . x) = 0 by A1, Def1;

          hence y in { 0 , r} by A1, TARSKI:def 2;

        end;

      end;

      hence ( rng ( chi (r,A,X))) c= { 0 , r};

      ( chi (A,X)) is without+infty without-infty by Th3;

      then (r (#) ( chi (A,X))) is without+infty without-infty;

      hence ( chi (r,A,X)) is without+infty without-infty by Th1;

    end;

    theorem :: MESFUN12:5

    

     Th5: for X be non empty set, S be SigmaField of X, f be non empty PartFunc of X, ExtREAL , M be sigma_Measure of S st f is_simple_func_in S holds ex E be non empty Finite_Sep_Sequence of S, a be FinSequence of ExtREAL , r be FinSequence of REAL st (E,a) are_Re-presentation_of f & for n be Nat holds (a . n) = (r . n) & (f | (E . n)) = (( chi ((r . n),(E . n),X)) | (E . n)) & ((E . n) = {} implies (r . n) = 0 )

    proof

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

      assume

       A1: f is_simple_func_in S;

      then

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

       A2: (E,b) are_Re-presentation_of f by MESFUNC3: 12;

      

       A3: ( dom f) = ( union ( rng E)) & ( dom E) = ( dom b) & for n be Nat st n in ( dom E) holds for x be object st x in (E . n) holds (f . x) = (b . n) by A2, MESFUNC3:def 1;

      reconsider E as non empty Finite_Sep_Sequence of S by A3, ZFMISC_1: 2;

      

       A4: for n be Nat st (E . n) <> {} holds (b . n) in REAL

      proof

        let n be Nat;

        assume

         A5: (E . n) <> {} ;

        then

        consider x be object such that

         A6: x in (E . n) by XBOOLE_0:def 1;

        

         A7: n in ( dom E) by A5, FUNCT_1:def 2;

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

        then x in ( dom f) by A3, A6, TARSKI:def 4;

        then

         A8: (f . x) in ( rng f) by FUNCT_1: 3;

        ( rng f) is Subset of REAL by A1, MESFUNC2:def 4, MESFUNC2: 32;

        then (f . x) in REAL by A8;

        hence (b . n) in REAL by A2, A6, A7, MESFUNC3:def 1;

      end;

      defpred P1[ Nat, object] means ((E . $1) <> {} implies $2 = (b . $1)) & ((E . $1) = {} implies $2 = 0 );

      

       A9: for n be Nat st n in ( Seg ( len E)) holds ex a be Element of ExtREAL st P1[n, a]

      proof

        let n be Nat;

        assume n in ( Seg ( len E));

        per cases ;

          suppose

           A10: (E . n) <> {} ;

          take a = (b . n);

          thus P1[n, a] by A10;

        end;

          suppose

           A11: (E . n) = {} ;

          take a = 0. ;

          thus P1[n, a] by A11;

        end;

      end;

      consider a be FinSequence of ExtREAL such that

       A12: ( dom a) = ( Seg ( len E)) & for n be Nat st n in ( Seg ( len E)) holds P1[n, (a . n)] from FINSEQ_1:sch 5( A9);

      defpred P2[ Nat, object] means $2 = (a . $1);

      

       A13: for n be Nat st n in ( Seg ( len E)) holds ex r be Element of REAL st P2[n, r]

      proof

        let n be Nat;

        assume

         A14: n in ( Seg ( len E));

        per cases ;

          suppose

           A15: (E . n) <> {} ;

          then (a . n) = (b . n) by A12, A14;

          then

          reconsider r = (a . n) as Element of REAL by A4, A15;

          take r;

          thus P2[n, r];

        end;

          suppose (E . n) = {} ;

          then (a . n) = 0 by A12, A14;

          then

          reconsider r = (a . n) as Element of REAL by XREAL_0:def 1;

          take r;

          thus P2[n, r];

        end;

      end;

      consider r be FinSequence of REAL such that

       A16: ( dom r) = ( Seg ( len E)) & for n be Nat st n in ( Seg ( len E)) holds P2[n, (r . n)] from FINSEQ_1:sch 5( A13);

      take E, a, r;

      

       A17: ( dom a) = ( dom E) by A12, FINSEQ_1:def 3;

      

       A18: for n be Nat st n in ( dom E) holds for x be object st x in (E . n) holds (f . x) = (a . n)

      proof

        let n be Nat;

        assume

         A19: n in ( dom E);

        then

         A20: n in ( Seg ( len E)) by FINSEQ_1:def 3;

        let x be object;

        assume

         A21: x in (E . n);

        then (f . x) = (b . n) by A2, A19, MESFUNC3:def 1;

        hence (f . x) = (a . n) by A12, A20, A21;

      end;

      hence (E,a) are_Re-presentation_of f by A3, A17, MESFUNC3:def 1;

      thus for n be Nat holds (a . n) = (r . n) & (f | (E . n)) = (( chi ((r . n),(E . n),X)) | (E . n)) & ((E . n) = {} implies (r . n) = 0 )

      proof

        let n be Nat;

        per cases ;

          suppose

           A22: (E . n) <> {} ;

          then

           A23: n in ( dom E) by FUNCT_1:def 2;

          then n in ( Seg ( len E)) by FINSEQ_1:def 3;

          hence

           A24: (a . n) = (r . n) by A16;

          (E . n) c= ( dom f) by A3, A23, FUNCT_1: 3, ZFMISC_1: 74;

          then

           A27: ( dom (f | (E . n))) = (E . n) by RELAT_1: 62;

          ( dom ( chi ((r . n),(E . n),X))) = X by FUNCT_2:def 1;

          then

           A28: ( dom (( chi ((r . n),(E . n),X)) | (E . n))) = ( dom (f | (E . n))) by A27, RELAT_1: 62;

          for x be Element of X st x in ( dom (f | (E . n))) holds ((f | (E . n)) . x) = ((( chi ((r . n),(E . n),X)) | (E . n)) . x)

          proof

            let x be Element of X;

            assume

             A29: x in ( dom (f | (E . n)));

            

            then ((( chi ((r . n),(E . n),X)) | (E . n)) . x) = (( chi ((r . n),(E . n),X)) . x) by A27, FUNCT_1: 49

            .= (a . n) by A24, A27, A29, Def1

            .= (f . x) by A18, A23, A27, A29;

            hence ((f | (E . n)) . x) = ((( chi ((r . n),(E . n),X)) | (E . n)) . x) by A29, FUNCT_1: 47;

          end;

          hence thesis by A22, A28, PARTFUN1: 5;

        end;

          suppose

           z1: (E . n) = {} ;

          now

            per cases ;

              suppose n in ( dom E);

              then

               A30: n in ( Seg ( len E)) by FINSEQ_1:def 3;

              hence (a . n) = 0 by A12, z1;

              hence (r . n) = 0 by A30, A16;

            end;

              suppose

               A31: not n in ( dom E);

              hence (a . n) = 0 by A17, FUNCT_1:def 2;

               not n in ( Seg ( len E)) by A31, FINSEQ_1:def 3;

              hence (r . n) = 0 by A16, FUNCT_1:def 2;

            end;

          end;

          hence thesis by z1;

        end;

      end;

    end;

    definition

      let F be FinSequence-like Function;

      :: original: disjoint_valued

      redefine

      :: MESFUN12:def2

      attr F is disjoint_valued means

      : Def2: for m,n be Nat st m in ( dom F) & n in ( dom F) & m <> n holds (F . m) misses (F . n);

      compatibility

      proof

        thus F is disjoint_valued implies (for m,n be Nat st m in ( dom F) & n in ( dom F) & m <> n holds (F . m) misses (F . n)) by PROB_2:def 2;

        assume

         A1: for m,n be Nat st m in ( dom F) & n in ( dom F) & m <> n holds (F . m) misses (F . n);

        now

          let m,n be object;

          assume

           A2: m <> n;

          per cases ;

            suppose not m in ( dom F) or not n in ( dom F);

            then (F . m) = {} or (F . n) = {} by FUNCT_1:def 2;

            hence (F . m) misses (F . n);

          end;

            suppose m in ( dom F) & n in ( dom F);

            hence (F . m) misses (F . n) by A1, A2;

          end;

        end;

        hence F is disjoint_valued by PROB_2:def 2;

      end;

    end

    theorem :: MESFUN12:6

    

     Th6: for X be non empty set, S be SigmaField of X, E1,E2 be Element of S st E1 misses E2 holds <*E1, E2*> is Finite_Sep_Sequence of S

    proof

      let X be non empty set, S be SigmaField of X, E1,E2 be Element of S;

      assume

       A0: E1 misses E2;

      

       A2: ( dom <*E1, E2*>) = {1, 2} by FINSEQ_1: 92;

      now

        let m,n be object;

        assume

         A3: m <> n;

        per cases ;

          suppose m in ( dom <*E1, E2*>) & n in ( dom <*E1, E2*>);

          then (m = 1 or m = 2) & (n = 1 or n = 2) by A2, TARSKI:def 2;

          then (( <*E1, E2*> . m) = E1 & ( <*E1, E2*> . n) = E2) or (( <*E1, E2*> . m) = E2 & ( <*E1, E2*> . n) = E1) by A3, FINSEQ_1: 44;

          hence ( <*E1, E2*> . m) misses ( <*E1, E2*> . n) by A0;

        end;

          suppose not m in ( dom <*E1, E2*>) or not n in ( dom <*E1, E2*>);

          then ( <*E1, E2*> . m) = {} or ( <*E1, E2*> . n) = {} by FUNCT_1:def 2;

          hence ( <*E1, E2*> . m) misses ( <*E1, E2*> . n);

        end;

      end;

      then <*E1, E2*> is disjoint_valued;

      hence <*E1, E2*> is Finite_Sep_Sequence of S;

    end;

    theorem :: MESFUN12:7

    

     Th7: for X be non empty set, A1,A2 be Subset of X, r1,r2 be Real holds <*( chi (r1,A1,X)), ( chi (r2,A2,X))*> is summable FinSequence of ( Funcs (X, ExtREAL ))

    proof

      let X be non empty set, A1,A2 be Subset of X, r1,r2 be Real;

      reconsider f1 = ( chi (r1,A1,X)), f2 = ( chi (r2,A2,X)) as Element of ( Funcs (X, ExtREAL )) by FUNCT_2: 8;

      reconsider F = <*f1, f2*> as FinSequence of ( Funcs (X, ExtREAL ));

      

       A1: f1 is without+infty without-infty & f2 is without+infty without-infty by Th4;

      

       A2: ( dom F) = {1, 2} by FINSEQ_1: 92;

      now

        let n be Nat;

        assume n in ( dom F);

        then n = 1 or n = 2 by A2, TARSKI:def 2;

        hence (F . n) is without-infty by A1, FINSEQ_1: 44;

      end;

      then F is without_-infty-valued;

      hence <*( chi (r1,A1,X)), ( chi (r2,A2,X))*> is summable FinSequence of ( Funcs (X, ExtREAL ));

    end;

    theorem :: MESFUN12:8

    

     Th8: for X be non empty set, F be summable FinSequence of ( Funcs (X, ExtREAL )) st ( len F) >= 2 holds (( Partial_Sums F) /. 2) = ((F /. 1) + (F /. 2))

    proof

      let X be non empty set, F be summable FinSequence of ( Funcs (X, ExtREAL ));

      assume

       A1: ( len F) >= 2;

      then (1 + 1) <= ( len F);

      then

       A3: 1 < ( len F) by NAT_1: 13;

      then

       A6: 1 in ( dom F) & 2 in ( dom F) by A1, FINSEQ_3: 25;

      ( len F) = ( len ( Partial_Sums F)) by MEASUR11:def 11;

      then

       A5: 1 in ( dom ( Partial_Sums F)) & 2 in ( dom ( Partial_Sums F)) by A1, A3, FINSEQ_3: 25;

      

      then

       A4: (( Partial_Sums F) /. 1) = (( Partial_Sums F) . 1) by PARTFUN1:def 6

      .= (F . 1) by MEASUR11:def 11

      .= (F /. 1) by A6, PARTFUN1:def 6;

      (( Partial_Sums F) . (1 + 1)) = ((( Partial_Sums F) /. 1) + (F /. (1 + 1))) by A1, NAT_1: 13, MEASUR11:def 11;

      hence (( Partial_Sums F) /. 2) = ((F /. 1) + (F /. 2)) by A4, A5, PARTFUN1:def 6;

    end;

    theorem :: MESFUN12:9

    

     Th9: for X be non empty set, f be Function of X, ExtREAL holds (f + (X --> 0. )) = f

    proof

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

      ( dom f) = X by FUNCT_2:def 1;

      hence thesis by MESFUN11: 27;

    end;

    theorem :: MESFUN12:10

    

     Th10: for X be non empty set, F be summable FinSequence of ( Funcs (X, ExtREAL )) holds ( dom F) = ( dom ( Partial_Sums F)) & (for n be Nat st n in ( dom F) holds (( Partial_Sums F) /. n) = (( Partial_Sums F) . n)) & (for n be Nat, x be Element of X st 1 <= n < ( len F) holds ((( Partial_Sums F) /. (n + 1)) . x) = (((( Partial_Sums F) /. n) . x) + ((F /. (n + 1)) . x)))

    proof

      let X be non empty set, F be summable FinSequence of ( Funcs (X, ExtREAL ));

      ( len F) = ( len ( Partial_Sums F)) by MEASUR11:def 11;

      hence

       A1: ( dom F) = ( dom ( Partial_Sums F)) by FINSEQ_3: 29;

      hence for n be Nat st n in ( dom F) holds (( Partial_Sums F) /. n) = (( Partial_Sums F) . n) by PARTFUN1:def 6;

      thus for n be Nat, x be Element of X st 1 <= n < ( len F) holds ((( Partial_Sums F) /. (n + 1)) . x) = (((( Partial_Sums F) /. n) . x) + ((F /. (n + 1)) . x))

      proof

        let n be Nat, x be Element of X;

        assume

         A3: 1 <= n < ( len F);

        then 1 <= (n + 1) <= ( len F) by NAT_1: 13;

        

        then

         A4: (( Partial_Sums F) /. (n + 1)) = (( Partial_Sums F) . (n + 1)) by A1, PARTFUN1:def 6, FINSEQ_3: 25

        .= ((( Partial_Sums F) /. n) + (F /. (n + 1))) by A3, MEASUR11:def 11;

        ( dom (( Partial_Sums F) /. (n + 1))) = X by FUNCT_2:def 1;

        hence ((( Partial_Sums F) /. (n + 1)) . x) = (((( Partial_Sums F) /. n) . x) + ((F /. (n + 1)) . x)) by A4, MESFUNC1:def 3;

      end;

    end;

    theorem :: MESFUN12:11

    

     Th11: for X be non empty set, S be SigmaField of X, f be Function of X, ExtREAL , E be Finite_Sep_Sequence of S, F be summable FinSequence of ( Funcs (X, ExtREAL )) st ( dom E) = ( dom F) & ( dom f) = ( union ( rng E)) & (for n be Nat st n in ( dom F) holds ex r be Real st (F /. n) = (r (#) ( chi ((E . n),X)))) & f = (( Partial_Sums F) /. ( len F)) holds (for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & m <> n holds ((F /. n) . x) = 0 ) & (for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & n < m holds ((( Partial_Sums F) /. n) . x) = 0 ) & (for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & n >= m holds ((( Partial_Sums F) /. n) . x) = (f . x)) & (for x be Element of X, m be Nat st m in ( dom F) & x in (E . m) holds ((F /. m) . x) = (f . x)) & f is_simple_func_in S

    proof

      let X be non empty set, S be SigmaField of X, f be Function of X, ExtREAL , E be Finite_Sep_Sequence of S, F be summable FinSequence of ( Funcs (X, ExtREAL ));

      assume that

       A1: ( dom E) = ( dom F) and

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

       A3: for n be Nat st n in ( dom F) holds ex r be Real st (F /. n) = (r (#) ( chi ((E . n),X))) and

       A4: f = (( Partial_Sums F) /. ( len F));

      E <> {} by A2, ZFMISC_1: 2;

      then 1 <= ( len E) by FINSEQ_1: 20;

      then 1 <= ( len F) by A1, FINSEQ_3: 29;

      then

       A5: ( len F) in ( dom F) by FINSEQ_3: 25;

      thus

       A6: for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & m <> n holds ((F /. n) . x) = 0

      proof

        let x be Element of X, m,n be Nat;

        assume

         A7: m in ( dom F) & n in ( dom F) & x in (E . m);

        then

        consider rn be Real such that

         A8: (F /. n) = (rn (#) ( chi ((E . n),X))) by A3;

        ( dom (F /. n)) = X by FUNCT_2:def 1;

        then

         A9: ((F /. n) . x) = (rn * (( chi ((E . n),X)) . x)) by A8, MESFUNC1:def 6;

        thus m <> n implies ((F /. n) . x) = 0

        proof

          assume m <> n;

          then not x in (E . n) by A7, XBOOLE_0: 3, PROB_2:def 2;

          then (( chi ((E . n),X)) . x) = 0 by FUNCT_3:def 3;

          hence ((F /. n) . x) = 0 by A9;

        end;

      end;

      thus

       A10: for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & n < m holds ((( Partial_Sums F) /. n) . x) = 0

      proof

        let x be Element of X, m,n be Nat;

        assume

         A11: m in ( dom F) & n in ( dom F) & x in (E . m) & n < m;

        defpred P[ Nat] means $1 in ( dom F) & $1 < m implies ((( Partial_Sums F) /. $1) . x) = 0 ;

        

         A12: P[ 0 ] by FINSEQ_3: 25;

        

         A13: for k be Nat st P[k] holds P[(k + 1)]

        proof

          let k be Nat;

          assume

           A14: P[k];

          assume

           A15: (k + 1) in ( dom F) & (k + 1) < m;

          then

           A16: ((F /. (k + 1)) . x) = 0 by A6, A11;

          per cases ;

            suppose

             A17: (k + 1) = 1;

            (( Partial_Sums F) /. (k + 1)) = (( Partial_Sums F) . (k + 1)) by A15, Th10

            .= (F . (k + 1)) by A17, MEASUR11:def 11

            .= (F /. (k + 1)) by A15, PARTFUN1:def 6;

            hence ((( Partial_Sums F) /. (k + 1)) . x) = 0 by A6, A11, A15;

          end;

            suppose

             A18: (k + 1) <> 1;

            1 <= (k + 1) <= ( len F) by A15, FINSEQ_3: 25;

            then 1 < (k + 1) <= ( len F) by A18, XXREAL_0: 1;

            then 1 <= k < ( len F) by NAT_1: 13;

            then ((( Partial_Sums F) /. (k + 1)) . x) = ( 0 + ((F /. (k + 1)) . x)) by A14, A15, NAT_1: 13, FINSEQ_3: 25, Th10;

            hence ((( Partial_Sums F) /. (k + 1)) . x) = 0 by A16;

          end;

        end;

        for k be Nat holds P[k] from NAT_1:sch 2( A12, A13);

        hence ((( Partial_Sums F) /. n) . x) = 0 by A11;

      end;

      thus

       A17: for x be Element of X, m,n be Nat st m in ( dom F) & n in ( dom F) & x in (E . m) & n >= m holds ((( Partial_Sums F) /. n) . x) = (f . x)

      proof

        let x be Element of X, m,n be Nat;

        assume

         A18: m in ( dom F) & n in ( dom F) & x in (E . m) & n >= m;

        then

         A24: 1 <= m by FINSEQ_3: 25;

        defpred P[ Nat] means $1 in ( dom F) & $1 >= m implies ((( Partial_Sums F) /. $1) . x) = ((F /. m) . x);

        

         A19: P[ 0 ] by FINSEQ_3: 25;

        

         A20: for k be Nat st P[k] holds P[(k + 1)]

        proof

          let k be Nat;

          assume

           A21: P[k];

          assume

           A22: (k + 1) in ( dom F) & (k + 1) >= m;

          per cases ;

            suppose

             A23: (k + 1) = 1;

            (( Partial_Sums F) /. (k + 1)) = (( Partial_Sums F) . (k + 1)) by A22, Th10

            .= (F . (k + 1)) by A23, MEASUR11:def 11

            .= (F /. (k + 1)) by A22, PARTFUN1:def 6;

            hence ((( Partial_Sums F) /. (k + 1)) . x) = ((F /. m) . x) by A22, A23, A24, XXREAL_0: 1;

          end;

            suppose

             A25: (k + 1) <> 1;

            1 <= (k + 1) <= ( len F) by A22, FINSEQ_3: 25;

            then 1 < (k + 1) <= ( len F) by A25, XXREAL_0: 1;

            then

             A26: 1 <= k < ( len F) by NAT_1: 13;

            then

             A27: k in ( dom F) by FINSEQ_3: 25;

            per cases ;

              suppose

               A28: (k + 1) = m;

              then k < m by NAT_1: 13;

              then ((( Partial_Sums F) /. k) . x) = 0 by A10, A18, A27;

              then ((( Partial_Sums F) /. (k + 1)) . x) = ( 0 + ((F /. (k + 1)) . x)) by A26, Th10;

              hence ((( Partial_Sums F) /. (k + 1)) . x) = ((F /. m) . x) by A28, XXREAL_3: 4;

            end;

              suppose

               A29: (k + 1) <> m;

              then m < (k + 1) by A22, XXREAL_0: 1;

              

              then ((( Partial_Sums F) /. (k + 1)) . x) = (((F /. m) . x) + ((F /. (k + 1)) . x)) by A21, A26, FINSEQ_3: 25, NAT_1: 13, Th10

              .= (((F /. m) . x) + 0 ) by A6, A18, A22, A29;

              hence ((( Partial_Sums F) /. (k + 1)) . x) = ((F /. m) . x) by XXREAL_3: 4;

            end;

          end;

        end;

        

         A30: for k be Nat holds P[k] from NAT_1:sch 2( A19, A20);

        then ((( Partial_Sums F) /. n) . x) = ((F /. m) . x) by A18;

        hence ((( Partial_Sums F) /. n) . x) = (f . x) by A4, A5, A18, A30, FINSEQ_3: 25;

      end;

      thus

       A31: for x be Element of X, m be Nat st m in ( dom F) & x in (E . m) holds ((F /. m) . x) = (f . x)

      proof

        let x be Element of X, m be Nat;

        assume

         A32: m in ( dom F) & x in (E . m);

        then

         A33: 1 <= m <= ( len F) by FINSEQ_3: 25;

        

         A34: ((( Partial_Sums F) /. m) . x) = (f . x) by A17, A32;

        per cases ;

          suppose m = 1;

          then (( Partial_Sums F) . m) = (F . m) by MEASUR11:def 11;

          then (( Partial_Sums F) /. m) = (F . m) by A32, Th10;

          hence ((F /. m) . x) = (f . x) by A32, A34, PARTFUN1:def 6;

        end;

          suppose m <> 1;

          then

           A35: m > 1 by A33, XXREAL_0: 1;

          reconsider m1 = (m - 1) as Nat by A33;

          

           A36: m = (m1 + 1);

          then

           A37: 1 <= m1 < ( len F) by A33, A35, NAT_1: 13;

          then m1 in ( dom F) & m1 < m by A36, NAT_1: 19, FINSEQ_3: 25;

          then ((( Partial_Sums F) /. m1) . x) = 0 by A10, A32;

          then (f . x) = ( 0 + ((F /. (m1 + 1)) . x)) by A34, A37, Th10;

          hence ((F /. m) . x) = (f . x) by XXREAL_3: 4;

        end;

      end;

      

       A38: for x be Element of X st x in ( dom f) holds |.(f . x).| < +infty

      proof

        let x be Element of X;

        assume x in ( dom f);

        then

        consider A be set such that

         A39: x in A & A in ( rng E) by A2, TARSKI:def 4;

        consider k be object such that

         A40: k in ( dom E) & A = (E . k) by A39, FUNCT_1:def 3;

        reconsider k as Nat by A40;

        consider r be Real such that

         A41: (F /. k) = (r (#) ( chi ((E . k),X))) by A1, A3, A40;

        ( dom ( chi ((E . k),X))) = X by FUNCT_2:def 1;

        then x in ( dom ( chi ((E . k),X)));

        then

         A42: x in ( dom (r (#) ( chi ((E . k),X)))) by MESFUNC1:def 6;

        

         A43: (( chi ((E . k),X)) . x) = 1 by A39, A40, FUNCT_3:def 3;

        (f . x) = ((r (#) ( chi ((E . k),X))) . x) by A31, A39, A1, A40, A41;

        then (f . x) = (r * (( chi ((E . k),X)) . x)) by A42, MESFUNC1:def 6;

        hence |.(f . x).| < +infty by A43, EXTREAL1: 41, XREAL_0:def 1;

      end;

      for n be Nat, x,y be Element of X st n in ( dom E) & x in (E . n) & y in (E . n) holds (f . x) = (f . y)

      proof

        let n be Nat, x,y be Element of X;

        assume

         A44: n in ( dom E) & x in (E . n) & y in (E . n);

        then

        consider r be Real such that

         A45: (F /. n) = (r (#) ( chi ((E . n),X))) by A3, A1;

        ( dom ( chi ((E . n),X))) = X by FUNCT_2:def 1;

        then x in ( dom ( chi ((E . n),X))) & y in ( dom ( chi ((E . n),X)));

        then

         A46: x in ( dom (r (#) ( chi ((E . n),X)))) & y in ( dom (r (#) ( chi ((E . n),X)))) by MESFUNC1:def 6;

        

         A47: (( chi ((E . n),X)) . x) = 1 & (( chi ((E . n),X)) . y) = 1 by A44, FUNCT_3:def 3;

        ((F /. n) . x) = (r * (( chi ((E . n),X)) . x)) & ((F /. n) . y) = (r * (( chi ((E . n),X)) . y)) by A45, A46, MESFUNC1:def 6;

        then ((F /. n) . x) = r & ((F /. n) . y) = r by A47, XXREAL_3: 81;

        then (f . x) = r & (f . y) = r by A1, A31, A44;

        hence thesis;

      end;

      hence f is_simple_func_in S by A2, A38, MESFUNC2:def 1, MESFUNC2:def 4;

    end;

    theorem :: MESFUN12:12

    

     Th12: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S holds ( chi (E,X)) is_simple_func_in S

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S;

      X in S by MEASURE1: 7;

      then

      reconsider E2 = (X \ E) as Element of S by MEASURE1: 6;

      E misses E2 by XBOOLE_1: 79;

      then

      reconsider EE = <*E, E2*> as Finite_Sep_Sequence of S by Th6;

      (1 (#) ( chi (E,X))) = ( chi (1,E,X)) & ( 0 (#) ( chi (E2,X))) = ( chi ( 0 ,E2,X)) by Th1;

      then

      reconsider F = <*(1 (#) ( chi (E,X))), ( 0 (#) ( chi (E2,X)))*> as summable FinSequence of ( Funcs (X, ExtREAL )) by Th7;

      

       A1: ( dom EE) = {1, 2} & ( dom F) = {1, 2} by FINSEQ_1: 92;

      ( rng EE) = (( rng <*E*>) \/ ( rng <*E2*>)) by FINSEQ_1: 31;

      then ( rng EE) = ( {E} \/ ( rng <*E2*>)) by FINSEQ_1: 38;

      then ( rng EE) = ( {E} \/ {E2}) by FINSEQ_1: 38;

      then ( rng EE) = {E, E2} by ENUMSET1: 1;

      then ( union ( rng EE)) = (E \/ E2) by ZFMISC_1: 75;

      then ( union ( rng EE)) = (E \/ X) by XBOOLE_1: 39;

      then ( union ( rng EE)) = X by XBOOLE_1: 12;

      then

       A2: ( dom ( chi (E,X))) = ( union ( rng EE)) by FUNCT_2:def 1;

      

       A3: for n be Nat st n in ( dom F) holds ex r be Real st (F /. n) = (r (#) ( chi ((EE . n),X)))

      proof

        let n be Nat;

        assume

         A4: n in ( dom F);

        per cases by A1, A4, TARSKI:def 2;

          suppose n = 1;

          then (F . n) = (1 (#) ( chi (E,X))) & (EE . n) = E by FINSEQ_1: 44;

          hence ex r be Real st (F /. n) = (r (#) ( chi ((EE . n),X))) by A4, PARTFUN1:def 6;

        end;

          suppose n = 2;

          then (F . n) = ( 0 (#) ( chi (E2,X))) & (EE . n) = E2 by FINSEQ_1: 44;

          hence ex r be Real st (F /. n) = (r (#) ( chi ((EE . n),X))) by A4, PARTFUN1:def 6;

        end;

      end;

      1 in ( dom F) & 2 in ( dom F) by A1, TARSKI:def 2;

      then (F /. 1) = (F . 1) & (F /. 2) = (F . 2) by PARTFUN1:def 6;

      then (F /. 1) = (1 (#) ( chi (E,X))) & (F /. 2) = ( 0 (#) ( chi (E2,X))) by FINSEQ_1: 44;

      then

       A4: (F /. 1) = ( chi (E,X)) & (F /. 2) = (X --> 0 ) by MESFUNC2: 1, MESFUN11: 22;

      ( len F) = 2 by FINSEQ_1: 44;

      then (( Partial_Sums F) /. ( len F)) = ((F /. 1) + (F /. 2)) by Th8;

      then (( Partial_Sums F) /. ( len F)) = ( chi (E,X)) by A4, Th9;

      hence ( chi (E,X)) is_simple_func_in S by A1, A2, A3, Th11;

    end;

    theorem :: MESFUN12:13

    

     Th13: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A,B be Element of S, er be ExtReal holds ( chi (er,A,X)) is B -measurable

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A,B be Element of S, er be ExtReal;

      

       a1: ( Xchi (A,X)) is B -measurable by MEASUR10: 32;

      

       a2: ( dom ( Xchi (A,X))) = X by FUNCT_2:def 1;

      per cases ;

        suppose er = +infty ;

        hence ( chi (er,A,X)) is B -measurable by a1, Th2;

      end;

        suppose er = -infty ;

        then

         W: ( chi (er,A,X)) = ( - ( Xchi (A,X))) by Th2;

        ( Xchi (A,X)) is B -measurable by MEASUR10: 32;

        then ( - ( Xchi (A,X))) is B -measurable by a2, MEASUR11: 63;

        hence ( chi (er,A,X)) is B -measurable by W;

      end;

        suppose er <> +infty & er <> -infty ;

        then er in REAL by XXREAL_0: 14;

        then

        reconsider r = er as Real;

        

         a3: ( chi (er,A,X)) = (r (#) ( chi (A,X))) by Th1;

        ( dom ( chi (A,X))) = X by FUNCT_3:def 3;

        hence ( chi (er,A,X)) is B -measurable by a3, MESFUNC1: 37, MESFUNC2: 29;

      end;

    end;

    theorem :: MESFUN12:14

    

     Th14: for X be set, A1,A2 be Subset of X, er be ExtReal holds (( chi (er,A1,X)) | A2) = (( chi (er,(A1 /\ A2),X)) | A2)

    proof

      let X be set, A1,A2 be Subset of X, er be ExtReal;

      

       a1: ( dom (( chi (er,A1,X)) | A2)) = (( dom ( chi (er,A1,X))) /\ A2) by RELAT_1: 61

      .= (X /\ A2) by FUNCT_2:def 1;

      

       a2: ( dom (( chi (er,(A1 /\ A2),X)) | A2)) = (( dom ( chi (er,(A1 /\ A2),X))) /\ A2) by RELAT_1: 61

      .= ( dom (( chi (er,A1,X)) | A2)) by a1, FUNCT_2:def 1;

      now

        let x be Element of X;

        assume

         b1: x in ( dom (( chi (er,A1,X)) | A2));

        then

         a3: x in X & x in A2 by a1, XBOOLE_0:def 4;

        then

         a4: ((( chi (er,A1,X)) | A2) . x) = (( chi (er,A1,X)) . x) & ((( chi (er,(A1 /\ A2),X)) | A2) . x) = (( chi (er,(A1 /\ A2),X)) . x) by FUNCT_1: 49;

        per cases ;

          suppose

           a5: x in A1;

          then

           a6: ((( chi (er,A1,X)) | A2) . x) = er by a4, Def1;

          x in (A1 /\ A2) by a3, a5, XBOOLE_0:def 4;

          hence ((( chi (er,(A1 /\ A2),X)) | A2) . x) = ((( chi (er,A1,X)) | A2) . x) by a4, a6, Def1;

        end;

          suppose

           a7: not x in A1;

          then

           a8: ((( chi (er,A1,X)) | A2) . x) = 0 by a4, Def1, b1;

           not x in (A1 /\ A2) by a7, XBOOLE_0:def 4;

          hence ((( chi (er,(A1 /\ A2),X)) | A2) . x) = ((( chi (er,A1,X)) | A2) . x) by b1, a4, a8, Def1;

        end;

      end;

      hence (( chi (er,A1,X)) | A2) = (( chi (er,(A1 /\ A2),X)) | A2) by a2, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:15

    

     Th15: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A,B,C be Element of S, er be ExtReal st C c= B holds (( chi (er,A,X)) | B) is C -measurable

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A,B,C be Element of S, er be ExtReal;

      assume

       a1: C c= B;

      ( dom ( chi (er,A,X))) = X by FUNCT_2:def 1;

      then

       B: B = (( dom ( chi (er,A,X))) /\ B) by XBOOLE_1: 28;

      ( chi (er,A,X)) is B -measurable by Th13;

      then (( chi (er,A,X)) | B) is B -measurable by MESFUNC5: 42, B;

      hence thesis by a1, MESFUNC1: 30;

    end;

    theorem :: MESFUN12:16

    

     Th16: for X be set, A1,A2 be Subset of X, er be ExtReal, x be object st A1 misses A2 holds ((( chi (er,A1,X)) | A2) . x) = 0

    proof

      let X be set, A1,A2 be Subset of X, er be ExtReal, x be object;

      assume

       a1: A1 misses A2;

      per cases ;

        suppose

         a2: x in ( dom (( chi (er,A1,X)) | A2));

        then x in (( dom ( chi (er,A1,X))) /\ A2) by RELAT_1: 61;

        then x in X & x in A2 by XBOOLE_0:def 4;

        then not x in A1 by a1, XBOOLE_0: 3;

        then (( chi (er,A1,X)) . x) = 0 by a2, Def1;

        hence ((( chi (er,A1,X)) | A2) . x) = 0 by a2, FUNCT_1: 47;

      end;

        suppose not x in ( dom (( chi (er,A1,X)) | A2));

        hence ((( chi (er,A1,X)) | A2) . x) = 0 by FUNCT_1:def 2;

      end;

    end;

    theorem :: MESFUN12:17

    

     Th17: for X be set, A be Subset of X, er be ExtReal holds (er >= 0 implies ( chi (er,A,X)) is nonnegative) & (er <= 0 implies ( chi (er,A,X)) is nonpositive)

    proof

      let X be set, A be Subset of X, er be ExtReal;

      hereby

        assume

         a1: er >= 0 ;

        now

          let x be object;

          assume

           a2: x in ( dom ( chi (er,A,X)));

          x in A implies (( chi (er,A,X)) . x) >= 0 by a1, Def1;

          hence (( chi (er,A,X)) . x) >= 0 by a2, Def1;

        end;

        hence ( chi (er,A,X)) is nonnegative by SUPINF_2: 52;

      end;

      assume

       a3: er <= 0 ;

      now

        let x be set;

        assume

         a4: x in ( dom ( chi (er,A,X)));

        x in A implies (( chi (er,A,X)) . x) <= 0 by a3, Def1;

        hence (( chi (er,A,X)) . x) <= 0 by a4, Def1;

      end;

      hence ( chi (er,A,X)) is nonpositive by MESFUNC5: 9;

    end;

    theorem :: MESFUN12:18

    

     Th18: for A,X be set, B be Subset of X holds ( dom (( chi (A,X)) | B)) = B

    proof

      let A,X be set, B be Subset of X;

      ( dom (( chi (A,X)) | B)) = (( dom ( chi (A,X))) /\ B) by RELAT_1: 61

      .= (X /\ B) by FUNCT_2:def 1;

      hence thesis by XBOOLE_1: 28;

    end;

    begin

    theorem :: MESFUN12:19

    

     Th19: for X be non empty set, S be SigmaField of X, f be PartFunc of X, ExtREAL st f is empty holds f is_simple_func_in S

    proof

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

      reconsider EMP = {} as Element of S by MEASURE1: 7;

      reconsider F = <*EMP*> as Finite_Sep_Sequence of S;

      assume

       A1: f is empty;

      then ( dom f) = {} & ( rng F) = {EMP} by FINSEQ_1: 38;

      then

       A2: ( dom f) = ( union ( rng F)) by ZFMISC_1: 25;

      for n be Nat, x,y be Element of X st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y)

      proof

        let n be Nat, x,y be Element of X;

        assume

         A3: n in ( dom F) & x in (F . n) & y in (F . n);

        then n in {1} by FINSEQ_1: 2, FINSEQ_1: 38;

        then n = 1 by TARSKI:def 1;

        hence (f . x) = (f . y) by A3, FINSEQ_1: 40;

      end;

      hence f is_simple_func_in S by A1, A2, MESFUNC2:def 4;

    end;

    theorem :: MESFUN12:20

    

     Th20: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S holds ( Integral (M,(( chi (E1,X)) | E2))) = (M . (E1 /\ E2))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S;

      reconsider XX = X as Element of S by MEASURE1: 7;

      

       A1: E2 = ((E1 /\ E2) \/ (E2 \ E1)) by XBOOLE_1: 51;

      set F = (E2 \ E1);

      

       A2: ( dom (( chi (E1,X)) | (E1 /\ E2))) = (( dom ( chi (E1,X))) /\ (E1 /\ E2)) by RELAT_1: 61

      .= (X /\ (E1 /\ E2)) by FUNCT_3:def 3;

      

       A3: ( dom (( chi ((E1 /\ E2),X)) | (E1 /\ E2))) = (( dom ( chi ((E1 /\ E2),X))) /\ (E1 /\ E2)) by RELAT_1: 61

      .= (X /\ (E1 /\ E2)) by FUNCT_3:def 3;

      now

        let x be Element of X;

        assume

         A4: x in ( dom (( chi (E1,X)) | (E1 /\ E2)));

        then

         A5: ((( chi ((E1 /\ E2),X)) | (E1 /\ E2)) . x) = (( chi ((E1 /\ E2),X)) . x) by A2, A3, FUNCT_1: 47;

        

         A6: x in (E1 /\ E2) by A2, A4, XBOOLE_0:def 4;

        then

         A7: x in E1 by XBOOLE_0:def 4;

        ((( chi (E1,X)) | (E1 /\ E2)) . x) = (( chi (E1,X)) . x) by A4, FUNCT_1: 47

        .= 1 by A7, FUNCT_3:def 3;

        hence ((( chi (E1,X)) | (E1 /\ E2)) . x) = ((( chi ((E1 /\ E2),X)) | (E1 /\ E2)) . x) by A6, A5, FUNCT_3:def 3;

      end;

      then (( chi (E1,X)) | (E1 /\ E2)) = (( chi ((E1 /\ E2),X)) | (E1 /\ E2)) by A2, A3, PARTFUN1: 5;

      then

       A9: ( Integral (M,(( chi (E1,X)) | (E1 /\ E2)))) = (M . (E1 /\ E2)) by MESFUNC9: 14;

      

       A10: XX = ( dom ( chi (E1,X))) by FUNCT_3:def 3;

      then

       A11: F = ( dom (( chi (E1,X)) | (E2 \ E1))) by RELAT_1: 62;

      then F = (( dom ( chi (E1,X))) /\ F) by RELAT_1: 61;

      then

       A12: (( chi (E1,X)) | (E2 \ E1)) is F -measurable by MESFUNC2: 29, MESFUNC5: 42;

      now

        let x be Element of X;

        assume

         A15: x in ( dom (( chi (E1,X)) | (E2 \ E1)));

        (E2 \ E1) c= (X \ E1) by XBOOLE_1: 33;

        then (( chi (E1,X)) . x) = 0 by A11, A15, FUNCT_3: 37;

        hence 0 = ((( chi (E1,X)) | (E2 \ E1)) . x) by A15, FUNCT_1: 47;

      end;

      then ( integral+ (M,(( chi (E1,X)) | (E2 \ E1)))) = 0 by A11, A12, MESFUNC5: 87;

      then

       A16: ( Integral (M,(( chi (E1,X)) | (E2 \ E1)))) = 0. by A11, A12, MESFUNC5: 15, MESFUNC5: 88;

      ( chi (E1,X)) is XX -measurable by MESFUNC2: 29;

      then ( Integral (M,(( chi (E1,X)) | E2))) = (( Integral (M,(( chi (E1,X)) | (E1 /\ E2)))) + ( Integral (M,(( chi (E1,X)) | (E2 \ E1))))) by A10, A1, MESFUNC5: 91, XBOOLE_1: 89;

      hence thesis by A9, A16, XXREAL_3: 4;

    end;

    theorem :: MESFUN12:21

    

     Th21: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S, f,g be PartFunc of X, ExtREAL st E1 = ( dom f) & f is nonnegative & f is E1 -measurable & E2 = ( dom g) & g is nonnegative & g is E2 -measurable holds ( Integral (M,(f + g))) = (( Integral (M,(f | ( dom (f + g))))) + ( Integral (M,(g | ( dom (f + g))))))

    proof

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

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonnegative and

       A6: g is B -measurable;

      set f1 = (f | (A /\ B)), g1 = (g | (A /\ B));

      

       A7: ( dom (f + g)) = (A /\ B) by A1, A2, A4, A5, MESFUNC5: 22;

      

       A8: ( dom f1) = (A /\ B) & ( dom g1) = (A /\ B) & (( dom f) /\ (A /\ B)) = (A /\ B) & (( dom g) /\ (A /\ B)) = (A /\ B) by A1, A4, XBOOLE_1: 17, XBOOLE_1: 28, RELAT_1: 62;

      

       A9: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      

       A10: (f + g) is nonnegative by A2, A5, MESFUNC5: 22;

      f1 is nonnegative & g1 is nonnegative by A2, A5, MESFUNC5: 15;

      then

       A11: ( Integral (M,f1)) = ( integral+ (M,f1)) & ( Integral (M,g1)) = ( integral+ (M,g1)) by A8, A9, MESFUNC5: 42, MESFUNC5: 88;

      ex C be Element of S st C = ( dom (f + g)) & ( integral+ (M,(f + g))) = (( integral+ (M,(f | C))) + ( integral+ (M,(g | C)))) by A1, A2, A3, A4, A5, A6, MESFUNC5: 78;

      hence thesis by A2, A5, A7, A9, A10, A11, MESFUNC5: 31, MESFUNC5: 88;

    end;

    theorem :: MESFUN12:22

    

     Th22: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S, f,g be PartFunc of X, ExtREAL st E1 = ( dom f) & f is nonpositive & f is E1 -measurable & E2 = ( dom g) & g is nonpositive & g is E2 -measurable holds ( Integral (M,(f + g))) = (( Integral (M,(f | ( dom (f + g))))) + ( Integral (M,(g | ( dom (f + g))))))

    proof

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

      assume that

       A1: A = ( dom f) and

       A2: f is nonpositive and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonpositive and

       A6: g is B -measurable;

      reconsider f1 = ( - f) as nonnegative PartFunc of X, ExtREAL by A2;

      reconsider g1 = ( - g) as nonnegative PartFunc of X, ExtREAL by A5;

      

       A7: (f1 + g1) = ( - (f + g)) by MEASUR11: 64;

      then

       A13: (f + g) = ( - (f1 + g1)) by MESFUN11: 36;

      

       A8: ( dom f1) = A & ( dom g1) = B by A1, A4, MESFUNC1:def 7;

      then

       A9: ( dom (f1 + g1)) = (A /\ B) by MESFUNC5: 22;

      then

       A10: ( dom (f + g)) = (A /\ B) by A7, MESFUNC1:def 7;

      then

       A11: ( dom (f | ( dom (f + g)))) = (A /\ B) & ( dom (g | ( dom (f + g)))) = (A /\ B) by A1, A4, XBOOLE_1: 17, RELAT_1: 62;

      

       A12: (( dom f) /\ (A /\ B)) = (A /\ B) & (( dom g) /\ (A /\ B)) = (A /\ B) by A1, A4, XBOOLE_1: 17, XBOOLE_1: 28;

      

       A14: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      then

       A15: (f | ( dom (f + g))) is (A /\ B) -measurable & (g | ( dom (f + g))) is (A /\ B) -measurable by A10, A12, MESFUNC5: 42;

      

       A16: (f | ( dom (f + g))) is nonpositive & (g | ( dom (f + g))) is nonpositive by A2, A5, MESFUN11: 1;

      (f1 | ( dom (f1 + g1))) = ( - (f | ( dom (f + g)))) & (g1 | ( dom (f1 + g1))) = ( - (g | ( dom (f + g)))) by A9, A10, MESFUN11: 3;

      then

       A17: ( Integral (M,(f | ( dom (f + g))))) = ( - ( Integral (M,(f1 | ( dom (f1 + g1)))))) & ( Integral (M,(g | ( dom (f + g))))) = ( - ( Integral (M,(g1 | ( dom (f1 + g1)))))) by A11, A15, A16, MESFUN11: 57;

      (f + g) = (( - 1) (#) (f1 + g1)) & (f1 + g1) is nonnegative by A13, MESFUNC2: 9, MESFUNC5: 19;

      then

       A18: (f + g) is nonpositive by MESFUNC5: 20;

      (f + g) is (A /\ B) -measurable by A2, A5, A10, A14, MEASUR11: 65;

      then

       A19: ( Integral (M,(f + g))) = ( - ( Integral (M,(f1 + g1)))) by A7, A10, A18, MESFUN11: 57;

      f1 is A -measurable & g1 is B -measurable by A1, A3, A4, A6, MEASUR11: 63;

      then ( Integral (M,(f1 + g1))) = (( Integral (M,(f1 | ( dom (f1 + g1))))) + ( Integral (M,(g1 | ( dom (f1 + g1)))))) by A8, Th21;

      hence thesis by A17, A19, XXREAL_3: 9;

    end;

    theorem :: MESFUN12:23

    for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S, f,g be PartFunc of X, ExtREAL st E1 = ( dom f) & f is nonnegative & f is E1 -measurable & E2 = ( dom g) & g is nonpositive & g is E2 -measurable holds ( Integral (M,(f - g))) = (( Integral (M,(f | ( dom (f - g))))) - ( Integral (M,(g | ( dom (f - g)))))) & ( Integral (M,(g - f))) = (( Integral (M,(g | ( dom (g - f))))) - ( Integral (M,(f | ( dom (g - f))))))

    proof

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

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonpositive and

       A6: g is B -measurable;

      reconsider g1 = ( - g) as nonnegative PartFunc of X, ExtREAL by A5;

      

       A7: B = ( dom g1) by A4, MESFUNC1:def 7;

      

       A8: g1 is B -measurable by A4, A6, MEASUR11: 63;

      

       A9: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      

       A10: ( dom (f - g)) = (A /\ B) by A1, A2, A4, A5, MESFUNC5: 17;

      then

       A11: (A /\ B) = ( dom (g | ( dom (f - g)))) by A4, XBOOLE_1: 17, RELAT_1: 62;

      then (A /\ B) = (( dom g) /\ ( dom (f - g))) by RELAT_1: 61;

      then

       A12: (g | ( dom (f - g))) is (A /\ B) -measurable by A9, A10, MESFUNC5: 42;

      (f + g1) = (f - g) by MESFUNC2: 8;

      then

       A14: ( Integral (M,(f - g))) = (( Integral (M,(f | ( dom (f - g))))) + ( Integral (M,(g1 | ( dom (f - g)))))) by A1, A2, A3, A7, A8, Th21;

      

       A15: (g | ( dom (f - g))) is nonpositive by A5, MESFUN11: 1;

      (g1 | ( dom (f - g))) = ( - (g | ( dom (f - g)))) by MESFUN11: 3;

      then ( Integral (M,(g | ( dom (f - g))))) = ( - ( Integral (M,(g1 | ( dom (f - g)))))) by A12, A11, A15, MESFUN11: 57;

      then ( - ( Integral (M,(g | ( dom (f - g)))))) = ( Integral (M,(g1 | ( dom (f - g)))));

      hence

       A20: ( Integral (M,(f - g))) = (( Integral (M,(f | ( dom (f - g))))) - ( Integral (M,(g | ( dom (f - g)))))) by A14, XXREAL_3:def 4;

      

       A16: (g - f) = ( - (f - g)) by MEASUR11: 64;

      then

       A17: ( dom (g - f)) = (A /\ B) by A10, MESFUNC1:def 7;

      (f - g) is (A /\ B) -measurable by A2, A5, A9, A10, MEASUR11: 67;

      then ( Integral (M,(g - f))) = ( - ( Integral (M,(f - g)))) by A10, A16, MESFUN11: 52;

      hence ( Integral (M,(g - f))) = (( Integral (M,(g | ( dom (g - f))))) - ( Integral (M,(f | ( dom (g - f)))))) by A20, A17, A10, XXREAL_3: 26;

    end;

    

     Lm1: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, f be PartFunc of X, ExtREAL , r be Real st E = ( dom f) & f is nonnegative & f is E -measurable holds ( Integral (M,(r (#) f))) = (r * ( Integral (M,f)))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, f be PartFunc of X, ExtREAL , r be Real;

      assume that

       A1: E = ( dom f) and

       A2: f is nonnegative and

       A3: f is E -measurable;

      

       A7: ( dom (r (#) f)) = E by A1, MESFUNC1:def 6;

      

       A8: (r (#) f) is E -measurable by A1, A3, MESFUNC1: 37;

      per cases ;

        suppose

         A9: r >= 0 ;

        ( Integral (M,(r (#) f))) = ( integral+ (M,(r (#) f))) by A2, A7, A8, A9, MESFUNC5: 20, MESFUNC5: 88

        .= (r * ( integral+ (M,f))) by A1, A3, A9, A2, MESFUNC5: 86

        .= (r * ( Integral (M,f))) by A1, A3, A2, MESFUNC5: 88;

        hence ( Integral (M,(r (#) f))) = (r * ( Integral (M,f)));

      end;

        suppose

         A10: r < 0 ;

        set r2 = ( - r);

        r = (( - 1) * r2);

        then (r (#) f) = (( - 1) (#) (r2 (#) f)) by MESFUN11: 35;

        then

         A11: (r (#) f) = ( - (r2 (#) f)) by MESFUNC2: 9;

        

         A12: (r (#) f) is nonpositive by A2, A10, MESFUNC5: 20;

        ( Integral (M,(r (#) f))) = ( - ( integral+ (M,( - (r (#) f))))) by A12, A7, A8, MESFUN11: 57

        .= ( - ( integral+ (M,(r2 (#) f)))) by A11, MESFUN11: 36

        .= ( - (r2 * ( integral+ (M,f)))) by A1, A3, A10, A2, MESFUNC5: 86

        .= (( - r2) * ( integral+ (M,f))) by XXREAL_3: 92

        .= (r * ( Integral (M,f))) by A1, A3, A2, MESFUNC5: 88;

        hence ( Integral (M,(r (#) f))) = (r * ( Integral (M,f)));

      end;

    end;

    

     Lm2: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, f be PartFunc of X, ExtREAL , r be Real st E = ( dom f) & f is nonpositive & f is E -measurable holds ( Integral (M,(r (#) f))) = (r * ( Integral (M,f)))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, f be PartFunc of X, ExtREAL , r be Real;

      assume that

       A1: E = ( dom f) and

       A2: f is nonpositive and

       A3: f is E -measurable;

      set f2 = ( - f);

      

       A4: ( dom f2) = E by A1, MESFUNC1:def 7;

      then

       A5: E = ( dom (r (#) f2)) by MESFUNC1:def 6;

      

       A6: f2 is E -measurable by A1, A3, MEASUR11: 63;

      f = ( - f2) by MESFUN11: 36;

      then f = (( - 1) (#) f2) by MESFUNC2: 9;

      then (r (#) f) = ((r * ( - 1)) (#) f2) by MESFUN11: 35;

      then (r (#) f) = (( - 1) (#) (r (#) f2)) by MESFUN11: 35;

      then (r (#) f) = ( - (r (#) f2)) by MESFUNC2: 9;

      

      then ( Integral (M,(r (#) f))) = ( - ( Integral (M,(r (#) f2)))) by A5, A6, A4, MESFUNC1: 37, MESFUN11: 52

      .= (( - 1) * ( Integral (M,(r (#) f2)))) by XXREAL_3: 91

      .= (( - 1) * (r * ( Integral (M,f2)))) by A2, A4, Lm1, A1, A3, MEASUR11: 63

      .= ((( - 1) * r) * ( Integral (M,f2))) by XXREAL_3: 66

      .= (( - r) * ( - ( Integral (M,f)))) by A1, A3, MESFUN11: 52

      .= (( - r) * (( - 1) * ( Integral (M,f)))) by XXREAL_3: 91

      .= ((( - r) * ( - 1)) * ( Integral (M,f))) by XXREAL_3: 66;

      hence ( Integral (M,(r (#) f))) = (r * ( Integral (M,f)));

    end;

    theorem :: MESFUN12:24

    for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, f be PartFunc of X, ExtREAL , r be Real st E = ( dom f) & (f is nonpositive or f is nonnegative) & f is E -measurable holds ( Integral (M,(r (#) f))) = (r * ( Integral (M,f))) by Lm1, Lm2;

    begin

    theorem :: MESFUN12:25

    

     Th25: for X,Y be non empty set, A be Element of ( bool [:X, Y:]), x,y be set st x in X & y in Y holds ( [x, y] in A iff x in ( Y-section (A,y))) & ( [x, y] in A iff y in ( X-section (A,x)))

    proof

      let X,Y be non empty set, E be Element of ( bool [:X, Y:]), x,y be set;

      assume

       A1: x in X & y in Y;

       A2:

      now

        assume y in ( X-section (E,x));

        then y in { y where y be Element of Y : [x, y] in E } by MEASUR11:def 4;

        then ex y1 be Element of Y st y = y1 & [x, y1] in E;

        hence [x, y] in E;

      end;

       A3:

      now

        assume [x, y] in E;

        then y in { y where y be Element of Y : [x, y] in E } by A1;

        hence y in ( X-section (E,x)) by MEASUR11:def 4;

      end;

       A4:

      now

        assume x in ( Y-section (E,y));

        then x in { x where x be Element of X : [x, y] in E } by MEASUR11:def 5;

        then ex x1 be Element of X st x = x1 & [x1, y] in E;

        hence [x, y] in E;

      end;

      now

        assume [x, y] in E;

        then x in { x where x be Element of X : [x, y] in E } by A1;

        hence x in ( Y-section (E,y)) by MEASUR11:def 5;

      end;

      hence thesis by A2, A3, A4;

    end;

    definition

      let X1,X2 be non empty set;

      let Y be set;

      let f be PartFunc of [:X1, X2:], Y;

      let x be Element of X1;

      :: MESFUN12:def3

      func ProjPMap1 (f,x) -> PartFunc of X2, Y means

      : Def3: ( dom it ) = ( X-section (( dom f),x)) & for y be Element of X2 st [x, y] in ( dom f) holds (it . y) = (f . (x,y));

      existence

      proof

        deffunc F( object) = (f . (x,$1));

        defpred P[ object] means $1 in ( X-section (( dom f),x));

         A1:

        now

          let d be object;

          assume d in ( X-section (( dom f),x));

          then d in { y where y be Element of X2 : [x, y] in ( dom f) } by MEASUR11:def 4;

          then ex d1 be Element of X2 st d = d1 & [x, d1] in ( dom f);

          hence d in X2 & [x, d] in ( dom f);

        end;

        

         A3: for d be object st P[d] holds F(d) in Y by A1, PARTFUN1: 4;

        consider P be PartFunc of X2, Y such that

         A4: (for x be object holds x in ( dom P) iff x in X2 & P[x]) & for x be object st x in ( dom P) holds (P . x) = F(x) from PARTFUN1:sch 3( A3);

        take P;

        

         A5: ( dom P) c= ( X-section (( dom f),x)) by A4;

        ( X-section (( dom f),x)) c= ( dom P) by A4;

        hence ( dom P) = ( X-section (( dom f),x)) by A5;

        thus for d be Element of X2 st [x, d] in ( dom f) holds (P . d) = (f . (x,d))

        proof

          let d be Element of X2;

          assume [x, d] in ( dom f);

          then d in { y where y be Element of X2 : [x, y] in ( dom f) };

          then d in ( X-section (( dom f),x)) by MEASUR11:def 4;

          hence (P . d) = (f . (x,d)) by A4;

        end;

      end;

      uniqueness

      proof

        let P1,P2 be PartFunc of X2, Y;

        assume that

         A1: ( dom P1) = ( X-section (( dom f),x)) and

         A2: for d be Element of X2 st [x, d] in ( dom f) holds (P1 . d) = (f . (x,d)) and

         A3: ( dom P2) = ( X-section (( dom f),x)) and

         A4: for d be Element of X2 st [x, d] in ( dom f) holds (P2 . d) = (f . (x,d));

         A5:

        now

          let d be object;

          assume d in ( X-section (( dom f),x));

          then d in { y where y be Element of X2 : [x, y] in ( dom f) } by MEASUR11:def 4;

          then ex d1 be Element of X2 st d = d1 & [x, d1] in ( dom f);

          hence d in X2 & [x, d] in ( dom f);

        end;

        now

          let d be Element of X2;

          assume d in ( dom P1);

          then (P1 . d) = (f . (x,d)) & (P2 . d) = (f . (x,d)) by A1, A2, A4, A5;

          hence (P1 . d) = (P2 . d);

        end;

        hence P1 = P2 by A1, A3, PARTFUN1: 5;

      end;

    end

    definition

      let X1,X2 be non empty set;

      let Y be set;

      let f be PartFunc of [:X1, X2:], Y;

      let y be Element of X2;

      :: MESFUN12:def4

      func ProjPMap2 (f,y) -> PartFunc of X1, Y means

      : Def4: ( dom it ) = ( Y-section (( dom f),y)) & for x be Element of X1 st [x, y] in ( dom f) holds (it . x) = (f . (x,y));

      existence

      proof

        deffunc F( object) = (f . ($1,y));

        defpred P[ object] means $1 in ( Y-section (( dom f),y));

         A1:

        now

          let c be object;

          assume c in ( Y-section (( dom f),y));

          then c in { x where x be Element of X1 : [x, y] in ( dom f) } by MEASUR11:def 5;

          then ex c1 be Element of X1 st c = c1 & [c1, y] in ( dom f);

          hence c in X1 & [c, y] in ( dom f);

        end;

        

         A3: for c be object st P[c] holds F(c) in Y by A1, PARTFUN1: 4;

        consider P be PartFunc of X1, Y such that

         A4: (for x be object holds x in ( dom P) iff x in X1 & P[x]) & for x be object st x in ( dom P) holds (P . x) = F(x) from PARTFUN1:sch 3( A3);

        take P;

        

         A5: ( dom P) c= ( Y-section (( dom f),y)) by A4;

        ( Y-section (( dom f),y)) c= ( dom P) by A4;

        hence ( dom P) = ( Y-section (( dom f),y)) by A5;

        thus for c be Element of X1 st [c, y] in ( dom f) holds (P . c) = (f . (c,y))

        proof

          let c be Element of X1;

          assume [c, y] in ( dom f);

          then c in { x where x be Element of X1 : [x, y] in ( dom f) };

          then c in ( Y-section (( dom f),y)) by MEASUR11:def 5;

          hence (P . c) = (f . (c,y)) by A4;

        end;

      end;

      uniqueness

      proof

        let P1,P2 be PartFunc of X1, Y;

        assume that

         A1: ( dom P1) = ( Y-section (( dom f),y)) and

         A2: for c be Element of X1 st [c, y] in ( dom f) holds (P1 . c) = (f . (c,y)) and

         A3: ( dom P2) = ( Y-section (( dom f),y)) and

         A4: for c be Element of X1 st [c, y] in ( dom f) holds (P2 . c) = (f . (c,y));

         A5:

        now

          let c be object;

          assume c in ( Y-section (( dom f),y));

          then c in { x where x be Element of X1 : [x, y] in ( dom f) } by MEASUR11:def 5;

          then ex c1 be Element of X1 st c = c1 & [c1, y] in ( dom f);

          hence c in X1 & [c, y] in ( dom f);

        end;

        now

          let c be Element of X1;

          assume c in ( dom P1);

          then (P1 . c) = (f . (c,y)) & (P2 . c) = (f . (c,y)) by A1, A2, A4, A5;

          hence (P1 . c) = (P2 . c);

        end;

        hence P1 = P2 by A1, A3, PARTFUN1: 5;

      end;

    end

    theorem :: MESFUN12:26

    

     Th26: for X1,X2 be non empty set, Y be set, f be PartFunc of [:X1, X2:], Y, x be Element of X1, y be Element of X2 holds (x in ( dom ( ProjPMap2 (f,y))) implies (( ProjPMap2 (f,y)) . x) = (f . (x,y))) & (y in ( dom ( ProjPMap1 (f,x))) implies (( ProjPMap1 (f,x)) . y) = (f . (x,y)))

    proof

      let X1,X2 be non empty set, Y be set, f be PartFunc of [:X1, X2:], Y, c be Element of X1, d be Element of X2;

      hereby

        assume c in ( dom ( ProjPMap2 (f,d)));

        then c in ( Y-section (( dom f),d)) by Def4;

        then c in { x where x be Element of X1 : [x, d] in ( dom f) } by MEASUR11:def 5;

        then ex x be Element of X1 st c = x & [x, d] in ( dom f);

        hence (( ProjPMap2 (f,d)) . c) = (f . (c,d)) by Def4;

      end;

      assume d in ( dom ( ProjPMap1 (f,c)));

      then d in ( X-section (( dom f),c)) by Def3;

      then d in { y where y be Element of X2 : [c, y] in ( dom f) } by MEASUR11:def 4;

      then ex y be Element of X2 st d = y & [c, y] in ( dom f);

      hence (( ProjPMap1 (f,c)) . d) = (f . (c,d)) by Def3;

    end;

    theorem :: MESFUN12:27

    

     Th27: for X1,X2,Y be non empty set, f be Function of [:X1, X2:], Y, x be Element of X1, y be Element of X2 holds ( ProjPMap1 (f,x)) = ( ProjMap1 (f,x)) & ( ProjPMap2 (f,y)) = ( ProjMap2 (f,y))

    proof

      let X1,X2,Y be non empty set, f be Function of [:X1, X2:], Y, x be Element of X1, y be Element of X2;

      ( dom f) = [:X1, X2:] by FUNCT_2:def 1;

      then

       A1: ( dom f) = ( [#] [:X1, X2:]) by SUBSET_1:def 3;

      then ( X-section (( dom f),x)) = X2 by MEASUR11: 24;

      then ( dom ( ProjPMap1 (f,x))) = X2 by Def3;

      then

       A2: ( dom ( ProjPMap1 (f,x))) = ( dom ( ProjMap1 (f,x))) by FUNCT_2:def 1;

      for y be Element of X2 st y in ( dom ( ProjPMap1 (f,x))) holds (( ProjPMap1 (f,x)) . y) = (( ProjMap1 (f,x)) . y)

      proof

        let y be Element of X2;

        assume y in ( dom ( ProjPMap1 (f,x)));

        then (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by Th26;

        hence (( ProjPMap1 (f,x)) . y) = (( ProjMap1 (f,x)) . y) by MESFUNC9:def 6;

      end;

      hence ( ProjPMap1 (f,x)) = ( ProjMap1 (f,x)) by A2, PARTFUN1: 5;

      ( Y-section (( dom f),y)) = X1 by A1, MEASUR11: 24;

      then ( dom ( ProjPMap2 (f,y))) = X1 by Def4;

      then

       A3: ( dom ( ProjPMap2 (f,y))) = ( dom ( ProjMap2 (f,y))) by FUNCT_2:def 1;

      for x be Element of X1 st x in ( dom ( ProjPMap2 (f,y))) holds (( ProjPMap2 (f,y)) . x) = (( ProjMap2 (f,y)) . x)

      proof

        let x be Element of X1;

        assume x in ( dom ( ProjPMap2 (f,y)));

        then (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by Th26;

        hence (( ProjPMap2 (f,y)) . x) = (( ProjMap2 (f,y)) . x) by MESFUNC9:def 7;

      end;

      hence ( ProjPMap2 (f,y)) = ( ProjMap2 (f,y)) by A3, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:28

    for X,Y,Z be non empty set, f be PartFunc of [:X, Y:], Z, x be Element of X, y be Element of Y, A be set holds ( X-section ((f " A),x)) = (( ProjPMap1 (f,x)) " A) & ( Y-section ((f " A),y)) = (( ProjPMap2 (f,y)) " A)

    proof

      let X,Y,Z be non empty set, f be PartFunc of [:X, Y:], Z, x be Element of X, y be Element of Y, A be set;

      reconsider E = (f " A) as Subset of [:X, Y:];

      now

        let y be object;

        assume y in ( X-section ((f " A),x));

        then y in { y where y be Element of Y : [x, y] in E } by MEASUR11:def 4;

        then

        consider y1 be Element of Y such that

         A1: y1 = y & [x, y1] in E;

        

         A2: [x, y] in ( dom f) & (f . [x, y]) in A by A1, FUNCT_1:def 7;

        then y in { y where y be Element of Y : [x, y] in ( dom f) } by A1;

        then y in ( X-section (( dom f),x)) by MEASUR11:def 4;

        then

         A3: y in ( dom ( ProjPMap1 (f,x))) by Def3;

        (( ProjPMap1 (f,x)) . y1) = (f . (x,y1)) by A1, A2, Def3;

        hence y in (( ProjPMap1 (f,x)) " A) by A1, A2, A3, FUNCT_1:def 7;

      end;

      then

       A4: ( X-section ((f " A),x)) c= (( ProjPMap1 (f,x)) " A);

      now

        let y be object;

        assume y in (( ProjPMap1 (f,x)) " A);

        then

         A5: y in ( dom ( ProjPMap1 (f,x))) & (( ProjPMap1 (f,x)) . y) in A by FUNCT_1:def 7;

        then y in ( X-section (( dom f),x)) by Def3;

        then y in { y where y be Element of Y : [x, y] in ( dom f) } by MEASUR11:def 4;

        then

        consider y1 be Element of Y such that

         A6: y1 = y & [x, y1] in ( dom f);

        (f . (x,y1)) in A by A5, A6, Def3;

        then [x, y1] in (f " A) by A6, FUNCT_1:def 7;

        then y in { y where y be Element of Y : [x, y] in (f " A) } by A6;

        hence y in ( X-section ((f " A),x)) by MEASUR11:def 4;

      end;

      then (( ProjPMap1 (f,x)) " A) c= ( X-section ((f " A),x));

      hence ( X-section ((f " A),x)) = (( ProjPMap1 (f,x)) " A) by A4;

      now

        let x be object;

        assume x in ( Y-section ((f " A),y));

        then x in { x where x be Element of X : [x, y] in E } by MEASUR11:def 5;

        then

        consider x1 be Element of X such that

         A7: x1 = x & [x1, y] in E;

        

         A8: [x, y] in ( dom f) & (f . [x, y]) in A by A7, FUNCT_1:def 7;

        then x in { x where x be Element of X : [x, y] in ( dom f) } by A7;

        then x in ( Y-section (( dom f),y)) by MEASUR11:def 5;

        then

         A9: x in ( dom ( ProjPMap2 (f,y))) by Def4;

        (( ProjPMap2 (f,y)) . x1) = (f . (x1,y)) by A7, A8, Def4;

        hence x in (( ProjPMap2 (f,y)) " A) by A7, A8, A9, FUNCT_1:def 7;

      end;

      then

       A10: ( Y-section ((f " A),y)) c= (( ProjPMap2 (f,y)) " A);

      now

        let x be object;

        assume x in (( ProjPMap2 (f,y)) " A);

        then

         A11: x in ( dom ( ProjPMap2 (f,y))) & (( ProjPMap2 (f,y)) . x) in A by FUNCT_1:def 7;

        then x in ( Y-section (( dom f),y)) by Def4;

        then x in { x where x be Element of X : [x, y] in ( dom f) } by MEASUR11:def 5;

        then

        consider x1 be Element of X such that

         A12: x1 = x & [x1, y] in ( dom f);

        (f . (x1,y)) in A by A11, A12, Def4;

        then [x1, y] in (f " A) by A12, FUNCT_1:def 7;

        then x in { x where x be Element of X : [x, y] in (f " A) } by A12;

        hence x in ( Y-section ((f " A),y)) by MEASUR11:def 5;

      end;

      then (( ProjPMap2 (f,y)) " A) c= ( Y-section ((f " A),y));

      hence ( Y-section ((f " A),y)) = (( ProjPMap2 (f,y)) " A) by A10;

    end;

    theorem :: MESFUN12:29

    

     Th29: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, r be Real, f be PartFunc of [:X1, X2:], ExtREAL holds ( ProjPMap1 ((r (#) f),x)) = (r (#) ( ProjPMap1 (f,x))) & ( ProjPMap2 ((r (#) f),y)) = (r (#) ( ProjPMap2 (f,y)))

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, r be Real, f be PartFunc of [:X1, X2:], ExtREAL ;

      ( dom ( ProjPMap1 ((r (#) f),x))) = ( X-section (( dom (r (#) f)),x)) & ( dom ( ProjPMap2 ((r (#) f),y))) = ( Y-section (( dom (r (#) f)),y)) by Def3, Def4;

      then

       A1: ( dom ( ProjPMap1 ((r (#) f),x))) = ( X-section (( dom f),x)) & ( dom ( ProjPMap2 ((r (#) f),y))) = ( Y-section (( dom f),y)) by MESFUNC1:def 6;

      ( dom (r (#) ( ProjPMap1 (f,x)))) = ( dom ( ProjPMap1 (f,x))) & ( dom (r (#) ( ProjPMap2 (f,y)))) = ( dom ( ProjPMap2 (f,y))) by MESFUNC1:def 6;

      then

       A2: ( dom (r (#) ( ProjPMap1 (f,x)))) = ( X-section (( dom f),x)) & ( dom (r (#) ( ProjPMap2 (f,y)))) = ( Y-section (( dom f),y)) by Def3, Def4;

      now

        let y be Element of X2;

        assume

         A3: y in ( dom ( ProjPMap1 ((r (#) f),x)));

        then y in { y where y be Element of X2 : [x, y] in ( dom f) } by A1, MEASUR11:def 4;

        then

         A4: ex y1 be Element of X2 st y1 = y & [x, y1] in ( dom f);

        then

         A5: [x, y] in ( dom (r (#) f)) by MESFUNC1:def 6;

        

         A6: (f . (x,y)) = (f . [x, y]);

        ((r (#) ( ProjPMap1 (f,x))) . y) = (r * (( ProjPMap1 (f,x)) . y)) by A1, A2, A3, MESFUNC1:def 6

        .= (r * (f . [x, y])) by A4, A6, Def3

        .= ((r (#) f) . (x,y)) by A5, MESFUNC1:def 6;

        hence (( ProjPMap1 ((r (#) f),x)) . y) = ((r (#) ( ProjPMap1 (f,x))) . y) by A5, Def3;

      end;

      hence ( ProjPMap1 ((r (#) f),x)) = (r (#) ( ProjPMap1 (f,x))) by A1, A2, PARTFUN1: 5;

      now

        let x be Element of X1;

        assume

         A7: x in ( dom ( ProjPMap2 ((r (#) f),y)));

        then x in { x where x be Element of X1 : [x, y] in ( dom f) } by A1, MEASUR11:def 5;

        then

         A8: ex x1 be Element of X1 st x1 = x & [x1, y] in ( dom f);

        then

         A9: [x, y] in ( dom (r (#) f)) by MESFUNC1:def 6;

        

         A10: (f . (x,y)) = (f . [x, y]);

        ((r (#) ( ProjPMap2 (f,y))) . x) = (r * (( ProjPMap2 (f,y)) . x)) by A1, A2, A7, MESFUNC1:def 6

        .= (r * (f . [x, y])) by A8, A10, Def4

        .= ((r (#) f) . (x,y)) by A9, MESFUNC1:def 6;

        hence (( ProjPMap2 ((r (#) f),y)) . x) = ((r (#) ( ProjPMap2 (f,y))) . x) by A9, Def4;

      end;

      hence ( ProjPMap2 ((r (#) f),y)) = (r (#) ( ProjPMap2 (f,y))) by A1, A2, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:30

    for X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2 st (for z be Element of [:X1, X2:] st z in ( dom f) holds (f . z) = 0 ) holds (( ProjPMap2 (f,y)) . x) = 0 & (( ProjPMap1 (f,x)) . y) = 0

    proof

      let X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2;

      assume

       A1: for z be Element of [:X1, X2:] st z in ( dom f) holds (f . z) = 0 ;

      now

        assume x in ( dom ( ProjPMap2 (f,y)));

        then x in ( Y-section (( dom f),y)) by Def4;

        then x in { x where x be Element of X1 : [x, y] in ( dom f) } by MEASUR11:def 5;

        then

        consider x1 be Element of X1 such that

         A2: x1 = x & [x1, y] in ( dom f);

        (f . (x1,y)) = 0 by A1, A2;

        hence (( ProjPMap2 (f,y)) . x) = 0 by A2, Def4;

      end;

      hence (( ProjPMap2 (f,y)) . x) = 0 by FUNCT_1:def 2;

      now

        assume y in ( dom ( ProjPMap1 (f,x)));

        then y in ( X-section (( dom f),x)) by Def3;

        then y in { y where y be Element of X2 : [x, y] in ( dom f) } by MEASUR11:def 4;

        then

        consider y1 be Element of X2 such that

         A3: y1 = y & [x, y1] in ( dom f);

        (f . (x,y1)) = 0 by A1, A3;

        hence (( ProjPMap1 (f,x)) . y) = 0 by A3, Def3;

      end;

      hence (( ProjPMap1 (f,x)) . y) = 0 by FUNCT_1:def 2;

    end;

    theorem :: MESFUN12:31

    

     Th31: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL st f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) holds ( ProjPMap1 (f,x)) is_simple_func_in S2 & ( ProjPMap2 (f,y)) is_simple_func_in S1

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL ;

      assume

       AS: f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2)));

      then

       A1: f is real-valued & ex F be Finite_Sep_Sequence of ( sigma ( measurable_rectangles (S1,S2))) st (( dom f) = ( union ( rng F)) & for n be Nat, x,y be Element of [:X1, X2:] st n in ( dom F) & x in (F . n) & y in (F . n) holds (f . x) = (f . y)) by MESFUNC2:def 4;

      consider F be Finite_Sep_Sequence of ( sigma ( measurable_rectangles (S1,S2))) such that

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

       A3: for n be Nat, z1,z2 be Element of [:X1, X2:] st n in ( dom F) & z1 in (F . n) & z2 in (F . n) holds (f . z1) = (f . z2) by AS, MESFUNC2:def 4;

      

       A4: ( rng f) c= REAL by A1, VALUED_0:def 3;

      now

        let a be object;

        assume a in ( rng ( ProjPMap1 (f,x)));

        then

        consider y1 be Element of X2 such that

         A5: y1 in ( dom ( ProjPMap1 (f,x))) & a = (( ProjPMap1 (f,x)) . y1) by PARTFUN1: 3;

        

         A6: a = (f . (x,y1)) by A5, Th26;

        y1 in ( X-section (( dom f),x)) by A5, Def3;

        then y1 in { y where y be Element of X2 : [x, y] in ( dom f) } by MEASUR11:def 4;

        then ex y be Element of X2 st y1 = y & [x, y] in ( dom f);

        hence a in REAL by A4, A6, FUNCT_1: 3;

      end;

      then ( rng ( ProjPMap1 (f,x))) c= REAL ;

      then

       A7: ( ProjPMap1 (f,x)) is real-valued by VALUED_0:def 3;

      deffunc F1( Nat) = ( Measurable-X-section ((F . $1),x));

      consider F1 be FinSequence of S2 such that

       A8: ( len F1) = ( len F) & for n be Nat st n in ( dom F1) holds (F1 . n) = F1(n) from FINSEQ_2:sch 1;

      

       A9: ( dom F1) = ( dom F) by A8, FINSEQ_3: 29;

      reconsider FF = F as FinSequence of ( bool [:X1, X2:]) by FINSEQ_2: 24;

      now

        let m,n be Nat;

        assume

         A10: m in ( dom F1) & n in ( dom F1) & m <> n;

        ( Measurable-X-section ((F . m),x)) = ( X-section ((F . m),x)) & ( Measurable-X-section ((F . n),x)) = ( X-section ((F . n),x)) by MEASUR11:def 6;

        then

         A11: (F1 . m) = ( X-section ((F . m),x)) & (F1 . n) = ( X-section ((F . n),x)) by A10, A8;

        (F . m) misses (F . n) by A9, A10, Def2;

        hence (F1 . m) misses (F1 . n) by A11, MEASUR11: 35;

      end;

      then F1 is disjoint_valued;

      then

      reconsider F1 as Finite_Sep_Sequence of S2;

      reconsider FF1 = F1 as FinSequence of ( bool X2) by FINSEQ_2: 24;

      

       A12: for n be Nat st n in ( dom FF1) holds (FF1 . n) = ( X-section ((FF . n),x))

      proof

        let n be Nat;

        assume n in ( dom FF1);

        then (FF1 . n) = ( Measurable-X-section ((F . n),x)) by A8;

        hence (FF1 . n) = ( X-section ((FF . n),x)) by MEASUR11:def 6;

      end;

      then ( X-section (( union ( rng FF)),x)) = ( union ( rng FF1)) by A9, MEASUR11: 28;

      then

       A13: ( dom ( ProjPMap1 (f,x))) = ( union ( rng F1)) by A2, Def3;

      for n be Nat, y1,y2 be Element of X2 st n in ( dom F1) & y1 in (F1 . n) & y2 in (F1 . n) holds (( ProjPMap1 (f,x)) . y1) = (( ProjPMap1 (f,x)) . y2)

      proof

        let n be Nat, y1,y2 be Element of X2;

        assume

         A14: n in ( dom F1) & y1 in (F1 . n) & y2 in (F1 . n);

        then

         A15: (F1 . n) = ( X-section ((FF . n),x)) by A12;

        

         A17: (FF . n) in ( rng F) by A9, A14, FUNCT_1: 3;

        then (FF . n) c= ( union ( rng F)) by TARSKI:def 4;

        then (F1 . n) c= ( X-section (( dom f),x)) by A2, A15, MEASUR11: 20;

        then y1 in ( X-section (( dom f),x)) & y2 in ( X-section (( dom f),x)) by A14;

        then y1 in ( dom ( ProjPMap1 (f,x))) & y2 in ( dom ( ProjPMap1 (f,x))) by Def3;

        then

         A16: (( ProjPMap1 (f,x)) . y1) = (f . (x,y1)) & (( ProjPMap1 (f,x)) . y2) = (f . (x,y2)) by Th26;

        

         A18: [x, y1] in ( union ( rng F)) implies [x, y1] in (F . n)

        proof

          assume [x, y1] in ( union ( rng F));

          then

          consider A be set such that

           A19: [x, y1] in A & A in ( rng F) by TARSKI:def 4;

          consider m be object such that

           A20: m in ( dom F) & A = (F . m) by A19, FUNCT_1:def 3;

          reconsider m as Nat by A20;

          now

            assume m <> n;

            then for y be Element of X2 st y1 = y holds not [x, y] in (F . n) by A19, A20, XBOOLE_0: 3, PROB_2:def 2;

            then not y1 in { y where y be Element of X2 : [x, y] in (F . n) };

            then not y1 in ( X-section ((F . n),x)) by MEASUR11:def 4;

            hence contradiction by A14, A12;

          end;

          hence [x, y1] in (F . n) by A19, A20;

        end;

        

         A21: [x, y1] in ( union ( rng F))

        proof

          assume not [x, y1] in ( union ( rng F));

          then for y be Element of X2 st y1 = y holds not [x, y] in (F . n) by A17, TARSKI:def 4;

          then not y1 in { y where y be Element of X2 : [x, y] in (F . n) };

          then not y1 in ( X-section ((F . n),x)) by MEASUR11:def 4;

          hence contradiction by A14, A12;

        end;

        now

          assume not [x, y2] in (F . n);

          then for y be Element of X2 st y2 = y holds not [x, y] in (F . n);

          then not y2 in { y where y be Element of X2 : [x, y] in (F . n) };

          then not y2 in ( X-section ((F . n),x)) by MEASUR11:def 4;

          hence contradiction by A14, A12;

        end;

        hence (( ProjPMap1 (f,x)) . y1) = (( ProjPMap1 (f,x)) . y2) by A3, A14, A9, A16, A18, A21;

      end;

      hence ( ProjPMap1 (f,x)) is_simple_func_in S2 by A7, A13, MESFUNC2:def 4;

      now

        let a be object;

        assume a in ( rng ( ProjPMap2 (f,y)));

        then

        consider x1 be Element of X1 such that

         A25: x1 in ( dom ( ProjPMap2 (f,y))) & a = (( ProjPMap2 (f,y)) . x1) by PARTFUN1: 3;

        

         A26: a = (f . (x1,y)) by A25, Th26;

        x1 in ( Y-section (( dom f),y)) by A25, Def4;

        then x1 in { x where x be Element of X1 : [x, y] in ( dom f) } by MEASUR11:def 5;

        then ex x be Element of X1 st x1 = x & [x, y] in ( dom f);

        hence a in REAL by A4, A26, FUNCT_1: 3;

      end;

      then ( rng ( ProjPMap2 (f,y))) c= REAL ;

      then

       A27: ( ProjPMap2 (f,y)) is real-valued by VALUED_0:def 3;

      deffunc G1( Nat) = ( Measurable-Y-section ((F . $1),y));

      consider G1 be FinSequence of S1 such that

       A28: ( len G1) = ( len F) & for n be Nat st n in ( dom G1) holds (G1 . n) = G1(n) from FINSEQ_2:sch 1;

      

       A29: ( dom G1) = ( dom F) by A28, FINSEQ_3: 29;

      now

        let m,n be Nat;

        assume

         A30: m in ( dom G1) & n in ( dom G1) & m <> n;

        ( Measurable-Y-section ((F . m),y)) = ( Y-section ((F . m),y)) & ( Measurable-Y-section ((F . n),y)) = ( Y-section ((F . n),y)) by MEASUR11:def 7;

        then

         A31: (G1 . m) = ( Y-section ((F . m),y)) & (G1 . n) = ( Y-section ((F . n),y)) by A30, A28;

        (F . m) misses (F . n) by A29, A30, Def2;

        hence (G1 . m) misses (G1 . n) by A31, MEASUR11: 35;

      end;

      then G1 is disjoint_valued;

      then

      reconsider G1 as Finite_Sep_Sequence of S1;

      reconsider GG1 = G1 as FinSequence of ( bool X1) by FINSEQ_2: 24;

      

       A32: for n be Nat st n in ( dom GG1) holds (GG1 . n) = ( Y-section ((FF . n),y))

      proof

        let n be Nat;

        assume n in ( dom GG1);

        then (GG1 . n) = ( Measurable-Y-section ((F . n),y)) by A28;

        hence (GG1 . n) = ( Y-section ((FF . n),y)) by MEASUR11:def 7;

      end;

      then ( Y-section (( union ( rng FF)),y)) = ( union ( rng GG1)) by A29, MEASUR11: 29;

      then

       A33: ( dom ( ProjPMap2 (f,y))) = ( union ( rng G1)) by A2, Def4;

      for n be Nat, x1,x2 be Element of X1 st n in ( dom G1) & x1 in (G1 . n) & x2 in (G1 . n) holds (( ProjPMap2 (f,y)) . x1) = (( ProjPMap2 (f,y)) . x2)

      proof

        let n be Nat, x1,x2 be Element of X1;

        assume

         A34: n in ( dom G1) & x1 in (G1 . n) & x2 in (G1 . n);

        then

         A35: (G1 . n) = ( Y-section ((FF . n),y)) by A32;

        

         A37: (FF . n) in ( rng F) by A29, A34, FUNCT_1: 3;

        then (FF . n) c= ( union ( rng F)) by TARSKI:def 4;

        then (G1 . n) c= ( Y-section (( dom f),y)) by A2, A35, MEASUR11: 21;

        then x1 in ( Y-section (( dom f),y)) & x2 in ( Y-section (( dom f),y)) by A34;

        then x1 in ( dom ( ProjPMap2 (f,y))) & x2 in ( dom ( ProjPMap2 (f,y))) by Def4;

        then

         A36: (( ProjPMap2 (f,y)) . x1) = (f . (x1,y)) & (( ProjPMap2 (f,y)) . x2) = (f . (x2,y)) by Th26;

        

         A38: [x1, y] in ( union ( rng F)) implies [x1, y] in (F . n)

        proof

          assume [x1, y] in ( union ( rng F));

          then

          consider A be set such that

           A39: [x1, y] in A & A in ( rng F) by TARSKI:def 4;

          consider m be object such that

           A40: m in ( dom F) & A = (F . m) by A39, FUNCT_1:def 3;

          reconsider m as Nat by A40;

          now

            assume m <> n;

            then for x be Element of X1 st x1 = x holds not [x, y] in (F . n) by A39, A40, XBOOLE_0: 3, PROB_2:def 2;

            then not x1 in { x where x be Element of X1 : [x, y] in (F . n) };

            then not x1 in ( Y-section ((F . n),y)) by MEASUR11:def 5;

            hence contradiction by A34, A32;

          end;

          hence [x1, y] in (F . n) by A39, A40;

        end;

        

         A41: [x1, y] in ( union ( rng F))

        proof

          assume not [x1, y] in ( union ( rng F));

          then for x be Element of X1 st x1 = x holds not [x, y] in (F . n) by A37, TARSKI:def 4;

          then not x1 in { x where x be Element of X1 : [x, y] in (F . n) };

          then not x1 in ( Y-section ((F . n),y)) by MEASUR11:def 5;

          hence contradiction by A34, A32;

        end;

        now

          assume not [x2, y] in (F . n);

          then for x be Element of X1 st x2 = x holds not [x, y] in (F . n);

          then not x2 in { x where x be Element of X1 : [x, y] in (F . n) };

          then not x2 in ( Y-section ((F . n),y)) by MEASUR11:def 5;

          hence contradiction by A34, A32;

        end;

        hence (( ProjPMap2 (f,y)) . x1) = (( ProjPMap2 (f,y)) . x2) by A3, A34, A29, A36, A38, A41;

      end;

      hence ( ProjPMap2 (f,y)) is_simple_func_in S1 by A27, A33, MESFUNC2:def 4;

    end;

    theorem :: MESFUN12:32

    

     Th32: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL st f is nonnegative holds ( ProjPMap1 (f,x)) is nonnegative & ( ProjPMap2 (f,y)) is nonnegative

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL ;

      assume

       A1: f is nonnegative;

      for q be object st q in ( dom ( ProjPMap1 (f,x))) holds 0 <= (( ProjPMap1 (f,x)) . q)

      proof

        let q be object;

        assume

         A2: q in ( dom ( ProjPMap1 (f,x)));

        then

        reconsider y1 = q as Element of X2;

        (( ProjPMap1 (f,x)) . q) = (f . (x,y1)) by A2, Th26;

        hence 0 <= (( ProjPMap1 (f,x)) . q) by A1, SUPINF_2: 51;

      end;

      hence ( ProjPMap1 (f,x)) is nonnegative by SUPINF_2: 52;

      for p be object st p in ( dom ( ProjPMap2 (f,y))) holds 0 <= (( ProjPMap2 (f,y)) . p)

      proof

        let p be object;

        assume

         A3: p in ( dom ( ProjPMap2 (f,y)));

        then

        reconsider x1 = p as Element of X1;

        (( ProjPMap2 (f,y)) . p) = (f . (x1,y)) by A3, Th26;

        hence 0 <= (( ProjPMap2 (f,y)) . p) by A1, SUPINF_2: 51;

      end;

      hence ( ProjPMap2 (f,y)) is nonnegative by SUPINF_2: 52;

    end;

    theorem :: MESFUN12:33

    

     Th33: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL st f is nonpositive holds ( ProjPMap1 (f,x)) is nonpositive & ( ProjPMap2 (f,y)) is nonpositive

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL ;

      assume

       A1: f is nonpositive;

      for q be set st q in ( dom ( ProjPMap1 (f,x))) holds 0 >= (( ProjPMap1 (f,x)) . q)

      proof

        let q be set;

        assume

         A2: q in ( dom ( ProjPMap1 (f,x)));

        then

        reconsider y1 = q as Element of X2;

        (( ProjPMap1 (f,x)) . q) = (f . (x,y1)) by A2, Th26;

        hence 0 >= (( ProjPMap1 (f,x)) . q) by A1, MESFUNC5: 8;

      end;

      hence ( ProjPMap1 (f,x)) is nonpositive by MESFUNC5: 9;

      for p be set st p in ( dom ( ProjPMap2 (f,y))) holds 0 >= (( ProjPMap2 (f,y)) . p)

      proof

        let p be set;

        assume

         A3: p in ( dom ( ProjPMap2 (f,y)));

        then

        reconsider x1 = p as Element of X1;

        (( ProjPMap2 (f,y)) . p) = (f . (x1,y)) by A3, Th26;

        hence 0 >= (( ProjPMap2 (f,y)) . p) by A1, MESFUNC5: 8;

      end;

      hence ( ProjPMap2 (f,y)) is nonpositive by MESFUNC5: 9;

    end;

    theorem :: MESFUN12:34

    

     Th34: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, A be Subset of [:X1, X2:], f be PartFunc of [:X1, X2:], ExtREAL holds ( ProjPMap1 ((f | A),x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) & ( ProjPMap2 ((f | A),y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y)))

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, A be Subset of [:X1, X2:], f be PartFunc of [:X1, X2:], ExtREAL ;

      set f1 = (f | A);

      

       A2: (( dom f) /\ A) c= ( dom f) by XBOOLE_1: 17;

      

       A4: ( dom f1) = (( dom f) /\ A) by RELAT_1: 61;

      

       A7: ( dom (( ProjPMap1 (f,x)) | ( X-section (A,x)))) = (( dom ( ProjPMap1 (f,x))) /\ ( X-section (A,x))) by RELAT_1: 61

      .= (( X-section (( dom f),x)) /\ ( X-section (A,x))) by Def3

      .= ( X-section ((( dom f) /\ A),x)) by MEASUR11: 27

      .= ( dom ( ProjPMap1 (f1,x))) by A4, Def3;

      now

        let y be Element of X2;

        assume y in ( dom ( ProjPMap1 (f1,x)));

        then

         A8: y in ( X-section ((( dom f) /\ A),x)) by A4, Def3;

        then

         A9: [x, y] in (( dom f) /\ A) by Th25;

        then (( ProjPMap1 (f1,x)) . y) = (f1 . (x,y)) by A4, Def3;

        then

         A10: (( ProjPMap1 (f1,x)) . y) = (f . (x,y)) by A9, FUNCT_1: 48;

        

         b3: (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by A2, A9, Def3;

        y in (( X-section (( dom f),x)) /\ ( X-section (A,x))) by A8, MEASUR11: 27;

        then y in ( X-section (A,x)) by XBOOLE_0:def 4;

        hence (( ProjPMap1 (f1,x)) . y) = ((( ProjPMap1 (f,x)) | ( X-section (A,x))) . y) by A10, b3, FUNCT_1: 49;

      end;

      hence ( ProjPMap1 (f1,x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) by A7, PARTFUN1: 5;

      

       A13: ( dom (( ProjPMap2 (f,y)) | ( Y-section (A,y)))) = (( dom ( ProjPMap2 (f,y))) /\ ( Y-section (A,y))) by RELAT_1: 61

      .= (( Y-section (( dom f),y)) /\ ( Y-section (A,y))) by Def4

      .= ( Y-section ((( dom f) /\ A),y)) by MEASUR11: 27

      .= ( dom ( ProjPMap2 (f1,y))) by A4, Def4;

      now

        let x be Element of X1;

        assume x in ( dom ( ProjPMap2 (f1,y)));

        then

         A14: x in ( Y-section ((( dom f) /\ A),y)) by A4, Def4;

        then

         A15: [x, y] in (( dom f) /\ A) by Th25;

        then (( ProjPMap2 (f1,y)) . x) = (f1 . (x,y)) by A4, Def4;

        then

         A16: (( ProjPMap2 (f1,y)) . x) = (f . (x,y)) by A15, FUNCT_1: 48;

        

         b4: (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by A2, A15, Def4;

        x in (( Y-section (( dom f),y)) /\ ( Y-section (A,y))) by A14, MEASUR11: 27;

        then x in ( Y-section (A,y)) by XBOOLE_0:def 4;

        hence (( ProjPMap2 (f1,y)) . x) = ((( ProjPMap2 (f,y)) | ( Y-section (A,y))) . x) by A16, b4, FUNCT_1: 49;

      end;

      hence ( ProjPMap2 (f1,y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y))) by A13, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:35

    

     Th35: for X1,X2 be non empty set, A be Subset of [:X1, X2:], x be Element of X1, y be Element of X2 holds ( ProjPMap1 (( Xchi (A, [:X1, X2:])),x)) = ( Xchi (( X-section (A,x)),X2)) & ( ProjPMap2 (( Xchi (A, [:X1, X2:])),y)) = ( Xchi (( Y-section (A,y)),X1))

    proof

      let X1,X2 be non empty set, A be Subset of [:X1, X2:], x be Element of X1, y be Element of X2;

      

       A3: ( ProjPMap1 (( Xchi (A, [:X1, X2:])),x)) = ( ProjMap1 (( Xchi (A, [:X1, X2:])),x)) & ( ProjPMap2 (( Xchi (A, [:X1, X2:])),y)) = ( ProjMap2 (( Xchi (A, [:X1, X2:])),y)) by Th27;

      for y be Element of X2 holds (( ProjMap1 (( Xchi (A, [:X1, X2:])),x)) . y) = (( Xchi (( X-section (A,x)),X2)) . y)

      proof

        let y be Element of X2;

        

         a5: [x, y] in [:X1, X2:] by ZFMISC_1:def 2;

        

         a4: (( ProjMap1 (( Xchi (A, [:X1, X2:])),x)) . y) = (( Xchi (A, [:X1, X2:])) . (x,y)) by MESFUNC9:def 6;

        per cases ;

          suppose

           b1: [x, y] in A;

          then y in ( X-section (A,x)) by Th25;

          then (( ProjMap1 (( Xchi (A, [:X1, X2:])),x)) . y) = +infty & (( Xchi (( X-section (A,x)),X2)) . y) = +infty by a4, b1, MEASUR10:def 7;

          hence thesis;

        end;

          suppose

           b2: not [x, y] in A;

          then not y in ( X-section (A,x)) by Th25;

          then (( ProjMap1 (( Xchi (A, [:X1, X2:])),x)) . y) = 0 & (( Xchi (( X-section (A,x)),X2)) . y) = 0 by a5, a4, b2, MEASUR10:def 7;

          hence thesis;

        end;

      end;

      hence ( ProjPMap1 (( Xchi (A, [:X1, X2:])),x)) = ( Xchi (( X-section (A,x)),X2)) by A3, FUNCT_2:def 8;

      for x be Element of X1 holds (( ProjMap2 (( Xchi (A, [:X1, X2:])),y)) . x) = (( Xchi (( Y-section (A,y)),X1)) . x)

      proof

        let x be Element of X1;

        

         a5: [x, y] in [:X1, X2:] by ZFMISC_1:def 2;

        

         a4: (( ProjMap2 (( Xchi (A, [:X1, X2:])),y)) . x) = (( Xchi (A, [:X1, X2:])) . (x,y)) by MESFUNC9:def 7;

        per cases ;

          suppose

           b1: [x, y] in A;

          then x in ( Y-section (A,y)) by Th25;

          then (( ProjMap2 (( Xchi (A, [:X1, X2:])),y)) . x) = +infty & (( Xchi (( Y-section (A,y)),X1)) . x) = +infty by a4, b1, MEASUR10:def 7;

          hence thesis;

        end;

          suppose

           b2: not [x, y] in A;

          then not x in ( Y-section (A,y)) by Th25;

          then (( ProjMap2 (( Xchi (A, [:X1, X2:])),y)) . x) = 0 & (( Xchi (( Y-section (A,y)),X1)) . x) = 0 by a5, a4, b2, MEASUR10:def 7;

          hence thesis;

        end;

      end;

      hence ( ProjPMap2 (( Xchi (A, [:X1, X2:])),y)) = ( Xchi (( Y-section (A,y)),X1)) by A3, FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:36

    

     Th36: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g be PartFunc of X, ExtREAL , E be Element of S st (f | E) = (g | E) & E c= ( dom f) & E c= ( dom g) & f is E -measurable holds g is E -measurable

    proof

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

      assume that

       A1: (f | A) = (g | A) and

       A2: A c= ( dom f) and

       A3: A c= ( dom g) and

       A4: f is A -measurable;

      now

        let r be Real;

        now

          let x be object;

          assume x in (A /\ ( less_dom (f,r)));

          then

           A5: x in A & x in ( less_dom (f,r)) by XBOOLE_0:def 4;

          then

           A6: x in ( dom f) & (f . x) < r by MESFUNC1:def 11;

          (f . x) = ((f | A) . x) by A5, FUNCT_1: 49;

          then (f . x) = (g . x) by A1, A5, FUNCT_1: 49;

          then x in ( less_dom (g,r)) by A3, A5, A6, MESFUNC1:def 11;

          hence x in (A /\ ( less_dom (g,r))) by A5, XBOOLE_0:def 4;

        end;

        then

         A7: (A /\ ( less_dom (f,r))) c= (A /\ ( less_dom (g,r)));

        now

          let x be object;

          assume x in (A /\ ( less_dom (g,r)));

          then

           A8: x in A & x in ( less_dom (g,r)) by XBOOLE_0:def 4;

          then

           A9: x in ( dom g) & (g . x) < r by MESFUNC1:def 11;

          (g . x) = ((g | A) . x) by A8, FUNCT_1: 49;

          then (g . x) = (f . x) by A1, A8, FUNCT_1: 49;

          then x in ( less_dom (f,r)) by A2, A8, A9, MESFUNC1:def 11;

          hence x in (A /\ ( less_dom (f,r))) by A8, XBOOLE_0:def 4;

        end;

        then (A /\ ( less_dom (g,r))) c= (A /\ ( less_dom (f,r)));

        then (A /\ ( less_dom (g,r))) = (A /\ ( less_dom (f,r))) by A7;

        hence (A /\ ( less_dom (g,r))) in S by A4, MESFUNC1:def 16;

      end;

      hence thesis by MESFUNC1:def 16;

    end;

    theorem :: MESFUN12:37

    

     Th37: for X1,X2 be non empty set, A be Subset of [:X1, X2:], f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, F be Functional_Sequence of [:X1, X2:], ExtREAL st A c= ( dom f) & (for n be Nat holds ( dom (F . n)) = A) & (for z be Element of [:X1, X2:] st z in A holds (F # z) is convergent & ( lim (F # z)) = (f . z)) holds (ex FX be with_the_same_dom Functional_Sequence of X1, ExtREAL st (for n be Nat holds (FX . n) = ( ProjPMap2 ((F . n),y))) & (for x be Element of X1 st x in ( Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f,y)) . x) = ( lim (FX # x)))) & (ex FY be with_the_same_dom Functional_Sequence of X2, ExtREAL st (for n be Nat holds (FY . n) = ( ProjPMap1 ((F . n),x))) & (for y be Element of X2 st y in ( X-section (A,x)) holds (FY # y) is convergent & (( ProjPMap1 (f,x)) . y) = ( lim (FY # y))))

    proof

      let X1,X2 be non empty set, A be Subset of [:X1, X2:], f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, F be Functional_Sequence of [:X1, X2:], ExtREAL ;

      assume that

       A1: A c= ( dom f) and

       A2: for n be Nat holds ( dom (F . n)) = A and

       A3: for x be Element of [:X1, X2:] st x in A holds (F # x) is convergent & ( lim (F # x)) = (f . x);

      set f1 = (f | A);

      

       A4: ( dom f1) = A by A1, RELAT_1: 62;

      defpred P2[ Element of NAT , object] means $2 = ( ProjPMap2 ((F . $1),y));

      

       A5: for n be Element of NAT holds ex f be Element of ( PFuncs (X1, ExtREAL )) st P2[n, f]

      proof

        let n be Element of NAT ;

        reconsider f = ( ProjPMap2 ((F . n),y)) as Element of ( PFuncs (X1, ExtREAL )) by PARTFUN1: 45;

        take f;

        thus thesis;

      end;

      thus ex FX be with_the_same_dom Functional_Sequence of X1, ExtREAL st (for n be Nat holds (FX . n) = ( ProjPMap2 ((F . n),y))) & (for x be Element of X1 st x in ( Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f,y)) . x) = ( lim (FX # x)))

      proof

        consider FX be sequence of ( PFuncs (X1, ExtREAL )) such that

         A6: for n be Element of NAT holds P2[n, (FX . n)] from FUNCT_2:sch 3( A5);

        

         A7: for n be Nat holds ( dom (FX . n)) = ( Y-section (A,y))

        proof

          let n be Nat;

          

           A8: ( dom (F . n)) = ( dom (f | A)) by A2, A4;

          n is Element of NAT by ORDINAL1:def 12;

          then (FX . n) = ( ProjPMap2 ((F . n),y)) by A6;

          hence ( dom (FX . n)) = ( Y-section (A,y)) by A4, A8, Def4;

        end;

        for m,n be Nat holds ( dom (FX . m)) = ( dom (FX . n))

        proof

          let m,n be Nat;

          ( dom (FX . m)) = ( Y-section (A,y)) by A7;

          hence ( dom (FX . m)) = ( dom (FX . n)) by A7;

        end;

        then

        reconsider FX as with_the_same_dom Functional_Sequence of X1, ExtREAL by MESFUNC8:def 2;

        take FX;

        thus for n be Nat holds (FX . n) = ( ProjPMap2 ((F . n),y))

        proof

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          hence (FX . n) = ( ProjPMap2 ((F . n),y)) by A6;

        end;

        thus for x be Element of X1 st x in ( Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f,y)) . x) = ( lim (FX # x))

        proof

          let x be Element of X1;

          reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

          assume x in ( Y-section (A,y));

          then

           A13: [x, y] in A by Th25;

          then

           A14: (F # z) is convergent & ( lim (F # z)) = (f . z) by A3;

          

           A15: for n be Element of NAT holds ((FX # x) . n) = ((F # z) . n)

          proof

            let n be Element of NAT ;

            

             A16: [x, y] in ( dom (F . n)) by A2, A13;

            ((FX # x) . n) = ((FX . n) . x) by MESFUNC5:def 13;

            then ((FX # x) . n) = (( ProjPMap2 ((F . n),y)) . x) by A6;

            then ((FX # x) . n) = ((F . n) . (x,y)) by A16, Def4;

            hence ((FX # x) . n) = ((F # z) . n) by MESFUNC5:def 13;

          end;

          hence (FX # x) is convergent by A14, FUNCT_2: 63;

          (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by A1, A13, Def4;

          hence (( ProjPMap2 (f,y)) . x) = ( lim (FX # x)) by A14, A15, FUNCT_2: 63;

        end;

      end;

      defpred P1[ Element of NAT , object] means $2 = ( ProjPMap1 ((F . $1),x));

      

       A9: for n be Element of NAT holds ex f be Element of ( PFuncs (X2, ExtREAL )) st P1[n, f]

      proof

        let n be Element of NAT ;

        reconsider f = ( ProjPMap1 ((F . n),x)) as Element of ( PFuncs (X2, ExtREAL )) by PARTFUN1: 45;

        take f;

        thus thesis;

      end;

      consider FY be sequence of ( PFuncs (X2, ExtREAL )) such that

       A10: for n be Element of NAT holds P1[n, (FY . n)] from FUNCT_2:sch 3( A9);

      

       A11: for n be Nat holds ( dom (FY . n)) = ( X-section (A,x))

      proof

        let n be Nat;

        

         A12: ( dom (F . n)) = ( dom (f | A)) by A2, A4;

        n is Element of NAT by ORDINAL1:def 12;

        then (FY . n) = ( ProjPMap1 ((F . n),x)) by A10;

        hence ( dom (FY . n)) = ( X-section (A,x)) by A4, A12, Def3;

      end;

      for m,n be Nat holds ( dom (FY . m)) = ( dom (FY . n))

      proof

        let m,n be Nat;

        ( dom (FY . m)) = ( X-section (A,x)) by A11;

        hence ( dom (FY . m)) = ( dom (FY . n)) by A11;

      end;

      then

      reconsider FY as with_the_same_dom Functional_Sequence of X2, ExtREAL by MESFUNC8:def 2;

      take FY;

      thus for n be Nat holds (FY . n) = ( ProjPMap1 ((F . n),x))

      proof

        let n be Nat;

        n is Element of NAT by ORDINAL1:def 12;

        hence (FY . n) = ( ProjPMap1 ((F . n),x)) by A10;

      end;

      thus for y be Element of X2 st y in ( X-section (A,x)) holds (FY # y) is convergent & (( ProjPMap1 (f,x)) . y) = ( lim (FY # y))

      proof

        let y be Element of X2;

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume y in ( X-section (A,x));

        then

         A17: [x, y] in A by Th25;

        then

         A18: (F # z) is convergent & ( lim (F # z)) = (f . z) by A3;

        

         A19: for n be Element of NAT holds ((FY # y) . n) = ((F # z) . n)

        proof

          let n be Element of NAT ;

          

           A20: [x, y] in ( dom (F . n)) by A2, A17;

          ((FY # y) . n) = ((FY . n) . y) by MESFUNC5:def 13;

          then ((FY # y) . n) = (( ProjPMap1 ((F . n),x)) . y) by A10;

          then ((FY # y) . n) = ((F . n) . (x,y)) by A20, Def3;

          hence ((FY # y) . n) = ((F # z) . n) by MESFUNC5:def 13;

        end;

        hence (FY # y) is convergent by A18, FUNCT_2: 63;

        (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by A1, A17, Def3;

        hence (( ProjPMap1 (f,x)) . y) = ( lim (FY # y)) by A18, A19, FUNCT_2: 63;

      end;

    end;

    

     Lm3: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2))) st (f is nonnegative or f is nonpositive) & A c= ( dom f) & f is A -measurable holds ( ProjPMap1 (f,x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap2 (f,y)) is ( Measurable-Y-section (A,y)) -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: (f is nonnegative or f is nonpositive) and

       A2: A c= ( dom f) and

       A3: f is A -measurable;

      reconsider X12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      reconsider S = ( sigma ( measurable_rectangles (S1,S2))) as SigmaField of [:X1, X2:];

      set f1 = (f | A);

      

       A4: ( dom f1) = A by A2, RELAT_1: 62;

      A = (( dom f) /\ A) by A2, XBOOLE_1: 28;

      then

       A5: f1 is A -measurable by A3, MESFUNC5: 42;

      

       A6: ( Measurable-X-section (A,x)) = ( X-section (A,x)) & ( Measurable-Y-section (A,y)) = ( Y-section (A,y)) by MEASUR11:def 6, MEASUR11:def 7;

      

       A7: ( dom ( ProjPMap1 (f,x))) = ( X-section (( dom f),x)) & ( dom ( ProjPMap1 (f1,x))) = ( X-section (A,x)) by A4, Def3;

      

       B7: ( dom ( ProjPMap2 (f,y))) = ( Y-section (( dom f),y)) & ( dom ( ProjPMap2 (f1,y))) = ( Y-section (A,y)) by A4, Def4;

      

       P1: ex F be Functional_Sequence of [:X1, X2:], ExtREAL st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f1)) & (for x be Element of [:X1, X2:] st x in ( dom f1) holds (F # x) is convergent & ( lim (F # x)) = (f1 . x))

      proof

        per cases by A1;

          suppose f is nonnegative;

          then ex F be Functional_Sequence of [:X1, X2:], ExtREAL st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f1)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for x be Element of [:X1, X2:] st x in ( dom f1) holds ((F . n) . x) <= ((F . m) . x)) & for x be Element of [:X1, X2:] st x in ( dom f1) holds (F # x) is convergent & ( lim (F # x)) = (f1 . x) by A4, A5, MESFUNC5: 15, MESFUNC5: 64;

          hence thesis;

        end;

          suppose f is nonpositive;

          then ex F be Functional_Sequence of [:X1, X2:], ExtREAL st (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f1)) & (for n be Nat holds (F . n) is nonpositive) & (for n,m be Nat st n <= m holds for x be Element of [:X1, X2:] st x in ( dom f1) holds ((F . n) . x) >= ((F . m) . x)) & for x be Element of [:X1, X2:] st x in ( dom f1) holds (F # x) is convergent & ( lim (F # x)) = (f1 . x) by A4, A5, MESFUN11: 1, MESFUN11: 56;

          hence thesis;

        end;

      end;

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL such that

       A8: (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f1)) and

       A9: for x be Element of [:X1, X2:] st x in ( dom f1) holds (F # x) is convergent & ( lim (F # x)) = (f1 . x) by P1;

      

       A10: for z be Element of [:X1, X2:] st z in A holds (F # z) is convergent & ( lim (F # z)) = (f . z)

      proof

        let z be Element of [:X1, X2:];

        assume

         A11: z in A;

        hence (F # z) is convergent by A4, A9;

        

        thus ( lim (F # z)) = (f1 . z) by A4, A9, A11

        .= (f . z) by A11, FUNCT_1: 49;

      end;

      consider FY be with_the_same_dom Functional_Sequence of X2, ExtREAL such that

       A12: (for n be Nat holds (FY . n) = ( ProjPMap1 ((F . n),x))) and

       A13: (for y be Element of X2 st y in ( X-section (A,x)) holds (FY # y) is convergent & (( ProjPMap1 (f,x)) . y) = ( lim (FY # y))) by A2, A4, A8, A10, Th37;

      for n be Nat holds ( dom (FY . n)) = ( X-section (A,x))

      proof

        let n be Nat;

        (FY . n) = ( ProjPMap1 ((F . n),x)) & ( dom (F . n)) = A by A4, A8, A12;

        hence ( dom (FY . n)) = ( X-section (A,x)) by Def3;

      end;

      then

       A14: ( dom (FY . 0 )) = ( Measurable-X-section (A,x)) by A6;

      

       A15: for n be Nat holds (FY . n) is ( Measurable-X-section (A,x)) -measurable

      proof

        let n be Nat;

        (FY . n) = ( ProjPMap1 ((F . n),x)) & (F . n) is_simple_func_in S by A8, A12;

        hence (FY . n) is ( Measurable-X-section (A,x)) -measurable by Th31, MESFUNC2: 34;

      end;

      

       A16: ( X-section (A,x)) c= ( dom ( ProjPMap1 (f,x))) by A2, A7, MEASUR11: 20;

      

       A17: for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (FY # y) is convergent & (( ProjPMap1 (f1,x)) . y) = ( lim (FY # y))

      proof

        let y be Element of X2;

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume

         A18: y in ( Measurable-X-section (A,x));

        hence (FY # y) is convergent by A6, A13;

        ( ProjPMap1 (f1,x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) by Th34;

        then (( ProjPMap1 (f1,x)) . y) = (( ProjPMap1 (f,x)) . y) by A6, A18, FUNCT_1: 49;

        hence (( ProjPMap1 (f1,x)) . y) = ( lim (FY # y)) by A6, A13, A18;

      end;

      ( ProjPMap1 (f1,x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) by Th34;

      then (( ProjPMap1 (f1,x)) | ( Measurable-X-section (A,x))) = (( ProjPMap1 (f,x)) | ( Measurable-X-section (A,x))) by A6;

      hence ( ProjPMap1 (f,x)) is ( Measurable-X-section (A,x)) -measurable by A6, A7, A14, A15, A16, A17, Th36, MESFUNC8: 26;

      consider FX be with_the_same_dom Functional_Sequence of X1, ExtREAL such that

       A19: (for n be Nat holds (FX . n) = ( ProjPMap2 ((F . n),y))) and

       A20: (for x be Element of X1 st x in ( Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f,y)) . x) = ( lim (FX # x))) by A2, A4, A8, A10, Th37;

      for n be Nat holds ( dom (FX . n)) = ( Y-section (A,y))

      proof

        let n be Nat;

        (FX . n) = ( ProjPMap2 ((F . n),y)) & ( dom (F . n)) = A by A4, A8, A19;

        hence ( dom (FX . n)) = ( Y-section (A,y)) by Def4;

      end;

      then

       A21: ( dom (FX . 0 )) = ( Measurable-Y-section (A,y)) by A6;

      

       A22: for n be Nat holds (FX . n) is ( Measurable-Y-section (A,y)) -measurable

      proof

        let n be Nat;

        (FX . n) = ( ProjPMap2 ((F . n),y)) & (F . n) is_simple_func_in S by A8, A19;

        hence (FX . n) is ( Measurable-Y-section (A,y)) -measurable by Th31, MESFUNC2: 34;

      end;

      

       A23: ( Y-section (A,y)) c= ( dom ( ProjPMap2 (f,y))) by A2, B7, MEASUR11: 21;

      

       A24: for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f1,y)) . x) = ( lim (FX # x))

      proof

        let x be Element of X1;

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume x in ( Measurable-Y-section (A,y));

        then

         A25: x in ( Y-section (A,y)) by MEASUR11:def 7;

        hence (FX # x) is convergent by A20;

        ( ProjPMap2 (f1,y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y))) by Th34;

        then (( ProjPMap2 (f1,y)) . x) = (( ProjPMap2 (f,y)) . x) by A25, FUNCT_1: 49;

        hence (( ProjPMap2 (f1,y)) . x) = ( lim (FX # x)) by A25, A20;

      end;

      ( ProjPMap2 (f1,y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y))) by Th34;

      then (( ProjPMap2 (f1,y)) | ( Measurable-Y-section (A,y))) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section (A,y))) by A6;

      hence ( ProjPMap2 (f,y)) is ( Measurable-Y-section (A,y)) -measurable by A6, B7, A21, A22, A23, A24, Th36, MESFUNC8: 26;

    end;

    theorem :: MESFUN12:38

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), M2 be sigma_Measure of S2, A be Element of S1, B be Element of S2, x be Element of X1 holds ((M2 . (B /\ ( Measurable-X-section (E,x)))) * (( chi (A,X1)) . x)) = ( Integral (M2,( ProjPMap1 ((( chi ( [:A, B:], [:X1, X2:])) | E),x))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), M2 be sigma_Measure of S2, A be Element of S1, B be Element of S2, x be Element of X1;

      set CAB = (( chi ( [:A, B:], [:X1, X2:])) | E);

      ( ProjPMap1 (( chi ( [:A, B:], [:X1, X2:])),x)) = ( ProjMap1 (( chi ( [:A, B:], [:X1, X2:])),x)) by Th27;

      then

       A0: ( dom ( ProjPMap1 (( chi ( [:A, B:], [:X1, X2:])),x))) = X2 by FUNCT_2:def 1;

      ( ProjPMap1 (CAB,x)) = (( ProjPMap1 (( chi ( [:A, B:], [:X1, X2:])),x)) | ( X-section (E,x))) by Th34;

      then ( dom ( ProjPMap1 (CAB,x))) = (X2 /\ ( X-section (E,x))) by A0, RELAT_1: 61;

      then

       A1: ( dom ( ProjPMap1 (CAB,x))) = ( X-section (E,x)) by XBOOLE_1: 28;

      

       A2: for y be Element of X2 holds (( ProjPMap1 (CAB,x)) . y) = (((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) * (( chi (B,X2)) . y))

      proof

        let y be Element of X2;

        per cases ;

          suppose

           A3: [x, y] in E;

          then y in ( X-section (E,x)) by Th25;

          then (( ProjPMap1 (CAB,x)) . y) = (CAB . (x,y)) by A1, Th26;

          then

           A4: (( ProjPMap1 (CAB,x)) . y) = (( chi ( [:A, B:], [:X1, X2:])) . (x,y)) by A3, FUNCT_1: 49;

          x in ( Y-section (E,y)) by A3, Th25;

          then x in ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

          then ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) = (( chi (A,X1)) . x) by FUNCT_1: 49;

          hence (( ProjPMap1 (CAB,x)) . y) = (((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) * (( chi (B,X2)) . y)) by A4, MEASUR10: 2;

        end;

          suppose

           A5: not [x, y] in E;

          then not y in ( X-section (E,x)) by Th25;

          then

           A6: (( ProjPMap1 (CAB,x)) . y) = 0 by A1, FUNCT_1:def 2;

           not x in ( Y-section (E,y)) by A5, Th25;

          then not x in ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

          then not x in ( dom (( chi (A,X1)) | ( Measurable-Y-section (E,y)))) by Th18;

          then ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) = 0 by FUNCT_1:def 2;

          hence (( ProjPMap1 (CAB,x)) . y) = (((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) * (( chi (B,X2)) . y)) by A6;

        end;

      end;

      per cases ;

        suppose x in A;

        then

         A7: (( chi (A,X1)) . x) = 1 by FUNCT_3:def 3;

        then

         A8: ((M2 . (B /\ ( Measurable-X-section (E,x)))) * (( chi (A,X1)) . x)) = (M2 . (B /\ ( Measurable-X-section (E,x)))) by XXREAL_3: 81;

        ( dom (( chi (B,X2)) | ( Measurable-X-section (E,x)))) = ( Measurable-X-section (E,x)) by Th18;

        then

         A9: ( dom ( ProjPMap1 (CAB,x))) = ( dom (( chi (B,X2)) | ( Measurable-X-section (E,x)))) by A1, MEASUR11:def 6;

        for y be Element of X2 st y in ( dom ( ProjPMap1 (CAB,x))) holds (( ProjPMap1 (CAB,x)) . y) = ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y)

        proof

          let y be Element of X2;

          assume

           A10: y in ( dom ( ProjPMap1 (CAB,x)));

          then

           A11: y in ( Measurable-X-section (E,x)) by A1, MEASUR11:def 6;

           [x, y] in E by A1, A10, Th25;

          then x in ( Y-section (E,y)) by Th25;

          then x in ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

          then

           A12: ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) = (( chi (A,X1)) . x) by FUNCT_1: 49;

          (( ProjPMap1 (CAB,x)) . y) = (((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) * (( chi (B,X2)) . y)) by A2;

          then (( ProjPMap1 (CAB,x)) . y) = (( chi (B,X2)) . y) by A7, A12, XXREAL_3: 81;

          hence (( ProjPMap1 (CAB,x)) . y) = ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) by A11, FUNCT_1: 49;

        end;

        then ( ProjPMap1 (CAB,x)) = (( chi (B,X2)) | ( Measurable-X-section (E,x))) by A9, PARTFUN1: 5;

        hence ((M2 . (B /\ ( Measurable-X-section (E,x)))) * (( chi (A,X1)) . x)) = ( Integral (M2,( ProjPMap1 (CAB,x)))) by A8, Th20;

      end;

        suppose not x in A;

        then

         A13: (( chi (A,X1)) . x) = 0 by FUNCT_3:def 3;

        then

         A14: ((M2 . (B /\ ( Measurable-X-section (E,x)))) * (( chi (A,X1)) . x)) = 0 ;

        

         A15: {} is Element of S2 by PROB_1: 4;

        

         A16: ( dom ( ProjPMap1 (CAB,x))) = ( Measurable-X-section (E,x)) by A1, MEASUR11:def 6

        .= ( dom (( chi ( {} ,X2)) | ( Measurable-X-section (E,x)))) by Th18;

        for y be Element of X2 st y in ( dom ( ProjPMap1 (CAB,x))) holds (( ProjPMap1 (CAB,x)) . y) = ((( chi ( {} ,X2)) | ( Measurable-X-section (E,x))) . y)

        proof

          let y be Element of X2;

          assume

           A17: y in ( dom ( ProjPMap1 (CAB,x)));

          then y in ( Measurable-X-section (E,x)) by A1, MEASUR11:def 6;

          then

           A18: ((( chi ( {} ,X2)) | ( Measurable-X-section (E,x))) . y) = (( chi ( {} ,X2)) . y) by FUNCT_1: 49;

           [x, y] in E by A1, A17, Th25;

          then x in ( Y-section (E,y)) by Th25;

          then x in ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

          then

           A19: ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) = (( chi (A,X1)) . x) by FUNCT_1: 49;

          (( ProjPMap1 (CAB,x)) . y) = (((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) * (( chi (B,X2)) . y)) by A2;

          then (( ProjPMap1 (CAB,x)) . y) = 0 by A13, A19;

          hence (( ProjPMap1 (CAB,x)) . y) = ((( chi ( {} ,X2)) | ( Measurable-X-section (E,x))) . y) by A18, FUNCT_3:def 3;

        end;

        then ( ProjPMap1 (CAB,x)) = (( chi ( {} ,X2)) | ( Measurable-X-section (E,x))) by A16, PARTFUN1: 5;

        then ( Integral (M2,( ProjPMap1 (CAB,x)))) = (M2 . ( {} /\ ( Measurable-X-section (E,x)))) by A15, Th20;

        hence ((M2 . (B /\ ( Measurable-X-section (E,x)))) * (( chi (A,X1)) . x)) = ( Integral (M2,( ProjPMap1 (CAB,x)))) by A14, VALUED_0:def 19;

      end;

    end;

    theorem :: MESFUN12:39

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), M1 be sigma_Measure of S1, A be Element of S1, B be Element of S2, y be Element of X2 holds ((M1 . (A /\ ( Measurable-Y-section (E,y)))) * (( chi (B,X2)) . y)) = ( Integral (M1,( ProjPMap2 ((( chi ( [:A, B:], [:X1, X2:])) | E),y))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), M1 be sigma_Measure of S1, A be Element of S1, B be Element of S2, y be Element of X2;

      set CAB = (( chi ( [:A, B:], [:X1, X2:])) | E);

      ( ProjPMap2 (( chi ( [:A, B:], [:X1, X2:])),y)) = ( ProjMap2 (( chi ( [:A, B:], [:X1, X2:])),y)) by Th27;

      then

       A0: ( dom ( ProjPMap2 (( chi ( [:A, B:], [:X1, X2:])),y))) = X1 by FUNCT_2:def 1;

      ( ProjPMap2 (CAB,y)) = (( ProjPMap2 (( chi ( [:A, B:], [:X1, X2:])),y)) | ( Y-section (E,y))) by Th34;

      then ( dom ( ProjPMap2 (CAB,y))) = (X1 /\ ( Y-section (E,y))) by A0, RELAT_1: 61;

      then

       A1: ( dom ( ProjPMap2 (CAB,y))) = ( Y-section (E,y)) by XBOOLE_1: 28;

      

       A2: for x be Element of X1 holds (( ProjPMap2 (CAB,y)) . x) = (((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) * (( chi (A,X1)) . x))

      proof

        let x be Element of X1;

        per cases ;

          suppose

           A3: [x, y] in E;

          then x in ( Y-section (E,y)) by Th25;

          then (( ProjPMap2 (CAB,y)) . x) = (CAB . (x,y)) by A1, Th26;

          then

           A4: (( ProjPMap2 (CAB,y)) . x) = (( chi ( [:A, B:], [:X1, X2:])) . (x,y)) by A3, FUNCT_1: 49;

          y in ( X-section (E,x)) by A3, Th25;

          then y in ( Measurable-X-section (E,x)) by MEASUR11:def 6;

          then ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) = (( chi (B,X2)) . y) by FUNCT_1: 49;

          hence (( ProjPMap2 (CAB,y)) . x) = (((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) * (( chi (A,X1)) . x)) by A4, MEASUR10: 2;

        end;

          suppose

           A5: not [x, y] in E;

          then not x in ( Y-section (E,y)) by Th25;

          then

           A6: (( ProjPMap2 (CAB,y)) . x) = 0 by A1, FUNCT_1:def 2;

           not y in ( X-section (E,x)) by A5, Th25;

          then not y in ( Measurable-X-section (E,x)) by MEASUR11:def 6;

          then not y in ( dom (( chi (B,X2)) | ( Measurable-X-section (E,x)))) by Th18;

          then ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) = 0 by FUNCT_1:def 2;

          hence (( ProjPMap2 (CAB,y)) . x) = (((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) * (( chi (A,X1)) . x)) by A6;

        end;

      end;

      per cases ;

        suppose y in B;

        then

         A7: (( chi (B,X2)) . y) = 1 by FUNCT_3:def 3;

        then

         A8: ((M1 . (A /\ ( Measurable-Y-section (E,y)))) * (( chi (B,X2)) . y)) = (M1 . (A /\ ( Measurable-Y-section (E,y)))) by XXREAL_3: 81;

        ( dom (( chi (A,X1)) | ( Measurable-Y-section (E,y)))) = ( Measurable-Y-section (E,y)) by Th18;

        then

         A9: ( dom ( ProjPMap2 (CAB,y))) = ( dom (( chi (A,X1)) | ( Measurable-Y-section (E,y)))) by A1, MEASUR11:def 7;

        for x be Element of X1 st x in ( dom ( ProjPMap2 (CAB,y))) holds (( ProjPMap2 (CAB,y)) . x) = ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x)

        proof

          let x be Element of X1;

          assume

           A10: x in ( dom ( ProjPMap2 (CAB,y)));

          then

           A11: x in ( Measurable-Y-section (E,y)) by A1, MEASUR11:def 7;

           [x, y] in E by A1, A10, Th25;

          then y in ( X-section (E,x)) by Th25;

          then y in ( Measurable-X-section (E,x)) by MEASUR11:def 6;

          then

           A12: ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) = (( chi (B,X2)) . y) by FUNCT_1: 49;

          (( ProjPMap2 (CAB,y)) . x) = (((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) * (( chi (A,X1)) . x)) by A2;

          then (( ProjPMap2 (CAB,y)) . x) = (( chi (A,X1)) . x) by A7, A12, XXREAL_3: 81;

          hence (( ProjPMap2 (CAB,y)) . x) = ((( chi (A,X1)) | ( Measurable-Y-section (E,y))) . x) by A11, FUNCT_1: 49;

        end;

        then ( ProjPMap2 (CAB,y)) = (( chi (A,X1)) | ( Measurable-Y-section (E,y))) by A9, PARTFUN1: 5;

        hence ((M1 . (A /\ ( Measurable-Y-section (E,y)))) * (( chi (B,X2)) . y)) = ( Integral (M1,( ProjPMap2 (CAB,y)))) by A8, Th20;

      end;

        suppose not y in B;

        then

         A13: (( chi (B,X2)) . y) = 0 by FUNCT_3:def 3;

        then

         A14: ((M1 . (A /\ ( Measurable-Y-section (E,y)))) * (( chi (B,X2)) . y)) = 0 ;

        

         A15: {} is Element of S1 by PROB_1: 4;

        

         A16: ( dom ( ProjPMap2 (CAB,y))) = ( Measurable-Y-section (E,y)) by A1, MEASUR11:def 7

        .= ( dom (( chi ( {} ,X1)) | ( Measurable-Y-section (E,y)))) by Th18;

        for x be Element of X1 st x in ( dom ( ProjPMap2 (CAB,y))) holds (( ProjPMap2 (CAB,y)) . x) = ((( chi ( {} ,X1)) | ( Measurable-Y-section (E,y))) . x)

        proof

          let x be Element of X1;

          assume

           A17: x in ( dom ( ProjPMap2 (CAB,y)));

          then x in ( Measurable-Y-section (E,y)) by A1, MEASUR11:def 7;

          then

           A18: ((( chi ( {} ,X1)) | ( Measurable-Y-section (E,y))) . x) = (( chi ( {} ,X1)) . x) by FUNCT_1: 49;

           [x, y] in E by A1, A17, Th25;

          then y in ( X-section (E,x)) by Th25;

          then y in ( Measurable-X-section (E,x)) by MEASUR11:def 6;

          then

           A19: ((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) = (( chi (B,X2)) . y) by FUNCT_1: 49;

          (( ProjPMap2 (CAB,y)) . x) = (((( chi (B,X2)) | ( Measurable-X-section (E,x))) . y) * (( chi (A,X1)) . x)) by A2;

          then (( ProjPMap2 (CAB,y)) . x) = 0 by A13, A19;

          hence (( ProjPMap2 (CAB,y)) . x) = ((( chi ( {} ,X1)) | ( Measurable-Y-section (E,y))) . x) by A18, FUNCT_3:def 3;

        end;

        then ( ProjPMap2 (CAB,y)) = (( chi ( {} ,X1)) | ( Measurable-Y-section (E,y))) by A16, PARTFUN1: 5;

        then ( Integral (M1,( ProjPMap2 (CAB,y)))) = (M1 . ( {} /\ ( Measurable-Y-section (E,y)))) by A15, Th20;

        hence ((M1 . (A /\ ( Measurable-Y-section (E,y)))) * (( chi (B,X2)) . y)) = ( Integral (M1,( ProjPMap2 (CAB,y)))) by A14, VALUED_0:def 19;

      end;

    end;

    theorem :: MESFUN12:40

    

     Th40: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL , er be ExtReal holds ( [x, y] in ( dom f) & (f . (x,y)) = er iff y in ( dom ( ProjPMap1 (f,x))) & (( ProjPMap1 (f,x)) . y) = er) & ( [x, y] in ( dom f) & (f . (x,y)) = er iff x in ( dom ( ProjPMap2 (f,y))) & (( ProjPMap2 (f,y)) . x) = er)

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, f be PartFunc of [:X1, X2:], ExtREAL , a be ExtReal;

      hereby

        assume that

         A2: [x, y] in ( dom f) and

         A3: (f . (x,y)) = a;

        y in ( X-section (( dom f),x)) by A2, Th25;

        hence y in ( dom ( ProjPMap1 (f,x))) by Def3;

        hence (( ProjPMap1 (f,x)) . y) = a by A3, Th26;

      end;

      hereby

        assume that

         A4: y in ( dom ( ProjPMap1 (f,x))) and

         A5: (( ProjPMap1 (f,x)) . y) = a;

        y in ( X-section (( dom f),x)) by A4, Def3;

        hence [x, y] in ( dom f) by Th25;

        thus (f . (x,y)) = a by A4, A5, Th26;

      end;

      hereby

        assume that

         A6: [x, y] in ( dom f) and

         A7: (f . (x,y)) = a;

        x in ( Y-section (( dom f),y)) by A6, Th25;

        hence x in ( dom ( ProjPMap2 (f,y))) by Def4;

        hence (( ProjPMap2 (f,y)) . x) = a by A7, Th26;

      end;

      assume that

       A8: x in ( dom ( ProjPMap2 (f,y))) and

       A9: (( ProjPMap2 (f,y)) . x) = a;

      x in ( Y-section (( dom f),y)) by A8, Def4;

      hence [x, y] in ( dom f) by Th25;

      thus (f . (x,y)) = a by A8, A9, Th26;

    end;

    theorem :: MESFUN12:41

    

     Th41: for X1,X2 be non empty set, A,Z be set, f be PartFunc of [:X1, X2:], Z, x be Element of X1 holds ( X-section ((f " A),x)) = (( ProjPMap1 (f,x)) " A)

    proof

      let X,Y be non empty set, A,Z be set, f be PartFunc of [:X, Y:], Z, x be Element of X;

      reconsider E = (f " A) as Subset of [:X, Y:];

      now

        let y be object;

        assume y in ( X-section ((f " A),x));

        then y in { y where y be Element of Y : [x, y] in E } by MEASUR11:def 4;

        then

        consider y1 be Element of Y such that

         A1: y1 = y & [x, y1] in E;

        

         A2: [x, y] in ( dom f) & (f . [x, y]) in A by A1, FUNCT_1:def 7;

        then y in { y where y be Element of Y : [x, y] in ( dom f) } by A1;

        then y in ( X-section (( dom f),x)) by MEASUR11:def 4;

        then

         A3: y in ( dom ( ProjPMap1 (f,x))) by Def3;

        (( ProjPMap1 (f,x)) . y1) = (f . (x,y1)) by A1, A2, Def3;

        hence y in (( ProjPMap1 (f,x)) " A) by A1, A2, A3, FUNCT_1:def 7;

      end;

      then

       A4: ( X-section ((f " A),x)) c= (( ProjPMap1 (f,x)) " A);

      now

        let y be object;

        assume y in (( ProjPMap1 (f,x)) " A);

        then

         A5: y in ( dom ( ProjPMap1 (f,x))) & (( ProjPMap1 (f,x)) . y) in A by FUNCT_1:def 7;

        then y in ( X-section (( dom f),x)) by Def3;

        then y in { y where y be Element of Y : [x, y] in ( dom f) } by MEASUR11:def 4;

        then

        consider y1 be Element of Y such that

         A6: y1 = y & [x, y1] in ( dom f);

        (f . (x,y1)) in A by A5, A6, Def3;

        then [x, y1] in (f " A) by A6, FUNCT_1:def 7;

        then y in { y where y be Element of Y : [x, y] in (f " A) } by A6;

        hence y in ( X-section ((f " A),x)) by MEASUR11:def 4;

      end;

      then (( ProjPMap1 (f,x)) " A) c= ( X-section ((f " A),x));

      hence ( X-section ((f " A),x)) = (( ProjPMap1 (f,x)) " A) by A4;

    end;

    theorem :: MESFUN12:42

    

     Th42: for X1,X2 be non empty set, A,Z be set, f be PartFunc of [:X1, X2:], Z, y be Element of X2 holds ( Y-section ((f " A),y)) = (( ProjPMap2 (f,y)) " A)

    proof

      let X,Y be non empty set, A,Z be set, f be PartFunc of [:X, Y:], Z, y be Element of Y;

      reconsider E = (f " A) as Subset of [:X, Y:];

      now

        let x be object;

        assume x in ( Y-section ((f " A),y));

        then x in { x where x be Element of X : [x, y] in E } by MEASUR11:def 5;

        then

        consider x1 be Element of X such that

         A1: x1 = x & [x1, y] in E;

        

         A2: [x, y] in ( dom f) & (f . [x, y]) in A by A1, FUNCT_1:def 7;

        then x in { x where x be Element of X : [x, y] in ( dom f) } by A1;

        then x in ( Y-section (( dom f),y)) by MEASUR11:def 5;

        then

         A3: x in ( dom ( ProjPMap2 (f,y))) by Def4;

        (( ProjPMap2 (f,y)) . x1) = (f . (x1,y)) by A1, A2, Def4;

        hence x in (( ProjPMap2 (f,y)) " A) by A1, A2, A3, FUNCT_1:def 7;

      end;

      then

       A4: ( Y-section ((f " A),y)) c= (( ProjPMap2 (f,y)) " A);

      now

        let x be object;

        assume x in (( ProjPMap2 (f,y)) " A);

        then

         A5: x in ( dom ( ProjPMap2 (f,y))) & (( ProjPMap2 (f,y)) . x) in A by FUNCT_1:def 7;

        then x in ( Y-section (( dom f),y)) by Def4;

        then x in { x where x be Element of X : [x, y] in ( dom f) } by MEASUR11:def 5;

        then

        consider x1 be Element of X such that

         A6: x1 = x & [x1, y] in ( dom f);

        (f . (x1,y)) in A by A5, A6, Def4;

        then [x1, y] in (f " A) by A6, FUNCT_1:def 7;

        then x in { x where x be Element of X : [x, y] in (f " A) } by A6;

        hence x in ( Y-section ((f " A),y)) by MEASUR11:def 5;

      end;

      then (( ProjPMap2 (f,y)) " A) c= ( Y-section ((f " A),y));

      hence ( Y-section ((f " A),y)) = (( ProjPMap2 (f,y)) " A) by A4;

    end;

    theorem :: MESFUN12:43

    

     Th43: for X1,X2 be non empty set, A,B be Subset of [:X1, X2:], p be set holds ( X-section ((A \ B),p)) = (( X-section (A,p)) \ ( X-section (B,p))) & ( Y-section ((A \ B),p)) = (( Y-section (A,p)) \ ( Y-section (B,p)))

    proof

      let X1,X2 be non empty set, E1,E2 be Subset of [:X1, X2:], p be set;

      now

        let q be set;

        assume q in ( X-section ((E1 \ E2),p));

        then q in { y where y be Element of X2 : [p, y] in (E1 \ E2) } by MEASUR11:def 4;

        then

         A1: ex y be Element of X2 st q = y & [p, y] in (E1 \ E2);

        then [p, q] in E1 & not [p, q] in E2 by XBOOLE_0:def 5;

        then q in { y where y be Element of X2 : [p, y] in E1 } by A1;

        then

         A3: q in ( X-section (E1,p)) by MEASUR11:def 4;

        now

          assume q in ( X-section (E2,p));

          then q in { y where y be Element of X2 : [p, y] in E2 } by MEASUR11:def 4;

          then ex y be Element of X2 st q = y & [p, y] in E2;

          hence contradiction by A1, XBOOLE_0:def 5;

        end;

        hence q in (( X-section (E1,p)) \ ( X-section (E2,p))) by A3, XBOOLE_0:def 5;

      end;

      then

       A4: ( X-section ((E1 \ E2),p)) c= (( X-section (E1,p)) \ ( X-section (E2,p)));

      now

        let q be set;

        assume q in (( X-section (E1,p)) \ ( X-section (E2,p)));

        then q in ( X-section (E1,p)) & not q in ( X-section (E2,p)) by XBOOLE_0:def 5;

        then

         A5: q in { y where y be Element of X2 : [p, y] in E1 } & not q in { y where y be Element of X2 : [p, y] in E2 } by MEASUR11:def 4;

        then

         A6: ex y be Element of X2 st q = y & [p, y] in E1;

        then not [p, q] in E2 by A5;

        then [p, q] in (E1 \ E2) by A6, XBOOLE_0:def 5;

        then q in { y where y be Element of X2 : [p, y] in (E1 \ E2) } by A6;

        hence q in ( X-section ((E1 \ E2),p)) by MEASUR11:def 4;

      end;

      then (( X-section (E1,p)) \ ( X-section (E2,p))) c= ( X-section ((E1 \ E2),p));

      hence ( X-section ((E1 \ E2),p)) = (( X-section (E1,p)) \ ( X-section (E2,p))) by A4;

      now

        let q be set;

        assume q in ( Y-section ((E1 \ E2),p));

        then q in { x where x be Element of X1 : [x, p] in (E1 \ E2) } by MEASUR11:def 5;

        then

         B1: ex x be Element of X1 st q = x & [x, p] in (E1 \ E2);

        then [q, p] in E1 & not [q, p] in E2 by XBOOLE_0:def 5;

        then q in { x where x be Element of X1 : [x, p] in E1 } by B1;

        then

         B3: q in ( Y-section (E1,p)) by MEASUR11:def 5;

        now

          assume q in ( Y-section (E2,p));

          then q in { x where x be Element of X1 : [x, p] in E2 } by MEASUR11:def 5;

          then ex x be Element of X1 st q = x & [x, p] in E2;

          hence contradiction by B1, XBOOLE_0:def 5;

        end;

        hence q in (( Y-section (E1,p)) \ ( Y-section (E2,p))) by B3, XBOOLE_0:def 5;

      end;

      then

       B4: ( Y-section ((E1 \ E2),p)) c= (( Y-section (E1,p)) \ ( Y-section (E2,p)));

      now

        let q be set;

        assume q in (( Y-section (E1,p)) \ ( Y-section (E2,p)));

        then q in ( Y-section (E1,p)) & not q in ( Y-section (E2,p)) by XBOOLE_0:def 5;

        then

         B5: q in { x where x be Element of X1 : [x, p] in E1 } & not q in { x where x be Element of X1 : [x, p] in E2 } by MEASUR11:def 5;

        then

         B6: ex x be Element of X1 st q = x & [x, p] in E1;

        then not [q, p] in E2 by B5;

        then [q, p] in (E1 \ E2) by B6, XBOOLE_0:def 5;

        then q in { x where x be Element of X1 : [x, p] in (E1 \ E2) } by B6;

        hence q in ( Y-section ((E1 \ E2),p)) by MEASUR11:def 5;

      end;

      then (( Y-section (E1,p)) \ ( Y-section (E2,p))) c= ( Y-section ((E1 \ E2),p));

      hence ( Y-section ((E1 \ E2),p)) = (( Y-section (E1,p)) \ ( Y-section (E2,p))) by B4;

    end;

    theorem :: MESFUN12:44

    

     Th44: for X1,X2 be non empty set, x be Element of X1, y be Element of X2, f1,f2 be PartFunc of [:X1, X2:], ExtREAL holds ( ProjPMap1 ((f1 + f2),x)) = (( ProjPMap1 (f1,x)) + ( ProjPMap1 (f2,x))) & ( ProjPMap1 ((f1 - f2),x)) = (( ProjPMap1 (f1,x)) - ( ProjPMap1 (f2,x))) & ( ProjPMap2 ((f1 + f2),y)) = (( ProjPMap2 (f1,y)) + ( ProjPMap2 (f2,y))) & ( ProjPMap2 ((f1 - f2),y)) = (( ProjPMap2 (f1,y)) - ( ProjPMap2 (f2,y)))

    proof

      let X1,X2 be non empty set, x be Element of X1, y be Element of X2, f1,f2 be PartFunc of [:X1, X2:], ExtREAL ;

      

       A1: ( dom (f1 + f2)) = ((( dom f1) /\ ( dom f2)) \ (((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty })))) by MESFUNC1:def 3;

      

       B1: ( dom (f1 - f2)) = ((( dom f1) /\ ( dom f2)) \ (((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty })))) by MESFUNC1:def 4;

      

       A2: ( dom ( ProjPMap1 (f1,x))) = ( X-section (( dom f1),x)) & ( dom ( ProjPMap1 (f2,x))) = ( X-section (( dom f2),x)) & ( dom ( ProjPMap2 (f1,y))) = ( Y-section (( dom f1),y)) & ( dom ( ProjPMap2 (f2,y))) = ( Y-section (( dom f2),y)) by Def3, Def4;

      

       A3: ( X-section ((f1 " { -infty }),x)) = (( ProjPMap1 (f1,x)) " { -infty }) & ( X-section ((f1 " { +infty }),x)) = (( ProjPMap1 (f1,x)) " { +infty }) & ( X-section ((f2 " { -infty }),x)) = (( ProjPMap1 (f2,x)) " { -infty }) & ( X-section ((f2 " { +infty }),x)) = (( ProjPMap1 (f2,x)) " { +infty }) & ( Y-section ((f1 " { -infty }),y)) = (( ProjPMap2 (f1,y)) " { -infty }) & ( Y-section ((f1 " { +infty }),y)) = (( ProjPMap2 (f1,y)) " { +infty }) & ( Y-section ((f2 " { -infty }),y)) = (( ProjPMap2 (f2,y)) " { -infty }) & ( Y-section ((f2 " { +infty }),y)) = (( ProjPMap2 (f2,y)) " { +infty }) by Th42, Th41;

      

       A4: ( dom ( ProjPMap1 ((f1 + f2),x))) = ( X-section (( dom (f1 + f2)),x)) by Def3

      .= (( X-section ((( dom f1) /\ ( dom f2)),x)) \ ( X-section ((((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty }))),x))) by A1, Th43

      .= ((( X-section (( dom f1),x)) /\ ( X-section (( dom f2),x))) \ ( X-section ((((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty }))),x))) by MEASUR11: 27

      .= ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ (( X-section (((f1 " { -infty }) /\ (f2 " { +infty })),x)) \/ ( X-section (((f1 " { +infty }) /\ (f2 " { -infty })),x)))) by A2, MEASUR11: 26;

      

      then

       A5: ( dom ( ProjPMap1 ((f1 + f2),x))) = ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ ((( X-section ((f1 " { -infty }),x)) /\ ( X-section ((f2 " { +infty }),x))) \/ ( X-section (((f1 " { +infty }) /\ (f2 " { -infty })),x)))) by MEASUR11: 27

      .= ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ (((( ProjPMap1 (f1,x)) " { -infty }) /\ (( ProjPMap1 (f2,x)) " { +infty })) \/ (( X-section ((f1 " { +infty }),x)) /\ ( X-section ((f2 " { -infty }),x))))) by A3, MEASUR11: 27

      .= ( dom (( ProjPMap1 (f1,x)) + ( ProjPMap1 (f2,x)))) by A3, MESFUNC1:def 3;

      for y be Element of X2 st y in ( dom ( ProjPMap1 ((f1 + f2),x))) holds (( ProjPMap1 ((f1 + f2),x)) . y) = ((( ProjPMap1 (f1,x)) + ( ProjPMap1 (f2,x))) . y)

      proof

        let y be Element of X2;

        assume

         A6: y in ( dom ( ProjPMap1 ((f1 + f2),x)));

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        

         A7: (( ProjPMap1 ((f1 + f2),x)) . y) = ((f1 + f2) . (x,y)) by A6, Th26;

        then [x, y] in ( dom (f1 + f2)) by A6, Th40;

        then

         A8: (( ProjPMap1 ((f1 + f2),x)) . y) = ((f1 . z) + (f2 . z)) by A7, MESFUNC1:def 3;

        y in (( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) by A4, A6, XBOOLE_0:def 5;

        then y in ( dom ( ProjPMap1 (f1,x))) & y in ( dom ( ProjPMap1 (f2,x))) by XBOOLE_0:def 4;

        then (( ProjPMap1 (f1,x)) . y) = (f1 . (x,y)) & (( ProjPMap1 (f2,x)) . y) = (f2 . (x,y)) by Th26;

        hence (( ProjPMap1 ((f1 + f2),x)) . y) = ((( ProjPMap1 (f1,x)) + ( ProjPMap1 (f2,x))) . y) by A8, A5, A6, MESFUNC1:def 3;

      end;

      hence ( ProjPMap1 ((f1 + f2),x)) = (( ProjPMap1 (f1,x)) + ( ProjPMap1 (f2,x))) by A5, PARTFUN1: 5;

      

       B4: ( dom ( ProjPMap1 ((f1 - f2),x))) = ( X-section (( dom (f1 - f2)),x)) by Def3

      .= (( X-section ((( dom f1) /\ ( dom f2)),x)) \ ( X-section ((((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty }))),x))) by B1, Th43

      .= ((( X-section (( dom f1),x)) /\ ( X-section (( dom f2),x))) \ ( X-section ((((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty }))),x))) by MEASUR11: 27

      .= ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ (( X-section (((f1 " { +infty }) /\ (f2 " { +infty })),x)) \/ ( X-section (((f1 " { -infty }) /\ (f2 " { -infty })),x)))) by A2, MEASUR11: 26;

      

      then

       B5: ( dom ( ProjPMap1 ((f1 - f2),x))) = ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ ((( X-section ((f1 " { +infty }),x)) /\ ( X-section ((f2 " { +infty }),x))) \/ ( X-section (((f1 " { -infty }) /\ (f2 " { -infty })),x)))) by MEASUR11: 27

      .= ((( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) \ (((( ProjPMap1 (f1,x)) " { +infty }) /\ (( ProjPMap1 (f2,x)) " { +infty })) \/ (( X-section ((f1 " { -infty }),x)) /\ ( X-section ((f2 " { -infty }),x))))) by A3, MEASUR11: 27

      .= ( dom (( ProjPMap1 (f1,x)) - ( ProjPMap1 (f2,x)))) by A3, MESFUNC1:def 4;

      for y be Element of X2 st y in ( dom ( ProjPMap1 ((f1 - f2),x))) holds (( ProjPMap1 ((f1 - f2),x)) . y) = ((( ProjPMap1 (f1,x)) - ( ProjPMap1 (f2,x))) . y)

      proof

        let y be Element of X2;

        assume

         A6: y in ( dom ( ProjPMap1 ((f1 - f2),x)));

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        

         A7: (( ProjPMap1 ((f1 - f2),x)) . y) = ((f1 - f2) . (x,y)) by A6, Th26;

        then [x, y] in ( dom (f1 - f2)) by A6, Th40;

        then

         A8: (( ProjPMap1 ((f1 - f2),x)) . y) = ((f1 . z) - (f2 . z)) by A7, MESFUNC1:def 4;

        y in (( dom ( ProjPMap1 (f1,x))) /\ ( dom ( ProjPMap1 (f2,x)))) by B4, A6, XBOOLE_0:def 5;

        then y in ( dom ( ProjPMap1 (f1,x))) & y in ( dom ( ProjPMap1 (f2,x))) by XBOOLE_0:def 4;

        then (( ProjPMap1 (f1,x)) . y) = (f1 . (x,y)) & (( ProjPMap1 (f2,x)) . y) = (f2 . (x,y)) by Th26;

        hence (( ProjPMap1 ((f1 - f2),x)) . y) = ((( ProjPMap1 (f1,x)) - ( ProjPMap1 (f2,x))) . y) by A8, B5, A6, MESFUNC1:def 4;

      end;

      hence ( ProjPMap1 ((f1 - f2),x)) = (( ProjPMap1 (f1,x)) - ( ProjPMap1 (f2,x))) by B5, PARTFUN1: 5;

      

       C4: ( dom ( ProjPMap2 ((f1 + f2),y))) = ( Y-section (( dom (f1 + f2)),y)) by Def4

      .= (( Y-section ((( dom f1) /\ ( dom f2)),y)) \ ( Y-section ((((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty }))),y))) by A1, Th43

      .= ((( Y-section (( dom f1),y)) /\ ( Y-section (( dom f2),y))) \ ( Y-section ((((f1 " { -infty }) /\ (f2 " { +infty })) \/ ((f1 " { +infty }) /\ (f2 " { -infty }))),y))) by MEASUR11: 27

      .= ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ (( Y-section (((f1 " { -infty }) /\ (f2 " { +infty })),y)) \/ ( Y-section (((f1 " { +infty }) /\ (f2 " { -infty })),y)))) by A2, MEASUR11: 26;

      

      then

       C5: ( dom ( ProjPMap2 ((f1 + f2),y))) = ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ ((( Y-section ((f1 " { -infty }),y)) /\ ( Y-section ((f2 " { +infty }),y))) \/ ( Y-section (((f1 " { +infty }) /\ (f2 " { -infty })),y)))) by MEASUR11: 27

      .= ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ (((( ProjPMap2 (f1,y)) " { -infty }) /\ (( ProjPMap2 (f2,y)) " { +infty })) \/ (( Y-section ((f1 " { +infty }),y)) /\ ( Y-section ((f2 " { -infty }),y))))) by A3, MEASUR11: 27

      .= ( dom (( ProjPMap2 (f1,y)) + ( ProjPMap2 (f2,y)))) by A3, MESFUNC1:def 3;

      for x be Element of X1 st x in ( dom ( ProjPMap2 ((f1 + f2),y))) holds (( ProjPMap2 ((f1 + f2),y)) . x) = ((( ProjPMap2 (f1,y)) + ( ProjPMap2 (f2,y))) . x)

      proof

        let x be Element of X1;

        assume

         C6: x in ( dom ( ProjPMap2 ((f1 + f2),y)));

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        

         C7: (( ProjPMap2 ((f1 + f2),y)) . x) = ((f1 + f2) . (x,y)) by C6, Th26;

        then [x, y] in ( dom (f1 + f2)) by C6, Th40;

        then

         C8: (( ProjPMap2 ((f1 + f2),y)) . x) = ((f1 . z) + (f2 . z)) by C7, MESFUNC1:def 3;

        x in (( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) by C4, C6, XBOOLE_0:def 5;

        then x in ( dom ( ProjPMap2 (f1,y))) & x in ( dom ( ProjPMap2 (f2,y))) by XBOOLE_0:def 4;

        then (( ProjPMap2 (f1,y)) . x) = (f1 . (x,y)) & (( ProjPMap2 (f2,y)) . x) = (f2 . (x,y)) by Th26;

        hence (( ProjPMap2 ((f1 + f2),y)) . x) = ((( ProjPMap2 (f1,y)) + ( ProjPMap2 (f2,y))) . x) by C8, C5, C6, MESFUNC1:def 3;

      end;

      hence ( ProjPMap2 ((f1 + f2),y)) = (( ProjPMap2 (f1,y)) + ( ProjPMap2 (f2,y))) by C5, PARTFUN1: 5;

      

       D4: ( dom ( ProjPMap2 ((f1 - f2),y))) = ( Y-section (( dom (f1 - f2)),y)) by Def4

      .= (( Y-section ((( dom f1) /\ ( dom f2)),y)) \ ( Y-section ((((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty }))),y))) by B1, Th43

      .= ((( Y-section (( dom f1),y)) /\ ( Y-section (( dom f2),y))) \ ( Y-section ((((f1 " { +infty }) /\ (f2 " { +infty })) \/ ((f1 " { -infty }) /\ (f2 " { -infty }))),y))) by MEASUR11: 27

      .= ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ (( Y-section (((f1 " { +infty }) /\ (f2 " { +infty })),y)) \/ ( Y-section (((f1 " { -infty }) /\ (f2 " { -infty })),y)))) by A2, MEASUR11: 26;

      

      then

       D5: ( dom ( ProjPMap2 ((f1 - f2),y))) = ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ ((( Y-section ((f1 " { +infty }),y)) /\ ( Y-section ((f2 " { +infty }),y))) \/ ( Y-section (((f1 " { -infty }) /\ (f2 " { -infty })),y)))) by MEASUR11: 27

      .= ((( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) \ (((( ProjPMap2 (f1,y)) " { +infty }) /\ (( ProjPMap2 (f2,y)) " { +infty })) \/ (( Y-section ((f1 " { -infty }),y)) /\ ( Y-section ((f2 " { -infty }),y))))) by A3, MEASUR11: 27

      .= ( dom (( ProjPMap2 (f1,y)) - ( ProjPMap2 (f2,y)))) by A3, MESFUNC1:def 4;

      for x be Element of X1 st x in ( dom ( ProjPMap2 ((f1 - f2),y))) holds (( ProjPMap2 ((f1 - f2),y)) . x) = ((( ProjPMap2 (f1,y)) - ( ProjPMap2 (f2,y))) . x)

      proof

        let x be Element of X1;

        assume

         D6: x in ( dom ( ProjPMap2 ((f1 - f2),y)));

        reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        

         D7: (( ProjPMap2 ((f1 - f2),y)) . x) = ((f1 - f2) . (x,y)) by D6, Th26;

        then [x, y] in ( dom (f1 - f2)) by D6, Th40;

        then

         D8: (( ProjPMap2 ((f1 - f2),y)) . x) = ((f1 . z) - (f2 . z)) by D7, MESFUNC1:def 4;

        x in (( dom ( ProjPMap2 (f1,y))) /\ ( dom ( ProjPMap2 (f2,y)))) by D4, D6, XBOOLE_0:def 5;

        then x in ( dom ( ProjPMap2 (f1,y))) & x in ( dom ( ProjPMap2 (f2,y))) by XBOOLE_0:def 4;

        then (( ProjPMap2 (f1,y)) . x) = (f1 . (x,y)) & (( ProjPMap2 (f2,y)) . x) = (f2 . (x,y)) by Th26;

        hence (( ProjPMap2 ((f1 - f2),y)) . x) = ((( ProjPMap2 (f1,y)) - ( ProjPMap2 (f2,y))) . x) by D8, D5, D6, MESFUNC1:def 4;

      end;

      hence ( ProjPMap2 ((f1 - f2),y)) = (( ProjPMap2 (f1,y)) - ( ProjPMap2 (f2,y))) by D5, PARTFUN1: 5;

    end;

    

     Lm4: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st E c= ( dom f) & f is E -measurable holds ( ProjPMap1 (( max+ f),x)) is ( Measurable-X-section (E,x)) -measurable & ( ProjPMap2 (( max+ f),y)) is ( Measurable-Y-section (E,y)) -measurable & ( ProjPMap1 (( max- f),x)) is ( Measurable-X-section (E,x)) -measurable & ( ProjPMap2 (( max- f),y)) is ( Measurable-Y-section (E,y)) -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A c= ( dom f) and

       A2: f is A -measurable;

      

       A3: ( max+ f) is nonnegative & ( max- f) is nonnegative by MESFUN11: 5;

      

       A4: ( max+ f) is A -measurable by A2, MESFUNC2: 25;

      

       A5: ( max- f) is A -measurable by A1, A2, MESFUNC2: 26;

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

      hence ( ProjPMap1 (( max+ f),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap2 (( max+ f),y)) is ( Measurable-Y-section (A,y)) -measurable by A1, A3, A4, Lm3;

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

      hence ( ProjPMap1 (( max- f),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap2 (( max- f),y)) is ( Measurable-Y-section (A,y)) -measurable by A1, A3, A5, Lm3;

    end;

    theorem :: MESFUN12:45

    

     Th45: for X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1 holds ( ProjPMap1 (( max+ f),x)) = ( max+ ( ProjPMap1 (f,x))) & ( ProjPMap1 (( max- f),x)) = ( max- ( ProjPMap1 (f,x)))

    proof

      let X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1;

      ( dom ( ProjPMap1 (( max+ f),x))) = ( X-section (( dom ( max+ f)),x)) & ( dom ( ProjPMap1 (( max- f),x))) = ( X-section (( dom ( max- f)),x)) by Def3;

      then

       A1: ( dom ( ProjPMap1 (( max+ f),x))) = ( X-section (( dom f),x)) & ( dom ( ProjPMap1 (( max- f),x))) = ( X-section (( dom f),x)) by MESFUNC2:def 2, MESFUNC2:def 3;

      ( dom ( max+ ( ProjPMap1 (f,x)))) = ( dom ( ProjPMap1 (f,x))) & ( dom ( max- ( ProjPMap1 (f,x)))) = ( dom ( ProjPMap1 (f,x))) by MESFUNC2:def 2, MESFUNC2:def 3;

      then

       A2: ( dom ( max+ ( ProjPMap1 (f,x)))) = ( X-section (( dom f),x)) & ( dom ( max- ( ProjPMap1 (f,x)))) = ( X-section (( dom f),x)) by Def3;

      for y be Element of X2 st y in ( dom ( ProjPMap1 (( max+ f),x))) holds (( ProjPMap1 (( max+ f),x)) . y) = (( max+ ( ProjPMap1 (f,x))) . y)

      proof

        let y be Element of X2;

        assume

         A3: y in ( dom ( ProjPMap1 (( max+ f),x)));

        then y in { y where y be Element of X2 : [x, y] in ( dom f) } by A1, MEASUR11:def 4;

        then

         A4: ex y1 be Element of X2 st y1 = y & [x, y1] in ( dom f);

        set z = [x, y];

        

         A5: [x, y] in ( dom ( max+ f)) by A4, MESFUNC2:def 2;

        

        then

         A6: (( ProjPMap1 (( max+ f),x)) . y) = (( max+ f) . (x,y)) by Def3

        .= ( max ((f . z), 0 )) by A5, MESFUNC2:def 2;

        (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by A4, Def3;

        hence thesis by A6, A1, A3, A2, MESFUNC2:def 2;

      end;

      hence ( ProjPMap1 (( max+ f),x)) = ( max+ ( ProjPMap1 (f,x))) by A1, A2, PARTFUN1: 5;

      for y be Element of X2 st y in ( dom ( ProjPMap1 (( max- f),x))) holds (( ProjPMap1 (( max- f),x)) . y) = (( max- ( ProjPMap1 (f,x))) . y)

      proof

        let y be Element of X2;

        assume

         A8: y in ( dom ( ProjPMap1 (( max- f),x)));

        then y in { y where y be Element of X2 : [x, y] in ( dom f) } by A1, MEASUR11:def 4;

        then

         A9: ex y1 be Element of X2 st y1 = y & [x, y1] in ( dom f);

        set z = [x, y];

        

         A10: [x, y] in ( dom ( max- f)) by A9, MESFUNC2:def 3;

        

        then

         A11: (( ProjPMap1 (( max- f),x)) . y) = (( max- f) . (x,y)) by Def3

        .= ( max (( - (f . z)), 0 )) by A10, MESFUNC2:def 3;

        (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by A9, Def3;

        hence thesis by A11, A1, A2, A8, MESFUNC2:def 3;

      end;

      hence ( ProjPMap1 (( max- f),x)) = ( max- ( ProjPMap1 (f,x))) by A1, A2, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:46

    

     Th46: for X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , y be Element of X2 holds ( ProjPMap2 (( max+ f),y)) = ( max+ ( ProjPMap2 (f,y))) & ( ProjPMap2 (( max- f),y)) = ( max- ( ProjPMap2 (f,y)))

    proof

      let X1,X2 be non empty set, f be PartFunc of [:X1, X2:], ExtREAL , y be Element of X2;

      ( dom ( ProjPMap2 (( max+ f),y))) = ( Y-section (( dom ( max+ f)),y)) & ( dom ( ProjPMap2 (( max- f),y))) = ( Y-section (( dom ( max- f)),y)) by Def4;

      then

       A1: ( dom ( ProjPMap2 (( max+ f),y))) = ( Y-section (( dom f),y)) & ( dom ( ProjPMap2 (( max- f),y))) = ( Y-section (( dom f),y)) by MESFUNC2:def 2, MESFUNC2:def 3;

      ( dom ( max+ ( ProjPMap2 (f,y)))) = ( dom ( ProjPMap2 (f,y))) & ( dom ( max- ( ProjPMap2 (f,y)))) = ( dom ( ProjPMap2 (f,y))) by MESFUNC2:def 2, MESFUNC2:def 3;

      then

       A2: ( dom ( max+ ( ProjPMap2 (f,y)))) = ( Y-section (( dom f),y)) & ( dom ( max- ( ProjPMap2 (f,y)))) = ( Y-section (( dom f),y)) by Def4;

      for x be Element of X1 st x in ( dom ( ProjPMap2 (( max+ f),y))) holds (( ProjPMap2 (( max+ f),y)) . x) = (( max+ ( ProjPMap2 (f,y))) . x)

      proof

        let x be Element of X1;

        assume

         A3: x in ( dom ( ProjPMap2 (( max+ f),y)));

        then x in { x where x be Element of X1 : [x, y] in ( dom f) } by A1, MEASUR11:def 5;

        then

         A4: ex x1 be Element of X1 st x1 = x & [x1, y] in ( dom f);

        set z = [x, y];

        

         A5: [x, y] in ( dom ( max+ f)) by A4, MESFUNC2:def 2;

        

        then

         A6: (( ProjPMap2 (( max+ f),y)) . x) = (( max+ f) . (x,y)) by Def4

        .= ( max ((f . z), 0 )) by A5, MESFUNC2:def 2;

        (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by A4, Def4;

        hence thesis by A6, A1, A3, A2, MESFUNC2:def 2;

      end;

      hence ( ProjPMap2 (( max+ f),y)) = ( max+ ( ProjPMap2 (f,y))) by A1, A2, PARTFUN1: 5;

      for x be Element of X1 st x in ( dom ( ProjPMap2 (( max- f),y))) holds (( ProjPMap2 (( max- f),y)) . x) = (( max- ( ProjPMap2 (f,y))) . x)

      proof

        let x be Element of X1;

        assume

         A8: x in ( dom ( ProjPMap2 (( max- f),y)));

        then x in { x where x be Element of X1 : [x, y] in ( dom f) } by A1, MEASUR11:def 5;

        then

         A9: ex x1 be Element of X1 st x1 = x & [x1, y] in ( dom f);

        set z = [x, y];

        

         A10: [x, y] in ( dom ( max- f)) by A9, MESFUNC2:def 3;

        

        then

         A11: (( ProjPMap2 (( max- f),y)) . x) = (( max- f) . (x,y)) by Def4

        .= ( max (( - (f . z)), 0 )) by A10, MESFUNC2:def 3;

        (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by A9, Def4;

        hence thesis by A11, A1, A8, A2, MESFUNC2:def 3;

      end;

      hence ( ProjPMap2 (( max- f),y)) = ( max- ( ProjPMap2 (f,y))) by A1, A2, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:47

    

     Th47: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st E c= ( dom f) & f is E -measurable holds ( ProjPMap1 (f,x)) is ( Measurable-X-section (E,x)) -measurable & ( ProjPMap2 (f,y)) is ( Measurable-Y-section (E,y)) -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, f be PartFunc of [:X1, X2:], ExtREAL , x be Element of X1, y be Element of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A c= ( dom f) and

       A2: f is A -measurable;

      ( X-section (A,x)) c= ( X-section (( dom f),x)) & ( Y-section (A,y)) c= ( Y-section (( dom f),y)) by A1, MEASUR11: 20, MEASUR11: 21;

      then ( Measurable-X-section (A,x)) c= ( X-section (( dom f),x)) & ( Measurable-Y-section (A,y)) c= ( Y-section (( dom f),y)) by MEASUR11:def 6, MEASUR11:def 7;

      then

       A3: ( Measurable-X-section (A,x)) c= ( dom ( ProjPMap1 (f,x))) & ( Measurable-Y-section (A,y)) c= ( dom ( ProjPMap2 (f,y))) by Def3, Def4;

      ( ProjPMap1 (( max+ f),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap2 (( max+ f),y)) is ( Measurable-Y-section (A,y)) -measurable & ( ProjPMap1 (( max- f),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap2 (( max- f),y)) is ( Measurable-Y-section (A,y)) -measurable by A1, A2, Lm4;

      then ( max+ ( ProjPMap1 (f,x))) is ( Measurable-X-section (A,x)) -measurable & ( max+ ( ProjPMap2 (f,y))) is ( Measurable-Y-section (A,y)) -measurable & ( max- ( ProjPMap1 (f,x))) is ( Measurable-X-section (A,x)) -measurable & ( max- ( ProjPMap2 (f,y))) is ( Measurable-Y-section (A,y)) -measurable by Th45, Th46;

      hence thesis by A3, MESFUN11: 10;

    end;

    definition

      let X1,X2,Y be non empty set;

      let F be Functional_Sequence of [:X1, X2:], Y;

      let x be Element of X1;

      :: MESFUN12:def5

      func ProjPMap1 (F,x) -> Functional_Sequence of X2, Y means

      : Def5: for n be Nat holds (it . n) = ( ProjPMap1 ((F . n),x));

      existence

      proof

        defpred P[ Nat, object] means $2 = ( ProjPMap1 ((F . $1),x));

        

         A1: for n be Element of NAT holds ex f be Element of ( PFuncs (X2,Y)) st P[n, f]

        proof

          let n be Element of NAT ;

          reconsider f = ( ProjPMap1 ((F . n),x)) as Element of ( PFuncs (X2,Y)) by PARTFUN1: 45;

          take f;

          thus thesis;

        end;

        consider IT be Function of NAT , ( PFuncs (X2,Y)) such that

         A2: for n be Element of NAT holds P[n, (IT . n)] from FUNCT_2:sch 3( A1);

        take IT;

        hereby

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          hence (IT . n) = ( ProjPMap1 ((F . n),x)) by A2;

        end;

      end;

      uniqueness

      proof

        let F1,F2 be Functional_Sequence of X2, Y;

        assume that

         A1: for n be Nat holds (F1 . n) = ( ProjPMap1 ((F . n),x)) and

         A2: for n be Nat holds (F2 . n) = ( ProjPMap1 ((F . n),x));

        now

          let n be Element of NAT ;

          (F1 . n) = ( ProjPMap1 ((F . n),x)) by A1;

          hence (F1 . n) = (F2 . n) by A2;

        end;

        hence thesis by FUNCT_2:def 8;

      end;

    end

    definition

      let X1,X2,Y be non empty set;

      let F be Functional_Sequence of [:X1, X2:], Y;

      let y be Element of X2;

      :: MESFUN12:def6

      func ProjPMap2 (F,y) -> Functional_Sequence of X1, Y means

      : Def6: for n be Nat holds (it . n) = ( ProjPMap2 ((F . n),y));

      existence

      proof

        defpred P[ Nat, object] means $2 = ( ProjPMap2 ((F . $1),y));

        

         A1: for n be Element of NAT holds ex f be Element of ( PFuncs (X1,Y)) st P[n, f]

        proof

          let n be Element of NAT ;

          reconsider f = ( ProjPMap2 ((F . n),y)) as Element of ( PFuncs (X1,Y)) by PARTFUN1: 45;

          take f;

          thus thesis;

        end;

        consider IT be Function of NAT , ( PFuncs (X1,Y)) such that

         A2: for n be Element of NAT holds P[n, (IT . n)] from FUNCT_2:sch 3( A1);

        take IT;

        hereby

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          hence (IT . n) = ( ProjPMap2 ((F . n),y)) by A2;

        end;

      end;

      uniqueness

      proof

        let F1,F2 be Functional_Sequence of X1, Y;

        assume that

         A1: for n be Nat holds (F1 . n) = ( ProjPMap2 ((F . n),y)) and

         A2: for n be Nat holds (F2 . n) = ( ProjPMap2 ((F . n),y));

        now

          let n be Element of NAT ;

          (F1 . n) = ( ProjPMap2 ((F . n),y)) by A1;

          hence (F1 . n) = (F2 . n) by A2;

        end;

        hence thesis by FUNCT_2:def 8;

      end;

    end

    theorem :: MESFUN12:48

    

     Th48: for X1,X2 be non empty set, E be Subset of [:X1, X2:], x be Element of X1, y be Element of X2 holds ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( X-section (E,x)),X2)) & ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Y-section (E,y)),X1))

    proof

      let X1,X2 be non empty set, E be Subset of [:X1, X2:], x be Element of X1, y be Element of X2;

      for y be Element of X2 holds (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = (( chi (( X-section (E,x)),X2)) . y)

      proof

        let y be Element of X2;

        

         A1: (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = (( chi (E, [:X1, X2:])) . (x,y)) by MESFUNC9:def 6;

        then

         A2: [x, y] in E implies (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = 1 by FUNCT_3:def 3;

         [x, y] is Element of [:X1, X2:] by ZFMISC_1:def 2;

        then

         A3: not [x, y] in E implies (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = 0 by A1, FUNCT_3:def 3;

        per cases ;

          suppose

           A4: [x, y] in E;

          then y in ( X-section (E,x)) by Th25;

          hence (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = (( chi (( X-section (E,x)),X2)) . y) by A2, A4, FUNCT_3:def 3;

        end;

          suppose

           A5: not [x, y] in E;

          then not y in ( X-section (E,x)) by Th25;

          hence (( ProjMap1 (( chi (E, [:X1, X2:])),x)) . y) = (( chi (( X-section (E,x)),X2)) . y) by A3, A5, FUNCT_3:def 3;

        end;

      end;

      then ( ProjMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( X-section (E,x)),X2)) by FUNCT_2:def 8;

      hence ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( X-section (E,x)),X2)) by Th27;

      for x be Element of X1 holds (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = (( chi (( Y-section (E,y)),X1)) . x)

      proof

        let x be Element of X1;

        

         A1: (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = (( chi (E, [:X1, X2:])) . (x,y)) by MESFUNC9:def 7;

        then

         A2: [x, y] in E implies (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = 1 by FUNCT_3:def 3;

         [x, y] is Element of [:X1, X2:] by ZFMISC_1:def 2;

        then

         A3: not [x, y] in E implies (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = 0 by A1, FUNCT_3:def 3;

        per cases ;

          suppose

           A4: [x, y] in E;

          then x in ( Y-section (E,y)) by Th25;

          hence (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = (( chi (( Y-section (E,y)),X1)) . x) by A2, A4, FUNCT_3:def 3;

        end;

          suppose

           A5: not [x, y] in E;

          then not x in ( Y-section (E,y)) by Th25;

          hence (( ProjMap2 (( chi (E, [:X1, X2:])),y)) . x) = (( chi (( Y-section (E,y)),X1)) . x) by A3, A5, FUNCT_3:def 3;

        end;

      end;

      then ( ProjMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Y-section (E,y)),X1)) by FUNCT_2:def 8;

      hence ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Y-section (E,y)),X1)) by Th27;

    end;

    theorem :: MESFUN12:49

    

     Th49: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, er be ExtReal holds ( Integral (M,( chi (er,E,X)))) = (er * (M . E))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, er be ExtReal;

      reconsider XX = X as Element of S by MEASURE1: 7;

      per cases ;

        suppose

         a1: er = +infty ;

        then

         a2: ( chi (er,E,X)) = ( Xchi (E,X)) by Th2;

        per cases ;

          suppose

           a3: (M . E) <> 0 ;

          then

           a4: (M . E) > 0 by MEASURE1:def 2;

          

          thus ( Integral (M,( chi (er,E,X)))) = +infty by a2, a3, MEASUR10: 33

          .= (er * (M . E)) by a1, a4, XXREAL_3:def 5;

        end;

          suppose

           a5: (M . E) = 0 ;

          then ( Integral (M,( chi (er,E,X)))) = 0 by a2, MEASUR10: 33;

          hence ( Integral (M,( chi (er,E,X)))) = (er * (M . E)) by a5;

        end;

      end;

        suppose

         a6: er = -infty ;

        then

         a7: ( chi (er,E,X)) = ( - ( Xchi (E,X))) by Th2;

        

         a10: ( dom ( Xchi (E,X))) = XX by FUNCT_2:def 1;

        

         W: ( Xchi (E,X)) is XX -measurable by MEASUR10: 32;

        per cases ;

          suppose

           a8: (M . E) <> 0 ;

          then

           a9: (M . E) > 0 by MEASURE1:def 2;

          

          thus ( Integral (M,( chi (er,E,X)))) = ( - ( Integral (M,( Xchi (E,X))))) by a10, a7, MESFUN11: 52, W

          .= ( - +infty ) by a8, MEASUR10: 33

          .= (er * (M . E)) by a6, a9, XXREAL_3:def 5, XXREAL_3: 6;

        end;

          suppose

           a12: (M . E) = 0 ;

          

          thus ( Integral (M,( chi (er,E,X)))) = ( - ( Integral (M,( Xchi (E,X))))) by a10, a7, MESFUN11: 52, W

          .= ( - 0 ) by a12, MEASUR10: 33

          .= (er * (M . E)) by a12;

        end;

      end;

        suppose er <> +infty & er <> -infty ;

        then er in REAL by XXREAL_0: 14;

        then

        reconsider r = er as Real;

        

         a14: ( chi (E,X)) is_simple_func_in S by Th12;

        ( chi (er,E,X)) = (r (#) ( chi (E,X))) by Th1;

        

        hence ( Integral (M,( chi (er,E,X)))) = (r * ( integral' (M,( chi (E,X))))) by Th12, MESFUN11: 59

        .= (r * ( Integral (M,( chi (E,X))))) by a14, MESFUNC5: 89

        .= (er * (M . E)) by MESFUNC9: 14;

      end;

    end;

    theorem :: MESFUN12:50

    

     Th50: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, er be ExtReal holds ( Integral (M,(( chi (er,E,X)) | E))) = (er * (M . E))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, er be ExtReal;

      reconsider XX = X as Element of S by MEASURE1: 7;

      

       A1: XX = ( dom ( chi (er,E,X))) by FUNCT_2:def 1;

      then ( dom (( chi (er,E,X)) | (XX \ E))) = (XX /\ (XX \ E)) by RELAT_1: 61;

      then

       A2: ( dom (( chi (er,E,X)) | (XX \ E))) = (XX \ E) by XBOOLE_1: 28;

      

       A3: (( chi (er,E,X)) | (XX \ E)) is (XX \ E) -measurable by Th15;

      

       A4: (E \/ (XX \ E)) = (X \/ E) by XBOOLE_1: 39

      .= X by XBOOLE_1: 12;

      

       A5: E misses (XX \ E) by XBOOLE_1: 79;

      

       A6: ( Integral (M,( chi (er,E,X)))) = ( Integral (M,(( chi (er,E,X)) | X)));

      per cases ;

        suppose er = +infty ;

        then

         A7: ( chi (er,E,X)) = ( Xchi (E,X)) by Th2;

        then

         A8: (( chi (er,E,X)) | (XX \ E)) is nonnegative by MESFUNC5: 15;

        ( chi (er,E,X)) is XX -measurable by Th13;

        then

         V: ex W be Element of S st W = ( dom ( chi (er,E,X))) & ( chi (er,E,X)) is W -measurable by A1;

        ( Integral (M,( chi (er,E,X)))) = (( Integral (M,(( chi (er,E,X)) | E))) + ( Integral (M,(( chi (er,E,X)) | (XX \ E))))) by A4, V, A5, A6, A7, MESFUNC5: 91;

        then

         A9: (( Integral (M,(( chi (er,E,X)) | E))) + ( Integral (M,(( chi (er,E,X)) | (XX \ E))))) = (er * (M . E)) by Th49;

        for x be Element of X st x in ( dom (( chi (er,E,X)) | (XX \ E))) holds ((( chi (er,E,X)) | (XX \ E)) . x) = 0 by A5, Th16;

        then ( integral+ (M,(( chi (er,E,X)) | (XX \ E)))) = 0 by A2, Th15, MESFUNC5: 87;

        then ( Integral (M,(( chi (er,E,X)) | (XX \ E)))) = 0 by A2, A8, Th15, MESFUNC5: 88;

        hence ( Integral (M,(( chi (er,E,X)) | E))) = (er * (M . E)) by A9, XXREAL_3: 4;

      end;

        suppose er = -infty ;

        then

         A10: ( chi (er,E,X)) = ( - ( Xchi (E,X))) by Th2;

        then

         A11: (( chi (er,E,X)) | (XX \ E)) is nonpositive by MESFUN11: 1;

        ( chi (er,E,X)) is XX -measurable by Th13;

        then ex W be Element of S st W = ( dom ( chi (er,E,X))) & ( chi (er,E,X)) is W -measurable by A1;

        then ( Integral (M,( chi (er,E,X)))) = (( Integral (M,(( chi (er,E,X)) | E))) + ( Integral (M,(( chi (er,E,X)) | (XX \ E))))) by A4, A5, A6, A10, MESFUN11: 62;

        then

         A12: (( Integral (M,(( chi (er,E,X)) | E))) + ( Integral (M,(( chi (er,E,X)) | (XX \ E))))) = (er * (M . E)) by Th49;

        

         A13: ( dom (( - ( chi (er,E,X))) | (XX \ E))) = ( dom ( - (( chi (er,E,X)) | (XX \ E)))) by MESFUN11: 3

        .= (XX \ E) by A2, MESFUNC1:def 7;

        ( - (( chi (er,E,X)) | (XX \ E))) is (XX \ E) -measurable by A2, Th15, MEASUR11: 63;

        then

         A14: (( - ( chi (er,E,X))) | (XX \ E)) is (XX \ E) -measurable by MESFUN11: 3;

        now

          let x be Element of X;

          assume

           A15: x in ( dom (( - ( chi (er,E,X))) | (XX \ E)));

          then x in (( dom ( - ( chi (er,E,X)))) /\ (XX \ E)) by RELAT_1: 61;

          then

           A16: x in ( dom ( - ( chi (er,E,X)))) & x in (XX \ E) by XBOOLE_0:def 4;

          then x in X & not x in E by XBOOLE_0:def 5;

          then (( chi (er,E,X)) . x) = 0 by Def1;

          then (( - ( chi (er,E,X))) . x) = ( - 0 ) by A16, MESFUNC1:def 7;

          hence ((( - ( chi (er,E,X))) | (XX \ E)) . x) = 0 by A15, FUNCT_1: 47;

        end;

        then ( integral+ (M,(( - ( chi (er,E,X))) | (XX \ E)))) = 0 by A13, A14, MESFUNC5: 87;

        then ( integral+ (M,( - (( chi (er,E,X)) | (XX \ E))))) = 0 by MESFUN11: 3;

        then ( Integral (M,(( chi (er,E,X)) | (XX \ E)))) = ( - 0 ) by A2, A3, A11, MESFUN11: 57;

        hence ( Integral (M,(( chi (er,E,X)) | E))) = (er * (M . E)) by A12, XXREAL_3: 4;

      end;

        suppose er <> +infty & er <> -infty ;

        then er in REAL by XXREAL_0: 14;

        then

        reconsider r = er as Real;

        ( chi (er,E,X)) = (r (#) ( chi (E,X))) by Th1;

        then

         A17: (( chi (er,E,X)) | E) = (r (#) (( chi (E,X)) | E)) by MESFUN11: 2;

        

         A18: (( chi (E,X)) | E) is nonnegative by MESFUNC5: 15;

        

         A19: (( chi (E,X)) | E) is_simple_func_in S by Th12, MESFUNC5: 34;

        

        hence ( Integral (M,(( chi (er,E,X)) | E))) = (r * ( integral' (M,(( chi (E,X)) | E)))) by A17, MESFUNC5: 15, MESFUN11: 59

        .= (r * ( Integral (M,(( chi (E,X)) | E)))) by A19, A18, MESFUNC5: 89

        .= (er * (M . E)) by MESFUNC9: 14;

      end;

    end;

    theorem :: MESFUN12:51

    for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S, er be ExtReal holds ( Integral (M,(( chi (er,E1,X)) | E2))) = (er * (M . (E1 /\ E2)))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E1,E2 be Element of S, er be ExtReal;

      reconsider XX = X as Element of S by MEASURE1: 7;

      set f = ( chi (er,(E1 /\ E2),X));

      

       A1: (( chi (er,E1,X)) | E2) = (f | E2) by Th14;

      

       A2: ( dom f) = XX by FUNCT_2:def 1;

      

       A3: (E1 /\ E2) misses (E2 \ E1) by XBOOLE_1: 89;

      

       A4: ((E1 /\ E2) \/ (E2 \ E1)) = E2 by XBOOLE_1: 51;

      f is XX -measurable by Th13;

      then

       X: ex W be Element of S st W = ( dom f) & f is W -measurable by A2;

      er >= 0 or er < 0 ;

      then f is nonnegative or f is nonpositive by Th17;

      then

       A5: ( Integral (M,(f | E2))) = (( Integral (M,(f | (E1 /\ E2)))) + ( Integral (M,(f | (E2 \ E1))))) by X, A3, A4, MESFUNC5: 91, MESFUN11: 62;

      ( dom (f | (E2 \ E1))) = (( dom f) /\ (E2 \ E1)) by RELAT_1: 61;

      then ( dom (f | (E2 \ E1))) = (X /\ (E2 \ E1)) by FUNCT_2:def 1;

      then

       A6: ( dom (f | (E2 \ E1))) = (E2 \ E1) by XBOOLE_1: 28;

      for x be object st x in ( dom (f | (E2 \ E1))) holds ((f | (E2 \ E1)) . x) >= 0 by Th16, XBOOLE_1: 89;

      then

       A7: (f | (E2 \ E1)) is nonnegative by SUPINF_2: 52;

      for x be Element of X st x in ( dom (f | (E2 \ E1))) holds ((f | (E2 \ E1)) . x) = 0 by Th16, XBOOLE_1: 89;

      then ( integral+ (M,(f | (E2 \ E1)))) = 0 by A6, Th15, MESFUNC5: 87;

      then ( Integral (M,(f | (E2 \ E1)))) = 0 by A6, A7, Th15, MESFUNC5: 88;

      then ( Integral (M,(f | E2))) = ((er * (M . (E1 /\ E2))) + 0 ) by A5, Th50;

      hence ( Integral (M,(( chi (er,E1,X)) | E2))) = (er * (M . (E1 /\ E2))) by A1, XXREAL_3: 4;

    end;

    theorem :: MESFUN12:52

    

     Th52: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M2 is sigma_finite holds (( Y-vol (E,M2)) . x) = ( Integral (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) & (( Y-vol (E,M2)) . x) = ( integral+ (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) & (( Y-vol (E,M2)) . x) = ( integral' (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume

       A1: M2 is sigma_finite;

      

       A2: ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( X-section (E,x)),X2)) by Th48;

      then ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( Measurable-X-section (E,x)),X2)) by MEASUR11:def 6;

      then

       A4: ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) is_simple_func_in S2 by Th12;

      (( Y-vol (E,M2)) . x) = (M2 . ( Measurable-X-section (E,x))) by A1, MEASUR11:def 13;

      then (( Y-vol (E,M2)) . x) = ( Integral (M2,( ProjMap1 (( chi (E, [:X1, X2:])),x)))) by MEASUR11: 72;

      hence (( Y-vol (E,M2)) . x) = ( Integral (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) by Th27;

      hence (( Y-vol (E,M2)) . x) = ( integral+ (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) & (( Y-vol (E,M2)) . x) = ( integral' (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) by A2, A4, MESFUNC5: 89;

    end;

    theorem :: MESFUN12:53

    

     Th53: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite holds (( X-vol (E,M1)) . y) = ( Integral (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) & (( X-vol (E,M1)) . y) = ( integral+ (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) & (( X-vol (E,M1)) . y) = ( integral' (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume

       A1: M1 is sigma_finite;

      

       A2: ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Y-section (E,y)),X1)) by Th48;

      then ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Measurable-Y-section (E,y)),X1)) by MEASUR11:def 7;

      then

       A4: ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) is_simple_func_in S1 by Th12;

      (( X-vol (E,M1)) . y) = (M1 . ( Measurable-Y-section (E,y))) by A1, MEASUR11:def 14;

      then (( X-vol (E,M1)) . y) = ( Integral (M1,( ProjMap2 (( chi (E, [:X1, X2:])),y)))) by MEASUR11: 72;

      hence (( X-vol (E,M1)) . y) = ( Integral (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) by Th27;

      hence (( X-vol (E,M1)) . y) = ( integral+ (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) & (( X-vol (E,M1)) . y) = ( integral' (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) by A2, A4, MESFUNC5: 89;

    end;

    theorem :: MESFUN12:54

    for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, r be Real holds ( Integral (M,(r (#) ( chi (E,X))))) = (r * ( Integral (M,( chi (E,X)))))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, E be Element of S, r be Real;

      

       A3: ( chi (E,X)) is_simple_func_in S by Th12;

      ( Integral (M,(r (#) ( chi (E,X))))) = (r * ( integral' (M,( chi (E,X))))) by Th12, MESFUN11: 59;

      hence ( Integral (M,(r (#) ( chi (E,X))))) = (r * ( Integral (M,( chi (E,X))))) by A3, MESFUNC5: 89;

    end;

    theorem :: MESFUN12:55

    

     Th55: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real st M1 is sigma_finite holds ((r (#) ( X-vol (E,M1))) . y) = ( Integral (M1,( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)))) & (r >= 0 implies ((r (#) ( X-vol (E,M1))) . y) = ( integral+ (M1,( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real;

      assume

       A1: M1 is sigma_finite;

      set p2 = ( ProjPMap2 (( chi (E, [:X1, X2:])),y));

      ( chi (r,E, [:X1, X2:])) = (r (#) ( chi (E, [:X1, X2:]))) by Th1;

      then

       A2: ( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)) = (r (#) p2) by Th29;

      

       A3: p2 is nonnegative by Th32;

      

       A4: ( dom (r (#) ( X-vol (E,M1)))) = X2 by FUNCT_2:def 1;

      

       A5: ( chi (E, [:X1, X2:])) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by Th12;

      

      then ( Integral (M1,( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)))) = (r * ( integral' (M1,p2))) by A2, A3, Th31, MESFUN11: 59

      .= (r * (( X-vol (E,M1)) . y)) by A1, Th53;

      hence

       A7: ((r (#) ( X-vol (E,M1))) . y) = ( Integral (M1,( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)))) by A4, MESFUNC1:def 6;

      thus (r >= 0 implies ((r (#) ( X-vol (E,M1))) . y) = ( integral+ (M1,( ProjPMap2 (( chi (r,E, [:X1, X2:])),y)))))

      proof

        assume r >= 0 ;

        then

         A8: (r (#) p2) is nonnegative by A3, MESFUNC5: 20;

        (r (#) p2) is_simple_func_in S1 by A5, Th31, MESFUNC5: 39;

        hence thesis by A2, A7, A8, MESFUNC5: 89;

      end;

    end;

    theorem :: MESFUN12:56

    

     Th56: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real st M2 is sigma_finite holds ((r (#) ( Y-vol (E,M2))) . x) = ( Integral (M2,( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)))) & (r >= 0 implies ((r (#) ( Y-vol (E,M2))) . x) = ( integral+ (M2,( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real;

      assume

       A1: M2 is sigma_finite;

      set p2 = ( ProjPMap1 (( chi (E, [:X1, X2:])),x));

      ( chi (r,E, [:X1, X2:])) = (r (#) ( chi (E, [:X1, X2:]))) by Th1;

      then

       A2: ( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)) = (r (#) p2) by Th29;

      

       A3: p2 is nonnegative by Th32;

      

       A4: ( dom (r (#) ( Y-vol (E,M2)))) = X1 by FUNCT_2:def 1;

      

       A5: ( chi (E, [:X1, X2:])) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by Th12;

      

      then ( Integral (M2,( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)))) = (r * ( integral' (M2,p2))) by A2, A3, Th31, MESFUN11: 59

      .= (r * (( Y-vol (E,M2)) . x)) by A1, Th52;

      hence

       A7: ((r (#) ( Y-vol (E,M2))) . x) = ( Integral (M2,( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)))) by A4, MESFUNC1:def 6;

      thus (r >= 0 implies ((r (#) ( Y-vol (E,M2))) . x) = ( integral+ (M2,( ProjPMap1 (( chi (r,E, [:X1, X2:])),x)))))

      proof

        assume r >= 0 ;

        then

         A8: (r (#) p2) is nonnegative by A3, MESFUNC5: 20;

        (r (#) p2) is_simple_func_in S2 by A5, Th31, MESFUNC5: 39;

        hence thesis by A2, A7, A8, MESFUNC5: 89;

      end;

    end;

    theorem :: MESFUN12:57

    

     Th57: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, ExtREAL st ( dom f) in S & (for x be Element of X st x in ( dom f) holds 0 = (f . x)) holds (for E be Element of S st E c= ( dom f) holds f is E -measurable) & ( Integral (M,f)) = 0

    proof

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

      assume that

       a1: ( dom f) in S and

       a2: for x be Element of X st x in ( dom f) holds (f . x) = 0 ;

      reconsider E = ( dom f) as Element of S by a1;

      ( dom (( chi ( 0 ,E,X)) | E)) = (( dom ( chi ( 0 ,E,X))) /\ E) by RELAT_1: 61;

      then ( dom (( chi ( 0 ,E,X)) | E)) = (X /\ E) by FUNCT_2:def 1;

      then

       a3: ( dom (( chi ( 0 ,E,X)) | E)) = E by XBOOLE_1: 28;

      now

        let x be Element of X;

        assume

         a4: x in ( dom f);

        then ((( chi ( 0 ,E,X)) | E) . x) = (( chi ( 0 ,E,X)) . x) by FUNCT_1: 49;

        then ((( chi ( 0 ,E,X)) | E) . x) = 0 by a4, Def1;

        hence (f . x) = ((( chi ( 0 ,E,X)) | E) . x) by a2, a4;

      end;

      then

       a4: f = (( chi ( 0 ,E,X)) | E) by a3, PARTFUN1: 5;

      hence for A be Element of S st A c= ( dom f) holds f is A -measurable by Th15;

      ( Integral (M,f)) = ( 0 * (M . E)) by a4, Th50;

      hence ( Integral (M,f)) = 0 ;

    end;

    

     Lm5: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & (f is nonnegative or f is nonpositive) & A = ( dom f) & f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) holds ex I1 be Function of X2, ExtREAL st (for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))) & (for V be Element of S2 holds I1 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A2: (f is nonnegative or f is nonpositive) and

       A3: A = ( dom f) and

       A4: f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      per cases ;

        suppose f = {} ;

        then

         A5: ( dom f) = ( {} [:X1, X2:]);

        reconsider E1 = {} as Element of S1 by MEASURE1: 7;

        reconsider E = {} as Element of S2 by MEASURE1: 7;

        reconsider I1 = ( chi (E,X2)) as Function of X2, ExtREAL ;

        take I1;

        thus for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))

        proof

          let y be Element of X2;

          ( dom ( ProjPMap2 (f,y))) = ( Y-section (( dom f),y)) by Def4;

          then

           A6: ( dom ( ProjPMap2 (f,y))) = E1 by A5, MEASUR11: 24;

          

           A7: ( ProjPMap2 (f,y)) is E1 -measurable by A4, Th31, MESFUNC2: 34;

          (M1 . E1) = 0 by VALUED_0:def 19;

          then ( Integral (M1,(( ProjPMap2 (f,y)) | E1))) = 0 by A6, A7, MESFUNC5: 94;

          hence (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A6, FUNCT_3:def 3;

        end;

        thus for V be Element of S2 holds I1 is V -measurable by MESFUNC2: 29;

      end;

        suppose f <> {} ;

        then

        consider E be non empty Finite_Sep_Sequence of ( sigma ( measurable_rectangles (S1,S2))), a be FinSequence of ExtREAL , r be FinSequence of REAL such that

         A8: (E,a) are_Re-presentation_of f and

         A9: for n be Nat holds (a . n) = (r . n) & (f | (E . n)) = (( chi ((r . n),(E . n), [:X1, X2:])) | (E . n)) & ((E . n) = {} implies (r . n) = 0 ) by A4, Th5;

        defpred Q[ Nat, object] means $2 = ((r . $1) (#) ( X-vol ((E . $1),M1)));

        

         A10: for k be Nat st k in ( Seg ( len E)) holds ex x be Element of ( Funcs (X2, ExtREAL )) st Q[k, x]

        proof

          let k be Nat;

          assume k in ( Seg ( len E));

          reconsider x = ((r . k) (#) ( X-vol ((E . k),M1))) as Element of ( Funcs (X2, ExtREAL )) by FUNCT_2: 8;

          take x;

          thus thesis;

        end;

        consider H be FinSequence of ( Funcs (X2, ExtREAL )) such that

         A11: ( dom H) = ( Seg ( len E)) and

         A12: for n be Nat st n in ( Seg ( len E)) holds Q[n, (H . n)] from FINSEQ_1:sch 5( A10);

        

         A13: ( dom H) = ( dom E) by A11, FINSEQ_1:def 3;

        

         A14: f is nonnegative implies for n be Nat holds (r . n) >= 0

        proof

          assume

           A15: f is nonnegative;

          hereby

            let n be Nat;

            now

              assume

               A16: (E . n) <> {} ;

              then

              consider x be object such that

               A17: x in (E . n) by XBOOLE_0:def 1;

              n in ( dom E) by A16, FUNCT_1:def 2;

              then (a . n) = (f . x) by A8, A17, MESFUNC3:def 1;

              then (a . n) >= 0 by A15, SUPINF_2: 51;

              hence (r . n) >= 0 by A9;

            end;

            hence (r . n) >= 0 by A9;

          end;

        end;

        

         A18: f is nonpositive implies for n be Nat holds (r . n) <= 0

        proof

          assume

           A19: f is nonpositive;

          hereby

            let n be Nat;

            now

              assume

               A20: (E . n) <> {} ;

              then

              consider x be object such that

               A21: x in (E . n) by XBOOLE_0:def 1;

              n in ( dom E) by A20, FUNCT_1:def 2;

              then (a . n) = (f . x) by A8, A21, MESFUNC3:def 1;

              then (a . n) <= 0 by A19, MESFUNC5: 8;

              hence (r . n) <= 0 by A9;

            end;

            hence (r . n) <= 0 by A9;

          end;

        end;

        

         A22: f is nonnegative implies H is without_-infty-valued

        proof

          assume

           A6: f is nonnegative;

          for n be Nat st n in ( dom H) holds (H . n) is without-infty

          proof

            let n be Nat;

            assume

             A23: n in ( dom H);

            then (H . n) = ((r . n) (#) ( X-vol ((E . n),M1))) by A11, A12;

            then (H . n) is nonnegative by A6, A14, MESFUNC5: 20;

            then (H /. n) is nonnegative Function of X2, ExtREAL by A23, PARTFUN1:def 6;

            hence (H . n) is without-infty by A23, PARTFUN1:def 6;

          end;

          hence H is without_-infty-valued;

        end;

        

         A24: f is nonpositive implies H is without_+infty-valued

        proof

          assume

           A6: f is nonpositive;

          for n be Nat st n in ( dom H) holds (H . n) is without+infty

          proof

            let n be Nat;

            assume

             A25: n in ( dom H);

            then (H . n) = ((r . n) (#) ( X-vol ((E . n),M1))) by A11, A12;

            then (H . n) is nonpositive by A6, A18, MESFUNC5: 20;

            then (H /. n) is nonpositive Function of X2, ExtREAL by A25, PARTFUN1:def 6;

            hence (H . n) is without+infty by A25, PARTFUN1:def 6;

          end;

          hence thesis;

        end;

        then

        reconsider H as summable FinSequence of ( Funcs (X2, ExtREAL )) by A2, A22;

        

         A26: f is nonnegative implies ( Partial_Sums H) is without_-infty-valued by A22, MEASUR11: 61;

        

         A27: f is nonpositive implies ( Partial_Sums H) is without_+infty-valued by A24, MEASUR11: 60;

        ( len H) = ( len ( Partial_Sums H)) by MEASUR11:def 11;

        then

         A28: ( dom H) = ( dom ( Partial_Sums H)) by FINSEQ_3: 29;

        

         A29: H <> {} by A11;

        then

         A30: ( len H) >= 1 by FINSEQ_1: 20;

        

         A31: for y be Element of X2, n be Nat st n in ( dom E) holds ((H . n) . y) = ( Integral (M1,( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)))) & ((H . n) . y) = ( Integral (M1,((r . n) (#) ( ProjPMap2 (( chi ((E . n), [:X1, X2:])),y)))))

        proof

          let y be Element of X2, n be Nat;

          assume n in ( dom E);

          then n in ( Seg ( len E)) by FINSEQ_1:def 3;

          then (H . n) = ((r . n) (#) ( X-vol ((E . n),M1))) by A12;

          hence ((H . n) . y) = ( Integral (M1,( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)))) by A1, Th55;

          then ((H . n) . y) = ( Integral (M1,( ProjPMap2 (((r . n) (#) ( chi ((E . n), [:X1, X2:]))),y)))) by Th1;

          hence ((H . n) . y) = ( Integral (M1,((r . n) (#) ( ProjPMap2 (( chi ((E . n), [:X1, X2:])),y))))) by Th29;

        end;

        reconsider I1 = (( Partial_Sums H) /. ( len H)) as Function of X2, ExtREAL ;

        take I1;

        for y be Element of X2 holds ((( Partial_Sums H) /. ( len H)) . y) = ( Integral (M1,( ProjPMap2 (f,y))))

        proof

          let y be Element of X2;

          f is A -measurable by A4, MESFUNC2: 34;

          then

           A32: ( ProjPMap2 (f,y)) is ( Measurable-Y-section (A,y)) -measurable by A3, Th47;

          ( dom ( ProjPMap2 (f,y))) = ( Y-section (( dom f),y)) by Def4;

          then

           A33: ( dom ( ProjPMap2 (f,y))) = ( Measurable-Y-section (A,y)) by A3, MEASUR11:def 7;

          

           A34: ( ProjPMap2 (f,y)) is nonnegative or ( ProjPMap2 (f,y)) is nonpositive by A2, Th32, Th33;

          

           A35: for n be Nat holds ( dom (( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) | (XX1 \ ( Measurable-Y-section ((E . n),y))))) = (XX1 \ ( Measurable-Y-section ((E . n),y))) & (( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) is nonnegative or ( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) is nonpositive) & ( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) is XX1 -measurable & (for x be Element of X1 st x in ( dom (( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) | (XX1 \ ( Measurable-Y-section ((E . n),y))))) holds ((( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) | (XX1 \ ( Measurable-Y-section ((E . n),y)))) . x) = 0 ) & ( Integral (M1,( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)))) = ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section ((E . n),y))))) & ( Measurable-Y-section (( Union (E | n)),y)) misses ( Measurable-Y-section ((E . (n + 1)),y)) & ( Measurable-Y-section (( Union (E | (n + 1))),y)) = (( Measurable-Y-section (( Union (E | n)),y)) \/ ( Measurable-Y-section ((E . (n + 1)),y)))

          proof

            let n be Nat;

            set pn = ( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y));

            set dn = (XX1 \ ( Measurable-Y-section ((E . n),y)));

            set fn = (( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) | (XX1 \ ( Measurable-Y-section ((E . n),y))));

            pn = ( ProjMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) by Th27;

            then

             A36: ( dom pn) = XX1 by FUNCT_2:def 1;

            hence

             A37: ( dom fn) = (XX1 \ ( Measurable-Y-section ((E . n),y))) by RELAT_1: 62;

            

             A38: ( chi ((r . n),(E . n), [:X1, X2:])) = ((r . n) (#) ( chi ((E . n), [:X1, X2:]))) by Th1;

            ( ProjPMap2 (( chi ((E . n), [:X1, X2:])),y)) = ( chi (( Y-section ((E . n),y)),X1)) by Th48;

            then ( ProjPMap2 (( chi ((E . n), [:X1, X2:])),y)) = ( chi (( Measurable-Y-section ((E . n),y)),X1)) by MEASUR11:def 7;

            then

             A39: pn = ((r . n) (#) ( chi (( Measurable-Y-section ((E . n),y)),X1))) by A38, Th29;

            hence

             A40: pn is nonnegative or pn is nonpositive by A2, A14, A18, MESFUNC5: 20;

            ( dom ( chi (( Measurable-Y-section ((E . n),y)),X1))) = XX1 by FUNCT_2:def 1;

            hence

             A41: pn is XX1 -measurable by A39, MESFUNC1: 37, MESFUNC2: 29;

            thus for x be Element of X1 st x in ( dom fn) holds (fn . x) = 0

            proof

              let x be Element of X1;

              assume

               A42: x in ( dom fn);

              then (( chi (( Measurable-Y-section ((E . n),y)),X1)) . x) = 0 by A37, FUNCT_3: 37;

              then (pn . x) = ((r . n) * 0 ) by A36, A39, MESFUNC1:def 6;

              hence (fn . x) = 0 by A42, FUNCT_1: 47;

            end;

            then ( Integral (M1,fn)) = 0 by A37, Th57;

            then ( Integral (M1,(pn | ((XX1 \ ( Measurable-Y-section ((E . n),y))) \/ ( Measurable-Y-section ((E . n),y)))))) = (( Integral (M1,(pn | ( Measurable-Y-section ((E . n),y))))) + 0 ) by A36, A40, A41, XBOOLE_1: 79, MESFUNC5: 91, MESFUN11: 62;

            then ( Integral (M1,(pn | ((XX1 \ ( Measurable-Y-section ((E . n),y))) \/ ( Measurable-Y-section ((E . n),y)))))) = ( Integral (M1,(pn | ( Measurable-Y-section ((E . n),y))))) by XXREAL_3: 4;

            then

             A43: ( Integral (M1,(pn | XX1))) = ( Integral (M1,(pn | ( Measurable-Y-section ((E . n),y))))) by XBOOLE_1: 45;

            (pn | ( Measurable-Y-section ((E . n),y))) = (( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)) | ( Y-section ((E . n),y))) by MEASUR11:def 7;

            then (pn | ( Measurable-Y-section ((E . n),y))) = ( ProjPMap2 ((( chi ((r . n),(E . n), [:X1, X2:])) | (E . n)),y)) by Th34;

            then (pn | ( Measurable-Y-section ((E . n),y))) = ( ProjPMap2 ((f | (E . n)),y)) by A9;

            then (pn | ( Measurable-Y-section ((E . n),y))) = (( ProjPMap2 (f,y)) | ( Y-section ((E . n),y))) by Th34;

            hence ( Integral (M1,( ProjPMap2 (( chi ((r . n),(E . n), [:X1, X2:])),y)))) = ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section ((E . n),y))))) by A43, MEASUR11:def 7;

            ( union ( rng (E | n))) misses (E . (n + 1)) by NAT_1: 16, MEASUR11: 1;

            then ( Union (E | n)) misses (E . (n + 1)) by CARD_3:def 4;

            then ( Y-section (( Union (E | n)),y)) misses ( Y-section ((E . (n + 1)),y)) by MEASUR11: 35;

            then ( Measurable-Y-section (( Union (E | n)),y)) misses ( Y-section ((E . (n + 1)),y)) by MEASUR11:def 7;

            hence ( Measurable-Y-section (( Union (E | n)),y)) misses ( Measurable-Y-section ((E . (n + 1)),y)) by MEASUR11:def 7;

            ( union ( rng (E | (n + 1)))) = (( union ( rng (E | n))) \/ (E . (n + 1))) by MEASUR11: 3;

            then ( Union (E | (n + 1))) = (( union ( rng (E | n))) \/ (E . (n + 1))) by CARD_3:def 4;

            then ( Union (E | (n + 1))) = (( Union (E | n)) \/ (E . (n + 1))) by CARD_3:def 4;

            then ( Y-section (( Union (E | (n + 1))),y)) = (( Y-section (( Union (E | n)),y)) \/ ( Y-section ((E . (n + 1)),y))) by MEASUR11: 26;

            

            then ( Measurable-Y-section (( Union (E | (n + 1))),y)) = (( Y-section (( Union (E | n)),y)) \/ ( Y-section ((E . (n + 1)),y))) by MEASUR11:def 7

            .= (( Measurable-Y-section (( Union (E | n)),y)) \/ ( Y-section ((E . (n + 1)),y))) by MEASUR11:def 7;

            hence ( Measurable-Y-section (( Union (E | (n + 1))),y)) = (( Measurable-Y-section (( Union (E | n)),y)) \/ ( Measurable-Y-section ((E . (n + 1)),y))) by MEASUR11:def 7;

          end;

          defpred P[ Nat] means $1 <= ( len H) implies ((( Partial_Sums H) /. $1) . y) = ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | $1)))),y))));

          

           A44: P[1]

          proof

            assume

             A45: 1 <= ( len H);

            then

             A46: 1 in ( dom H) by FINSEQ_3: 25;

            ( len H) = ( len ( Partial_Sums H)) by MEASUR11:def 11;

            then ( dom H) = ( dom ( Partial_Sums H)) by FINSEQ_3: 29;

            then (( Partial_Sums H) /. 1) = (( Partial_Sums H) . 1) by A45, FINSEQ_3: 25, PARTFUN1:def 6;

            then (( Partial_Sums H) /. 1) = (H . 1) by MEASUR11:def 11;

            then

             A47: ((( Partial_Sums H) /. 1) . y) = ( Integral (M1,( ProjPMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y)))) by A13, A31, A46;

            (E | 1) = <*(E . 1)*> by FINSEQ_5: 20;

            then ( rng (E | 1)) = {(E . 1)} by FINSEQ_1: 39;

            then ( union ( rng (E | 1))) = (E . 1) by ZFMISC_1: 25;

            then ( ProjPMap2 ((f | ( union ( rng (E | 1)))),y)) = ( ProjPMap2 ((( chi ((r . 1),(E . 1), [:X1, X2:])) | (E . 1)),y)) by A9;

            then ( ProjPMap2 ((f | ( union ( rng (E | 1)))),y)) = (( ProjPMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y)) | ( Y-section ((E . 1),y))) by Th34;

            then

             A48: ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | 1)))),y)))) = ( Integral (M1,(( ProjPMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y)) | ( Measurable-Y-section ((E . 1),y))))) by MEASUR11:def 7;

            set p1 = ( ProjPMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y));

            set d1 = (XX1 \ ( Measurable-Y-section ((E . 1),y)));

            set f1 = (( ProjPMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y)) | (XX1 \ ( Measurable-Y-section ((E . 1),y))));

            

             A49: ( dom f1) = (XX1 \ ( Measurable-Y-section ((E . 1),y))) & (p1 is nonnegative or p1 is nonpositive) by A35;

            p1 = ( ProjMap2 (( chi ((r . 1),(E . 1), [:X1, X2:])),y)) by Th27;

            then

             A50: ( dom p1) = X1 by FUNCT_2:def 1;

            

             A51: (XX1 \ ( Measurable-Y-section ((E . 1),y))) misses ( Measurable-Y-section ((E . 1),y)) by XBOOLE_1: 79;

            

             A52: ((XX1 \ ( Measurable-Y-section ((E . 1),y))) \/ ( Measurable-Y-section ((E . 1),y))) = XX1 by XBOOLE_1: 45;

            for x be Element of X1 st x in ( dom f1) holds (f1 . x) = 0 by A35;

            then ( Integral (M1,f1)) = 0 by A49, Th57;

            then ( Integral (M1,(p1 | ((XX1 \ ( Measurable-Y-section ((E . 1),y))) \/ ( Measurable-Y-section ((E . 1),y)))))) = (( Integral (M1,(p1 | ( Measurable-Y-section ((E . 1),y))))) + 0 ) by A35, A49, A50, A51, MESFUNC5: 91, MESFUN11: 62;

            hence ((( Partial_Sums H) /. 1) . y) = ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | 1)))),y)))) by A47, A48, A52, XXREAL_3: 4;

          end;

          

           A54: for n be non zero Nat st P[n] holds P[(n + 1)]

          proof

            let n be non zero Nat;

            assume

             A55: P[n];

            assume

             A56: (n + 1) <= ( len H);

            then n < ( len H) by NAT_1: 13;

            then

             A57: n <= ( len ( Partial_Sums H)) & (n + 1) <= ( len ( Partial_Sums H)) by A56, MEASUR11:def 11;

            

             A58: 1 <= (n + 1) by NAT_1: 12;

            

             A59: n >= 1 by NAT_1: 14;

            then

             A60: n in ( dom ( Partial_Sums H)) & (n + 1) in ( dom ( Partial_Sums H)) & (n + 1) in ( dom H) by A56, A57, NAT_1: 12, FINSEQ_3: 25;

            then

             A61: (( Partial_Sums H) /. (n + 1)) = (( Partial_Sums H) . (n + 1)) & (( Partial_Sums H) /. n) = (( Partial_Sums H) . n) & (H /. (n + 1)) = (H . (n + 1)) by PARTFUN1:def 6;

            

             A62: ((( Partial_Sums H) /. n) is without-infty & (H /. (n + 1)) is without-infty) or ((( Partial_Sums H) /. n) is without+infty & (H /. (n + 1)) is without+infty)

            proof

              per cases by A2;

                suppose f is nonnegative;

                hence thesis by A22, A26, A56, A57, A58, A59, A61, FINSEQ_3: 25;

              end;

                suppose f is nonpositive;

                hence thesis by A24, A27, A56, A57, A58, A59, A61, FINSEQ_3: 25;

              end;

            end;

            

             A63: ( Y-section (( Union (E | n)),y)) = ( Measurable-Y-section (( Union (E | n)),y)) by MEASUR11:def 7;

            (( Partial_Sums H) . (n + 1)) = ((( Partial_Sums H) /. n) + (H /. (n + 1))) by A56, A59, NAT_1: 13, MEASUR11:def 11;

            then ((( Partial_Sums H) . (n + 1)) . y) = (((( Partial_Sums H) /. n) . y) + ((H /. (n + 1)) . y)) by A62, DBLSEQ_3: 7;

            then ((( Partial_Sums H) . (n + 1)) . y) = (( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | n)))),y)))) + ( Integral (M1,( ProjPMap2 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),y))))) by A13, A55, A56, A60, A61, A31, NAT_1: 13;

            then ((( Partial_Sums H) . (n + 1)) . y) = (( Integral (M1,( ProjPMap2 ((f | ( Union (E | n))),y)))) + ( Integral (M1,( ProjPMap2 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),y))))) by CARD_3:def 4;

            then ((( Partial_Sums H) . (n + 1)) . y) = (( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section (( Union (E | n)),y))))) + ( Integral (M1,( ProjPMap2 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),y))))) by Th34;

            then ((( Partial_Sums H) . (n + 1)) . y) = (( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section (( Union (E | n)),y))))) + ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section ((E . (n + 1)),y)))))) by A35;

            then ((( Partial_Sums H) . (n + 1)) . y) = ( Integral (M1,(( ProjPMap2 (f,y)) | (( Measurable-Y-section (( Union (E | n)),y)) \/ ( Measurable-Y-section ((E . (n + 1)),y)))))) by A32, A33, A34, A35, A63, MESFUNC5: 91, MESFUN11: 62;

            then ((( Partial_Sums H) . (n + 1)) . y) = ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section (( Union (E | (n + 1))),y))))) by A35;

            then ((( Partial_Sums H) . (n + 1)) . y) = ( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section (( Union (E | (n + 1))),y))))) by MEASUR11:def 7;

            then ((( Partial_Sums H) . (n + 1)) . y) = ( Integral (M1,( ProjPMap2 ((f | ( Union (E | (n + 1)))),y)))) by Th34;

            then ((( Partial_Sums H) . (n + 1)) . y) = ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | (n + 1))))),y)))) by CARD_3:def 4;

            hence ((( Partial_Sums H) /. (n + 1)) . y) = ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | (n + 1))))),y)))) by A60, PARTFUN1:def 6;

          end;

          ( len H) = ( len E) by A11, FINSEQ_1:def 3;

          then (E | ( len H)) = (E | ( dom E)) by FINSEQ_1:def 3;

          then ( union ( rng (E | ( len H)))) = ( dom f) by A8, MESFUNC3:def 1;

          then

           A64: (f | ( union ( rng (E | ( len H))))) = f;

          for n be non zero Nat holds P[n] from NAT_1:sch 10( A44, A54);

          hence ((( Partial_Sums H) /. ( len H)) . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A29, A64;

        end;

        hence for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))));

        thus for V be Element of S2 holds I1 is V -measurable

        proof

          let V be Element of S2;

          

           A65: for n be Nat st n in ( dom H) holds (H /. n) is V -measurable

          proof

            let n be Nat;

            assume n in ( dom H);

            then

             A66: (H /. n) = (H . n) & (H . n) = ((r . n) (#) ( X-vol ((E . n),M1))) by A11, A12, PARTFUN1:def 6;

            

             A67: ( dom ( X-vol ((E . n),M1))) = XX2 by FUNCT_2:def 1;

            ( X-vol ((E . n),M1)) is V -measurable by A1, MEASUR11:def 14;

            hence (H /. n) is V -measurable by A66, A67, MESFUNC1: 37;

          end;

          defpred P2[ Nat] means $1 <= ( len H) implies (( Partial_Sums H) /. $1) is V -measurable;

          (( Partial_Sums H) /. 1) = (( Partial_Sums H) . 1) by A28, A30, FINSEQ_3: 25, PARTFUN1:def 6;

          then (( Partial_Sums H) /. 1) = (H . 1) by MEASUR11:def 11;

          then (( Partial_Sums H) /. 1) = (H /. 1) by A30, A28, FINSEQ_3: 25, PARTFUN1:def 6;

          then

           A68: P2[1] by A65, FINSEQ_3: 25;

          

           A69: for n be non zero Nat st P2[n] holds P2[(n + 1)]

          proof

            let n be non zero Nat;

            assume

             A70: P2[n];

            assume

             A71: (n + 1) <= ( len H);

            then

             A72: 1 <= n < ( len H) by NAT_1: 13, NAT_1: 14;

            then

             A73: n in ( dom H) & (n + 1) in ( dom H) by A71, NAT_1: 11, FINSEQ_3: 25;

            then

             A74: (( Partial_Sums H) /. n) = (( Partial_Sums H) . n) & (H . (n + 1)) = (H /. (n + 1)) & (( Partial_Sums H) /. (n + 1)) = (( Partial_Sums H) . (n + 1)) by A28, PARTFUN1:def 6;

            then

             A75: (( Partial_Sums H) /. (n + 1)) = ((( Partial_Sums H) /. n) + (H /. (n + 1))) by A72, MEASUR11:def 11;

            

             A76: ( dom (H /. (n + 1))) = XX2 & ( dom (( Partial_Sums H) /. n)) = XX2 by FUNCT_2:def 1;

            

             A77: (H /. (n + 1)) is V -measurable by A73, A65;

            per cases by A2;

              suppose f is nonnegative;

              then (H /. (n + 1)) is without-infty & (( Partial_Sums H) /. n) is without-infty by A22, A26, A28, A73, A74;

              hence (( Partial_Sums H) /. (n + 1)) is V -measurable by A70, A71, A75, A77, NAT_1: 13, MESFUNC5: 31;

            end;

              suppose f is nonpositive;

              then

               A78: (H /. (n + 1)) is without+infty & (( Partial_Sums H) /. n) is without+infty by A24, A27, A28, A73, A74;

              then ( dom ((( Partial_Sums H) /. n) + (H /. (n + 1)))) = (( dom (( Partial_Sums H) /. n)) /\ ( dom (H /. (n + 1)))) by MESFUNC9: 1;

              hence (( Partial_Sums H) /. (n + 1)) is V -measurable by A70, A71, A75, A77, A76, A78, NAT_1: 13, MEASUR11: 65;

            end;

          end;

          for n be non zero Nat holds P2[n] from NAT_1:sch 10( A68, A69);

          hence thesis by A29;

        end;

      end;

    end;

    

     Lm6: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M2 is sigma_finite & (f is nonnegative or f is nonpositive) & A = ( dom f) & f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) holds ex I2 be Function of X1, ExtREAL st (for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))) & (for V be Element of S1 holds I2 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M2 is sigma_finite and

       A2: (f is nonnegative or f is nonpositive) and

       A3: A = ( dom f) and

       A4: f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      per cases ;

        suppose f = {} ;

        then

         A5: ( dom f) = ( {} [:X1, X2:]);

        reconsider E2 = {} as Element of S2 by MEASURE1: 7;

        reconsider E = {} as Element of S1 by MEASURE1: 7;

        reconsider I2 = ( chi (E,X1)) as Function of X1, ExtREAL ;

        take I2;

        thus for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))

        proof

          let x be Element of X1;

          ( dom ( ProjPMap1 (f,x))) = ( X-section (( dom f),x)) by Def3;

          then

           A6: ( dom ( ProjPMap1 (f,x))) = E2 by A5, MEASUR11: 24;

          

           A7: ( ProjPMap1 (f,x)) is E2 -measurable by A4, Th31, MESFUNC2: 34;

          (M2 . E2) = 0 by VALUED_0:def 19;

          then ( Integral (M2,(( ProjPMap1 (f,x)) | E2))) = 0 by A6, A7, MESFUNC5: 94;

          hence (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A6, FUNCT_3:def 3;

        end;

        thus for V be Element of S1 holds I2 is V -measurable by MESFUNC2: 29;

      end;

        suppose f <> {} ;

        then

        consider E be non empty Finite_Sep_Sequence of ( sigma ( measurable_rectangles (S1,S2))), a be FinSequence of ExtREAL , r be FinSequence of REAL such that

         A8: (E,a) are_Re-presentation_of f and

         A9: for n be Nat holds (a . n) = (r . n) & (f | (E . n)) = (( chi ((r . n),(E . n), [:X1, X2:])) | (E . n)) & ((E . n) = {} implies (r . n) = 0 ) by A4, Th5;

        defpred Q[ Nat, object] means $2 = ((r . $1) (#) ( Y-vol ((E . $1),M2)));

        

         A10: for k be Nat st k in ( Seg ( len E)) holds ex x be Element of ( Funcs (X1, ExtREAL )) st Q[k, x]

        proof

          let k be Nat;

          assume k in ( Seg ( len E));

          reconsider x = ((r . k) (#) ( Y-vol ((E . k),M2))) as Element of ( Funcs (X1, ExtREAL )) by FUNCT_2: 8;

          take x;

          thus thesis;

        end;

        consider H be FinSequence of ( Funcs (X1, ExtREAL )) such that

         A11: ( dom H) = ( Seg ( len E)) and

         A12: for n be Nat st n in ( Seg ( len E)) holds Q[n, (H . n)] from FINSEQ_1:sch 5( A10);

        

         A13: ( dom H) = ( dom E) by A11, FINSEQ_1:def 3;

        

         A14: f is nonnegative implies for n be Nat holds (r . n) >= 0

        proof

          assume

           A15: f is nonnegative;

          hereby

            let n be Nat;

            now

              assume

               A16: (E . n) <> {} ;

              then

              consider x be object such that

               A17: x in (E . n) by XBOOLE_0:def 1;

              n in ( dom E) by A16, FUNCT_1:def 2;

              then (a . n) = (f . x) by A8, A17, MESFUNC3:def 1;

              then (a . n) >= 0 by A15, SUPINF_2: 51;

              hence (r . n) >= 0 by A9;

            end;

            hence (r . n) >= 0 by A9;

          end;

        end;

        

         A18: f is nonpositive implies for n be Nat holds (r . n) <= 0

        proof

          assume

           A19: f is nonpositive;

          hereby

            let n be Nat;

            now

              assume

               A20: (E . n) <> {} ;

              then

              consider x be object such that

               A21: x in (E . n) by XBOOLE_0:def 1;

              n in ( dom E) by A20, FUNCT_1:def 2;

              then (a . n) = (f . x) by A8, A21, MESFUNC3:def 1;

              then (a . n) <= 0 by A19, MESFUNC5: 8;

              hence (r . n) <= 0 by A9;

            end;

            hence (r . n) <= 0 by A9;

          end;

        end;

        

         A22: f is nonnegative implies H is without_-infty-valued

        proof

          assume

           A6: f is nonnegative;

          for n be Nat st n in ( dom H) holds (H . n) is without-infty

          proof

            let n be Nat;

            assume

             A23: n in ( dom H);

            then (H . n) = ((r . n) (#) ( Y-vol ((E . n),M2))) by A11, A12;

            then (H . n) is nonnegative by A6, A14, MESFUNC5: 20;

            then (H /. n) is nonnegative Function of X1, ExtREAL by A23, PARTFUN1:def 6;

            hence (H . n) is without-infty by A23, PARTFUN1:def 6;

          end;

          hence H is without_-infty-valued;

        end;

        

         A24: f is nonpositive implies H is without_+infty-valued

        proof

          assume

           A6: f is nonpositive;

          for n be Nat st n in ( dom H) holds (H . n) is without+infty

          proof

            let n be Nat;

            assume

             A25: n in ( dom H);

            then (H . n) = ((r . n) (#) ( Y-vol ((E . n),M2))) by A11, A12;

            then (H . n) is nonpositive by A6, A18, MESFUNC5: 20;

            then (H /. n) is nonpositive Function of X1, ExtREAL by A25, PARTFUN1:def 6;

            hence (H . n) is without+infty by A25, PARTFUN1:def 6;

          end;

          hence thesis;

        end;

        then

        reconsider H as summable FinSequence of ( Funcs (X1, ExtREAL )) by A2, A22;

        

         A26: f is nonnegative implies ( Partial_Sums H) is without_-infty-valued by A22, MEASUR11: 61;

        

         A27: f is nonpositive implies ( Partial_Sums H) is without_+infty-valued by A24, MEASUR11: 60;

        ( len H) = ( len ( Partial_Sums H)) by MEASUR11:def 11;

        then

         A28: ( dom H) = ( dom ( Partial_Sums H)) by FINSEQ_3: 29;

        

         A29: H <> {} by A11;

        then

         A30: ( len H) >= 1 by FINSEQ_1: 20;

        

         A31: for x be Element of X1, n be Nat st n in ( dom E) holds ((H . n) . x) = ( Integral (M2,( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)))) & ((H . n) . x) = ( Integral (M2,((r . n) (#) ( ProjPMap1 (( chi ((E . n), [:X1, X2:])),x)))))

        proof

          let x be Element of X1, n be Nat;

          assume n in ( dom E);

          then n in ( Seg ( len E)) by FINSEQ_1:def 3;

          then (H . n) = ((r . n) (#) ( Y-vol ((E . n),M2))) by A12;

          hence ((H . n) . x) = ( Integral (M2,( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)))) by A1, Th56;

          then ((H . n) . x) = ( Integral (M2,( ProjPMap1 (((r . n) (#) ( chi ((E . n), [:X1, X2:]))),x)))) by Th1;

          hence ((H . n) . x) = ( Integral (M2,((r . n) (#) ( ProjPMap1 (( chi ((E . n), [:X1, X2:])),x))))) by Th29;

        end;

        reconsider I2 = (( Partial_Sums H) /. ( len H)) as Function of X1, ExtREAL ;

        take I2;

        for x be Element of X1 holds ((( Partial_Sums H) /. ( len H)) . x) = ( Integral (M2,( ProjPMap1 (f,x))))

        proof

          let x be Element of X1;

          f is A -measurable by A4, MESFUNC2: 34;

          then

           A32: ( ProjPMap1 (f,x)) is ( Measurable-X-section (A,x)) -measurable by A3, Th47;

          ( dom ( ProjPMap1 (f,x))) = ( X-section (( dom f),x)) by Def3;

          then

           A33: ( dom ( ProjPMap1 (f,x))) = ( Measurable-X-section (A,x)) by A3, MEASUR11:def 6;

          

           A34: ( ProjPMap1 (f,x)) is nonnegative or ( ProjPMap1 (f,x)) is nonpositive by A2, Th32, Th33;

          

           A35: for n be Nat holds ( dom (( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) | (XX2 \ ( Measurable-X-section ((E . n),x))))) = (XX2 \ ( Measurable-X-section ((E . n),x))) & (( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) is nonnegative or ( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) is nonpositive) & ( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) is XX2 -measurable & (for y be Element of X2 st y in ( dom (( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) | (XX2 \ ( Measurable-X-section ((E . n),x))))) holds ((( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) | (XX2 \ ( Measurable-X-section ((E . n),x)))) . y) = 0 ) & ( Integral (M2,( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)))) = ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section ((E . n),x))))) & ( Measurable-X-section (( Union (E | n)),x)) misses ( Measurable-X-section ((E . (n + 1)),x)) & ( Measurable-X-section (( Union (E | (n + 1))),x)) = (( Measurable-X-section (( Union (E | n)),x)) \/ ( Measurable-X-section ((E . (n + 1)),x)))

          proof

            let n be Nat;

            set pn = ( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x));

            set dn = (XX2 \ ( Measurable-X-section ((E . n),x)));

            set fn = (( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) | (XX2 \ ( Measurable-X-section ((E . n),x))));

            pn = ( ProjMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) by Th27;

            then

             A36: ( dom pn) = XX2 by FUNCT_2:def 1;

            hence

             A37: ( dom fn) = (XX2 \ ( Measurable-X-section ((E . n),x))) by RELAT_1: 62;

            

             A38: ( chi ((r . n),(E . n), [:X1, X2:])) = ((r . n) (#) ( chi ((E . n), [:X1, X2:]))) by Th1;

            ( ProjPMap1 (( chi ((E . n), [:X1, X2:])),x)) = ( chi (( X-section ((E . n),x)),X2)) by Th48;

            then ( ProjPMap1 (( chi ((E . n), [:X1, X2:])),x)) = ( chi (( Measurable-X-section ((E . n),x)),X2)) by MEASUR11:def 6;

            then

             A39: pn = ((r . n) (#) ( chi (( Measurable-X-section ((E . n),x)),X2))) by A38, Th29;

            hence

             A40: pn is nonnegative or pn is nonpositive by A2, A14, A18, MESFUNC5: 20;

            ( dom ( chi (( Measurable-X-section ((E . n),x)),X2))) = XX2 by FUNCT_2:def 1;

            hence

             A41: pn is XX2 -measurable by A39, MESFUNC1: 37, MESFUNC2: 29;

            thus for y be Element of X2 st y in ( dom fn) holds (fn . y) = 0

            proof

              let y be Element of X2;

              assume

               A42: y in ( dom fn);

              then (( chi (( Measurable-X-section ((E . n),x)),X2)) . y) = 0 by A37, FUNCT_3: 37;

              then (pn . y) = ((r . n) * 0 ) by A36, A39, MESFUNC1:def 6;

              hence (fn . y) = 0 by A42, FUNCT_1: 47;

            end;

            then ( Integral (M2,fn)) = 0 by A37, Th57;

            then ( Integral (M2,(pn | ((XX2 \ ( Measurable-X-section ((E . n),x))) \/ ( Measurable-X-section ((E . n),x)))))) = (( Integral (M2,(pn | ( Measurable-X-section ((E . n),x))))) + 0 ) by A36, A40, A41, XBOOLE_1: 79, MESFUNC5: 91, MESFUN11: 62;

            then ( Integral (M2,(pn | ((XX2 \ ( Measurable-X-section ((E . n),x))) \/ ( Measurable-X-section ((E . n),x)))))) = ( Integral (M2,(pn | ( Measurable-X-section ((E . n),x))))) by XXREAL_3: 4;

            then

             A43: ( Integral (M2,(pn | XX2))) = ( Integral (M2,(pn | ( Measurable-X-section ((E . n),x))))) by XBOOLE_1: 45;

            (pn | ( Measurable-X-section ((E . n),x))) = (( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)) | ( X-section ((E . n),x))) by MEASUR11:def 6;

            then (pn | ( Measurable-X-section ((E . n),x))) = ( ProjPMap1 ((( chi ((r . n),(E . n), [:X1, X2:])) | (E . n)),x)) by Th34;

            then (pn | ( Measurable-X-section ((E . n),x))) = ( ProjPMap1 ((f | (E . n)),x)) by A9;

            then (pn | ( Measurable-X-section ((E . n),x))) = (( ProjPMap1 (f,x)) | ( X-section ((E . n),x))) by Th34;

            hence ( Integral (M2,( ProjPMap1 (( chi ((r . n),(E . n), [:X1, X2:])),x)))) = ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section ((E . n),x))))) by A43, MEASUR11:def 6;

            ( union ( rng (E | n))) misses (E . (n + 1)) by NAT_1: 16, MEASUR11: 1;

            then ( Union (E | n)) misses (E . (n + 1)) by CARD_3:def 4;

            then ( X-section (( Union (E | n)),x)) misses ( X-section ((E . (n + 1)),x)) by MEASUR11: 35;

            then ( Measurable-X-section (( Union (E | n)),x)) misses ( X-section ((E . (n + 1)),x)) by MEASUR11:def 6;

            hence ( Measurable-X-section (( Union (E | n)),x)) misses ( Measurable-X-section ((E . (n + 1)),x)) by MEASUR11:def 6;

            ( union ( rng (E | (n + 1)))) = (( union ( rng (E | n))) \/ (E . (n + 1))) by MEASUR11: 3;

            then ( Union (E | (n + 1))) = (( union ( rng (E | n))) \/ (E . (n + 1))) by CARD_3:def 4;

            then ( Union (E | (n + 1))) = (( Union (E | n)) \/ (E . (n + 1))) by CARD_3:def 4;

            then ( X-section (( Union (E | (n + 1))),x)) = (( X-section (( Union (E | n)),x)) \/ ( X-section ((E . (n + 1)),x))) by MEASUR11: 26;

            

            then ( Measurable-X-section (( Union (E | (n + 1))),x)) = (( X-section (( Union (E | n)),x)) \/ ( X-section ((E . (n + 1)),x))) by MEASUR11:def 6

            .= (( Measurable-X-section (( Union (E | n)),x)) \/ ( X-section ((E . (n + 1)),x))) by MEASUR11:def 6;

            hence ( Measurable-X-section (( Union (E | (n + 1))),x)) = (( Measurable-X-section (( Union (E | n)),x)) \/ ( Measurable-X-section ((E . (n + 1)),x))) by MEASUR11:def 6;

          end;

          defpred P[ Nat] means $1 <= ( len H) implies ((( Partial_Sums H) /. $1) . x) = ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | $1)))),x))));

          

           A44: P[1]

          proof

            assume

             A45: 1 <= ( len H);

            then

             A46: 1 in ( dom H) by FINSEQ_3: 25;

            ( len H) = ( len ( Partial_Sums H)) by MEASUR11:def 11;

            then ( dom H) = ( dom ( Partial_Sums H)) by FINSEQ_3: 29;

            then (( Partial_Sums H) /. 1) = (( Partial_Sums H) . 1) by A45, FINSEQ_3: 25, PARTFUN1:def 6;

            then (( Partial_Sums H) /. 1) = (H . 1) by MEASUR11:def 11;

            then

             A47: ((( Partial_Sums H) /. 1) . x) = ( Integral (M2,( ProjPMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x)))) by A13, A31, A46;

            (E | 1) = <*(E . 1)*> by FINSEQ_5: 20;

            then ( rng (E | 1)) = {(E . 1)} by FINSEQ_1: 39;

            then ( union ( rng (E | 1))) = (E . 1) by ZFMISC_1: 25;

            then ( ProjPMap1 ((f | ( union ( rng (E | 1)))),x)) = ( ProjPMap1 ((( chi ((r . 1),(E . 1), [:X1, X2:])) | (E . 1)),x)) by A9;

            then ( ProjPMap1 ((f | ( union ( rng (E | 1)))),x)) = (( ProjPMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x)) | ( X-section ((E . 1),x))) by Th34;

            then

             A48: ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | 1)))),x)))) = ( Integral (M2,(( ProjPMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x)) | ( Measurable-X-section ((E . 1),x))))) by MEASUR11:def 6;

            set p1 = ( ProjPMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x));

            set d1 = (XX2 \ ( Measurable-X-section ((E . 1),x)));

            set f1 = (( ProjPMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x)) | (XX2 \ ( Measurable-X-section ((E . 1),x))));

            

             A49: ( dom f1) = (XX2 \ ( Measurable-X-section ((E . 1),x))) & (p1 is nonnegative or p1 is nonpositive) by A35;

            p1 = ( ProjMap1 (( chi ((r . 1),(E . 1), [:X1, X2:])),x)) by Th27;

            then

             A50: ( dom p1) = X2 by FUNCT_2:def 1;

            

             A51: (XX2 \ ( Measurable-X-section ((E . 1),x))) misses ( Measurable-X-section ((E . 1),x)) by XBOOLE_1: 79;

            

             A52: ((XX2 \ ( Measurable-X-section ((E . 1),x))) \/ ( Measurable-X-section ((E . 1),x))) = XX2 by XBOOLE_1: 45;

            for y be Element of X2 st y in ( dom f1) holds (f1 . y) = 0 by A35;

            then ( Integral (M2,f1)) = 0 by A49, Th57;

            then ( Integral (M2,(p1 | ((XX2 \ ( Measurable-X-section ((E . 1),x))) \/ ( Measurable-X-section ((E . 1),x)))))) = (( Integral (M2,(p1 | ( Measurable-X-section ((E . 1),x))))) + 0 ) by A35, A49, A50, A51, MESFUNC5: 91, MESFUN11: 62;

            hence ((( Partial_Sums H) /. 1) . x) = ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | 1)))),x)))) by A47, A48, A52, XXREAL_3: 4;

          end;

          

           A54: for n be non zero Nat st P[n] holds P[(n + 1)]

          proof

            let n be non zero Nat;

            assume

             A55: P[n];

            assume

             A56: (n + 1) <= ( len H);

            then n < ( len H) by NAT_1: 13;

            then

             A57: n <= ( len ( Partial_Sums H)) & (n + 1) <= ( len ( Partial_Sums H)) by A56, MEASUR11:def 11;

            

             A58: 1 <= (n + 1) by NAT_1: 12;

            

             A59: n >= 1 by NAT_1: 14;

            then

             A60: n in ( dom ( Partial_Sums H)) & (n + 1) in ( dom ( Partial_Sums H)) & (n + 1) in ( dom H) by A56, A57, NAT_1: 12, FINSEQ_3: 25;

            then

             A61: (( Partial_Sums H) /. (n + 1)) = (( Partial_Sums H) . (n + 1)) & (( Partial_Sums H) /. n) = (( Partial_Sums H) . n) & (H /. (n + 1)) = (H . (n + 1)) by PARTFUN1:def 6;

            

             A62: ((( Partial_Sums H) /. n) is without-infty & (H /. (n + 1)) is without-infty) or ((( Partial_Sums H) /. n) is without+infty & (H /. (n + 1)) is without+infty)

            proof

              per cases by A2;

                suppose f is nonnegative;

                hence thesis by A22, A26, A56, A57, A58, A59, A61, FINSEQ_3: 25;

              end;

                suppose f is nonpositive;

                hence thesis by A24, A27, A56, A57, A58, A59, A61, FINSEQ_3: 25;

              end;

            end;

            

             A63: ( X-section (( Union (E | n)),x)) = ( Measurable-X-section (( Union (E | n)),x)) by MEASUR11:def 6;

            (( Partial_Sums H) . (n + 1)) = ((( Partial_Sums H) /. n) + (H /. (n + 1))) by A56, A59, NAT_1: 13, MEASUR11:def 11;

            then ((( Partial_Sums H) . (n + 1)) . x) = (((( Partial_Sums H) /. n) . x) + ((H /. (n + 1)) . x)) by A62, DBLSEQ_3: 7;

            then ((( Partial_Sums H) . (n + 1)) . x) = (( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | n)))),x)))) + ( Integral (M2,( ProjPMap1 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),x))))) by A13, A55, A56, A60, A61, A31, NAT_1: 13;

            then ((( Partial_Sums H) . (n + 1)) . x) = (( Integral (M2,( ProjPMap1 ((f | ( Union (E | n))),x)))) + ( Integral (M2,( ProjPMap1 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),x))))) by CARD_3:def 4;

            then ((( Partial_Sums H) . (n + 1)) . x) = (( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section (( Union (E | n)),x))))) + ( Integral (M2,( ProjPMap1 (( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])),x))))) by Th34;

            then ((( Partial_Sums H) . (n + 1)) . x) = (( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section (( Union (E | n)),x))))) + ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section ((E . (n + 1)),x)))))) by A35;

            then ((( Partial_Sums H) . (n + 1)) . x) = ( Integral (M2,(( ProjPMap1 (f,x)) | (( Measurable-X-section (( Union (E | n)),x)) \/ ( Measurable-X-section ((E . (n + 1)),x)))))) by A32, A33, A34, A35, A63, MESFUNC5: 91, MESFUN11: 62;

            then ((( Partial_Sums H) . (n + 1)) . x) = ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section (( Union (E | (n + 1))),x))))) by A35;

            then ((( Partial_Sums H) . (n + 1)) . x) = ( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section (( Union (E | (n + 1))),x))))) by MEASUR11:def 6;

            then ((( Partial_Sums H) . (n + 1)) . x) = ( Integral (M2,( ProjPMap1 ((f | ( Union (E | (n + 1)))),x)))) by Th34;

            then ((( Partial_Sums H) . (n + 1)) . x) = ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | (n + 1))))),x)))) by CARD_3:def 4;

            hence ((( Partial_Sums H) /. (n + 1)) . x) = ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | (n + 1))))),x)))) by A60, PARTFUN1:def 6;

          end;

          ( len H) = ( len E) by A11, FINSEQ_1:def 3;

          then (E | ( len H)) = (E | ( dom E)) by FINSEQ_1:def 3;

          then ( union ( rng (E | ( len H)))) = ( dom f) by A8, MESFUNC3:def 1;

          then

           A64: (f | ( union ( rng (E | ( len H))))) = f;

          for n be non zero Nat holds P[n] from NAT_1:sch 10( A44, A54);

          hence ((( Partial_Sums H) /. ( len H)) . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A29, A64;

        end;

        hence for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))));

        thus for V be Element of S1 holds I2 is V -measurable

        proof

          let V be Element of S1;

          

           A65: for n be Nat st n in ( dom H) holds (H /. n) is V -measurable

          proof

            let n be Nat;

            assume n in ( dom H);

            then

             A66: (H /. n) = (H . n) & (H . n) = ((r . n) (#) ( Y-vol ((E . n),M2))) by A11, A12, PARTFUN1:def 6;

            

             A67: ( dom ( Y-vol ((E . n),M2))) = XX1 by FUNCT_2:def 1;

            ( Y-vol ((E . n),M2)) is V -measurable by A1, MEASUR11:def 13;

            hence (H /. n) is V -measurable by A66, A67, MESFUNC1: 37;

          end;

          defpred P2[ Nat] means $1 <= ( len H) implies (( Partial_Sums H) /. $1) is V -measurable;

          (( Partial_Sums H) /. 1) = (( Partial_Sums H) . 1) by A28, A30, FINSEQ_3: 25, PARTFUN1:def 6;

          then (( Partial_Sums H) /. 1) = (H . 1) by MEASUR11:def 11;

          then (( Partial_Sums H) /. 1) = (H /. 1) by A30, A28, FINSEQ_3: 25, PARTFUN1:def 6;

          then

           A68: P2[1] by A65, FINSEQ_3: 25;

          

           A69: for n be non zero Nat st P2[n] holds P2[(n + 1)]

          proof

            let n be non zero Nat;

            assume

             A70: P2[n];

            assume

             A71: (n + 1) <= ( len H);

            then

             A72: 1 <= n < ( len H) by NAT_1: 13, NAT_1: 14;

            then

             A73: n in ( dom H) & (n + 1) in ( dom H) by A71, NAT_1: 11, FINSEQ_3: 25;

            then

             A74: (( Partial_Sums H) /. n) = (( Partial_Sums H) . n) & (H . (n + 1)) = (H /. (n + 1)) & (( Partial_Sums H) /. (n + 1)) = (( Partial_Sums H) . (n + 1)) by A28, PARTFUN1:def 6;

            then

             A75: (( Partial_Sums H) /. (n + 1)) = ((( Partial_Sums H) /. n) + (H /. (n + 1))) by A72, MEASUR11:def 11;

            

             A76: ( dom (H /. (n + 1))) = XX1 & ( dom (( Partial_Sums H) /. n)) = XX1 by FUNCT_2:def 1;

            

             A77: (H /. (n + 1)) is V -measurable by A73, A65;

            per cases by A2;

              suppose f is nonnegative;

              then (H /. (n + 1)) is without-infty & (( Partial_Sums H) /. n) is without-infty by A22, A26, A28, A73, A74;

              hence (( Partial_Sums H) /. (n + 1)) is V -measurable by A70, A71, A75, A77, NAT_1: 13, MESFUNC5: 31;

            end;

              suppose f is nonpositive;

              then

               A78: (H /. (n + 1)) is without+infty & (( Partial_Sums H) /. n) is without+infty by A24, A27, A28, A73, A74;

              then ( dom ((( Partial_Sums H) /. n) + (H /. (n + 1)))) = (( dom (( Partial_Sums H) /. n)) /\ ( dom (H /. (n + 1)))) by MESFUNC9: 1;

              hence (( Partial_Sums H) /. (n + 1)) is V -measurable by A70, A71, A75, A77, A76, A78, NAT_1: 13, MEASUR11: 65;

            end;

          end;

          for n be non zero Nat holds P2[n] from NAT_1:sch 10( A68, A69);

          hence thesis by A29;

        end;

      end;

    end;

    theorem :: MESFUN12:58

    

     Th58: for X1,X2,Y be non empty set, F be Functional_Sequence of [:X1, X2:], Y, x be Element of X1, y be Element of X2 st F is with_the_same_dom holds ( ProjPMap1 (F,x)) is with_the_same_dom & ( ProjPMap2 (F,y)) is with_the_same_dom

    proof

      let X1,X2,Y be non empty set, F be Functional_Sequence of [:X1, X2:], Y, x1 be Element of X1, x2 be Element of X2;

      assume

       A1: F is with_the_same_dom;

      now

        let m,n be Nat;

        ( dom (( ProjPMap1 (F,x1)) . m)) = ( dom ( ProjPMap1 ((F . m),x1))) by Def5

        .= ( X-section (( dom (F . m)),x1)) by Def3

        .= ( X-section (( dom (F . n)),x1)) by A1, MESFUNC8:def 2

        .= ( dom ( ProjPMap1 ((F . n),x1))) by Def3;

        hence ( dom (( ProjPMap1 (F,x1)) . m)) = ( dom (( ProjPMap1 (F,x1)) . n)) by Def5;

      end;

      hence ( ProjPMap1 (F,x1)) is with_the_same_dom by MESFUNC8:def 2;

      now

        let m,n be Nat;

        ( dom (( ProjPMap2 (F,x2)) . m)) = ( dom ( ProjPMap2 ((F . m),x2))) by Def6

        .= ( Y-section (( dom (F . m)),x2)) by Def4

        .= ( Y-section (( dom (F . n)),x2)) by A1, MESFUNC8:def 2

        .= ( dom ( ProjPMap2 ((F . n),x2))) by Def4;

        hence ( dom (( ProjPMap2 (F,x2)) . m)) = ( dom (( ProjPMap2 (F,x2)) . n)) by Def6;

      end;

      hence ( ProjPMap2 (F,x2)) is with_the_same_dom by MESFUNC8:def 2;

    end;

    begin

    

     Lm7: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & f is nonnegative & A = ( dom f) & f is A -measurable holds ex I1 be Function of X2, ExtREAL st (for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))) & (for V be Element of S2 holds I1 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A3: f is nonnegative & A = ( dom f) & f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider M = ( product_sigma_Measure (M1,M2)) as sigma_Measure of S by MEASUR11: 8;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL such that

       A4: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

       A5: for n be Nat holds (F . n) is nonnegative and

       A6: for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) <= ((F . m) . z) and

       A7: for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z) by A3, MESFUNC5: 64;

      now

        let m,n be Nat;

        ( dom (F . m)) = ( dom f) by A4;

        hence ( dom (F . m)) = ( dom (F . n)) by A4;

      end;

      then

       A8: F is with_the_same_dom by MESFUNC8:def 2;

      defpred P[ Nat, object] means ex Fy be Function of X2, ExtREAL st $2 = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . $1),y1)))));

      

       A10: for n be Element of NAT holds ex FI1 be Element of ( PFuncs (X2, ExtREAL )) st P[n, FI1]

      proof

        let n be Element of NAT ;

        deffunc F( Element of X2) = ( Integral (M1,( ProjPMap2 ((F . n),$1))));

        consider FI1 be Function such that

         A11: ( dom FI1) = X2 & for y1 be Element of X2 holds (FI1 . y1) = F(y1) from FUNCT_1:sch 4;

        

         A12: for y2 be object st y2 in X2 holds (FI1 . y2) in ExtREAL

        proof

          let y2 be object;

          assume y2 in X2;

          then

          reconsider y1 = y2 as Element of X2;

          (FI1 . y2) = ( Integral (M1,( ProjPMap2 ((F . n),y1)))) by A11;

          hence (FI1 . y2) in ExtREAL ;

        end;

        then FI1 is Function of X2, ExtREAL by A11, FUNCT_2: 3;

        then

        reconsider FI1 as Element of ( PFuncs (X2, ExtREAL )) by PARTFUN1: 45;

        take FI1;

        reconsider Fy = FI1 as Function of X2, ExtREAL by A12, A11, FUNCT_2: 3;

        for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1)))) by A11;

        hence ex Fy be Function of X2, ExtREAL st FI1 = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))) by A11;

      end;

      consider FI1 be Function of NAT , ( PFuncs (X2, ExtREAL )) such that

       A13: for n be Element of NAT holds P[n, (FI1 . n)] from FUNCT_2:sch 3( A10);

      

       A14: for n be Nat holds ( dom (FI1 . n)) = X2

      proof

        let n be Nat;

        n is Element of NAT by ORDINAL1:def 12;

        then ex Fy be Function of X2, ExtREAL st (FI1 . n) = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))) by A13;

        hence ( dom (FI1 . n)) = X2;

      end;

      

       A15: for n be Nat, y1 be Element of X2 st y1 in ( dom (FI1 . n)) holds ((FI1 . n) . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))

      proof

        let n be Nat, y1 be Element of X2;

        assume y1 in ( dom (FI1 . n));

        n is Element of NAT by ORDINAL1:def 12;

        then P[n, (FI1 . n)] by A13;

        hence ((FI1 . n) . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))));

      end;

      

       A16: for y1 be Element of X2, x1 be Element of X1 st x1 in ( dom ( ProjPMap2 (f,y1))) holds (( ProjPMap2 (F,y1)) # x1) is convergent & ( lim (( ProjPMap2 (F,y1)) # x1)) = (( ProjPMap2 (f,y1)) . x1)

      proof

        let y1 be Element of X2, x1 be Element of X1;

        reconsider z1 = [x1, y1] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume x1 in ( dom ( ProjPMap2 (f,y1)));

        then x1 in ( Y-section (A,y1)) by A3, Def4;

        then

         A17: z1 in ( dom f) by A3, Th25;

        then

         A18: (F # z1) is convergent by A7;

        

         A19: for n be Element of NAT holds ((F # z1) . n) = ((( ProjPMap2 (F,y1)) # x1) . n)

        proof

          let n be Element of NAT ;

          

           A20: [x1, y1] in ( dom (F . n)) by A4, A17;

          ((F # z1) . n) = ((F . n) . (x1,y1)) by MESFUNC5:def 13;

          then ((F # z1) . n) = (( ProjPMap2 ((F . n),y1)) . x1) by A20, Def4;

          then ((F # z1) . n) = ((( ProjPMap2 (F,y1)) . n) . x1) by Def6;

          hence ((F # z1) . n) = ((( ProjPMap2 (F,y1)) # x1) . n) by MESFUNC5:def 13;

        end;

        hence (( ProjPMap2 (F,y1)) # x1) is convergent by A18, FUNCT_2:def 8;

        (F # z1) = (( ProjPMap2 (F,y1)) # x1) by A19, FUNCT_2:def 8;

        then ( lim (( ProjPMap2 (F,y1)) # x1)) = (f . (x1,y1)) by A7, A17;

        hence ( lim (( ProjPMap2 (F,y1)) # x1)) = (( ProjPMap2 (f,y1)) . x1) by A17, Def4;

      end;

      

       A21: for y be Element of X2 holds ( lim ( ProjPMap2 (F,y))) = ( ProjPMap2 (f,y)) & (FI1 # y) is convergent & ( lim (FI1 # y)) = ( Integral (M1,( lim ( ProjPMap2 (F,y)))))

      proof

        let y be Element of X2;

        ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom (( ProjPMap2 (F,y)) . 0 )) by MESFUNC8:def 9;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom ( ProjPMap2 ((F . 0 ),y))) by Def6;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( Y-section (( dom (F . 0 )),y)) by Def4;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( Y-section (( dom f),y)) by A4;

        then

         A22: ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom ( ProjPMap2 (f,y))) by Def4;

        for x be Element of X1 st x in ( dom ( lim ( ProjPMap2 (F,y)))) holds (( lim ( ProjPMap2 (F,y))) . x) = (( ProjPMap2 (f,y)) . x)

        proof

          let x be Element of X1;

          assume

           A23: x in ( dom ( lim ( ProjPMap2 (F,y))));

          then (( lim ( ProjPMap2 (F,y))) . x) = ( lim (( ProjPMap2 (F,y)) # x)) by MESFUNC8:def 9;

          hence (( lim ( ProjPMap2 (F,y))) . x) = (( ProjPMap2 (f,y)) . x) by A16, A22, A23;

        end;

        hence ( lim ( ProjPMap2 (F,y))) = ( ProjPMap2 (f,y)) by A22, PARTFUN1: 5;

        

         A24: (( ProjPMap2 (F,y)) . 0 ) = ( ProjPMap2 ((F . 0 ),y)) by Def6;

        then ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Y-section (( dom (F . 0 )),y)) by Def4;

        then ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Y-section (A,y)) by A4, A3;

        then

         A25: ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

        (F . 0 ) is nonnegative by A5;

        then

         A26: (( ProjPMap2 (F,y)) . 0 ) is nonnegative by A24, Th32;

        

         A27: for n be Nat holds (( ProjPMap2 (F,y)) . n) is ( Measurable-Y-section (A,y)) -measurable

        proof

          let n be Nat;

          

           A28: ( dom (F . n)) = A by A3, A4;

          (F . n) is A -measurable by A4, MESFUNC2: 34;

          then ( ProjPMap2 ((F . n),y)) is ( Measurable-Y-section (A,y)) -measurable by A28, Th47;

          hence (( ProjPMap2 (F,y)) . n) is ( Measurable-Y-section (A,y)) -measurable by Def6;

        end;

        

         A29: for n,m be Nat st n <= m holds for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds ((( ProjPMap2 (F,y)) . n) . x) <= ((( ProjPMap2 (F,y)) . m) . x)

        proof

          let n,m be Nat;

          assume

           A30: n <= m;

          let x be Element of X1;

          assume

           A31: x in ( Measurable-Y-section (A,y));

          then x in ( dom ( ProjPMap2 ((F . 0 ),y))) by A25, Def6;

          then x in ( Y-section (( dom (F . 0 )),y)) by Def4;

          then x in ( Y-section (( dom f),y)) by A4;

          then

           A32: [x, y] in ( dom f) by Th25;

          

           A33: ( dom (( ProjPMap2 (F,y)) . n)) = ( dom (( ProjPMap2 (F,y)) . 0 )) & ( dom (( ProjPMap2 (F,y)) . m)) = ( dom (( ProjPMap2 (F,y)) . 0 )) by A8, Th58, MESFUNC8:def 2;

          (( ProjPMap2 (F,y)) . n) = ( ProjPMap2 ((F . n),y)) & (( ProjPMap2 (F,y)) . m) = ( ProjPMap2 ((F . m),y)) by Def6;

          then ((( ProjPMap2 (F,y)) . n) . x) = ((F . n) . (x,y)) & ((( ProjPMap2 (F,y)) . m) . x) = ((F . m) . (x,y)) by A25, A31, A33, Th26;

          hence ((( ProjPMap2 (F,y)) . n) . x) <= ((( ProjPMap2 (F,y)) . m) . x) by A6, A30, A32;

        end;

        for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (( ProjPMap2 (F,y)) # x) is convergent

        proof

          let x be Element of X1;

          assume x in ( Measurable-Y-section (A,y));

          then x in ( Y-section (( dom f),y)) by A3, MEASUR11:def 7;

          then x in ( dom ( ProjPMap2 (f,y))) by Def4;

          hence (( ProjPMap2 (F,y)) # x) is convergent by A16;

        end;

        then

        consider J be ExtREAL_sequence such that

         A34: (for n be Nat holds (J . n) = ( Integral (M1,(( ProjPMap2 (F,y)) . n)))) and

         A35: J is convergent and

         A36: ( Integral (M1,( lim ( ProjPMap2 (F,y))))) = ( lim J) by A8, A25, A26, A27, A29, Th58, MESFUNC9: 52;

        for n be Element of NAT holds (J . n) = ((FI1 # y) . n)

        proof

          let n be Element of NAT ;

          

           A37: ( dom (FI1 . n)) = X2 by A14;

          ((FI1 # y) . n) = ((FI1 . n) . y) by MESFUNC5:def 13;

          then ((FI1 # y) . n) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by A15, A37;

          then ((FI1 # y) . n) = ( Integral (M1,(( ProjPMap2 (F,y)) . n))) by Def6;

          hence (J . n) = ((FI1 # y) . n) by A34;

        end;

        hence (FI1 # y) is convergent & ( lim (FI1 # y)) = ( Integral (M1,( lim ( ProjPMap2 (F,y))))) by A35, A36, FUNCT_2: 63;

      end;

      ( dom ( lim FI1)) = ( dom (FI1 . 0 )) by MESFUNC8:def 9;

      then

       A38: ( dom ( lim FI1)) = X2 by A14;

      then

      reconsider I1 = ( lim FI1) as Function of X2, ExtREAL by FUNCT_2:def 1;

      take I1;

      for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))

      proof

        let y be Element of X2;

        (I1 . y) = ( lim (FI1 # y)) by A38, MESFUNC8:def 9;

        then (I1 . y) = ( Integral (M1,( lim ( ProjPMap2 (F,y))))) by A21;

        hence (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A21;

      end;

      hence for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))));

      thus for V be Element of S2 holds I1 is V -measurable

      proof

        let V be Element of S2;

        now

          let m,n be Nat;

          ( dom (FI1 . m)) = X2 & ( dom (FI1 . n)) = X2 by A14;

          hence ( dom (FI1 . m)) = ( dom (FI1 . n));

        end;

        then

         A39: FI1 is with_the_same_dom by MESFUNC8:def 2;

        

         A40: ( dom (FI1 . 0 )) = XX2 by A14;

        

         A41: for n be Nat holds (FI1 . n) is XX2 -measurable

        proof

          let n be Nat;

          ( dom (F . n)) = A & (F . n) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by A4, A3;

          then

          consider L be Function of X2, ExtREAL such that

           A42: (for y be Element of X2 holds (L . y) = ( Integral (M1,( ProjPMap2 ((F . n),y))))) & (for W be Element of S2 holds L is W -measurable) by A1, A5, Lm5;

          

           A43: ( dom (FI1 . n)) = X2 by A14;

          then

           A44: (FI1 . n) is Function of X2, ExtREAL by FUNCT_2:def 1;

          for y be Element of X2 holds ((FI1 . n) . y) = (L . y)

          proof

            let y be Element of X2;

            ((FI1 . n) . y) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by A15, A43;

            hence ((FI1 . n) . y) = (L . y) by A42;

          end;

          then (FI1 . n) = L by A44, FUNCT_2: 63;

          hence (FI1 . n) is XX2 -measurable by A42;

        end;

        for y be Element of X2 st y in XX2 holds (FI1 # y) is convergent by A21;

        hence I1 is V -measurable by A39, A40, A41, MESFUNC8: 25, MESFUNC1: 30;

      end;

    end;

    

     Lm8: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & f is nonpositive & A = ( dom f) & f is A -measurable holds ex I1 be Function of X2, ExtREAL st (for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))) & (for V be Element of S2 holds I1 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A3: f is nonpositive & A = ( dom f) & f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider M = ( product_sigma_Measure (M1,M2)) as sigma_Measure of S by MEASUR11: 8;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL such that

       A4: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

       A5: for n be Nat holds (F . n) is nonpositive and

       A6: for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) >= ((F . m) . z) and

       A7: for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z) by A3, MESFUN11: 56;

      now

        let m,n be Nat;

        ( dom (F . m)) = ( dom f) by A4;

        hence ( dom (F . m)) = ( dom (F . n)) by A4;

      end;

      then

       A8: F is with_the_same_dom by MESFUNC8:def 2;

      defpred P[ Nat, object] means ex Fy be Function of X2, ExtREAL st $2 = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . $1),y1)))));

      

       A10: for n be Element of NAT holds ex FI1 be Element of ( PFuncs (X2, ExtREAL )) st P[n, FI1]

      proof

        let n be Element of NAT ;

        deffunc F( Element of X2) = ( Integral (M1,( ProjPMap2 ((F . n),$1))));

        consider FI1 be Function such that

         A11: ( dom FI1) = X2 & for y1 be Element of X2 holds (FI1 . y1) = F(y1) from FUNCT_1:sch 4;

        

         A12: for y2 be object st y2 in X2 holds (FI1 . y2) in ExtREAL

        proof

          let y2 be object;

          assume y2 in X2;

          then

          reconsider y1 = y2 as Element of X2;

          (FI1 . y2) = ( Integral (M1,( ProjPMap2 ((F . n),y1)))) by A11;

          hence (FI1 . y2) in ExtREAL ;

        end;

        then FI1 is Function of X2, ExtREAL by A11, FUNCT_2: 3;

        then

        reconsider FI1 as Element of ( PFuncs (X2, ExtREAL )) by PARTFUN1: 45;

        take FI1;

        reconsider Fy = FI1 as Function of X2, ExtREAL by A12, A11, FUNCT_2: 3;

        for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1)))) by A11;

        hence ex Fy be Function of X2, ExtREAL st FI1 = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))) by A11;

      end;

      consider FI1 be Function of NAT , ( PFuncs (X2, ExtREAL )) such that

       A13: for n be Element of NAT holds P[n, (FI1 . n)] from FUNCT_2:sch 3( A10);

      

       A14: for n be Nat holds ( dom (FI1 . n)) = X2

      proof

        let n be Nat;

        n is Element of NAT by ORDINAL1:def 12;

        then ex Fy be Function of X2, ExtREAL st (FI1 . n) = Fy & ( dom Fy) = X2 & (for y1 be Element of X2 st y1 in ( dom Fy) holds (Fy . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))) by A13;

        hence ( dom (FI1 . n)) = X2;

      end;

      

       A15: for n be Nat, y1 be Element of X2 st y1 in ( dom (FI1 . n)) holds ((FI1 . n) . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))))

      proof

        let n be Nat, y1 be Element of X2;

        assume y1 in ( dom (FI1 . n));

        n is Element of NAT by ORDINAL1:def 12;

        then P[n, (FI1 . n)] by A13;

        hence ((FI1 . n) . y1) = ( Integral (M1,( ProjPMap2 ((F . n),y1))));

      end;

      

       A16: for y1 be Element of X2, x1 be Element of X1 st x1 in ( dom ( ProjPMap2 (f,y1))) holds (( ProjPMap2 (F,y1)) # x1) is convergent & ( lim (( ProjPMap2 (F,y1)) # x1)) = (( ProjPMap2 (f,y1)) . x1)

      proof

        let y1 be Element of X2, x1 be Element of X1;

        reconsider z1 = [x1, y1] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume x1 in ( dom ( ProjPMap2 (f,y1)));

        then x1 in ( Y-section (A,y1)) by A3, Def4;

        then

         A17: z1 in ( dom f) by A3, Th25;

        then

         A18: (F # z1) is convergent by A7;

        

         A19: for n be Element of NAT holds ((F # z1) . n) = ((( ProjPMap2 (F,y1)) # x1) . n)

        proof

          let n be Element of NAT ;

          

           A20: [x1, y1] in ( dom (F . n)) by A4, A17;

          ((F # z1) . n) = ((F . n) . (x1,y1)) by MESFUNC5:def 13;

          then ((F # z1) . n) = (( ProjPMap2 ((F . n),y1)) . x1) by A20, Def4;

          then ((F # z1) . n) = ((( ProjPMap2 (F,y1)) . n) . x1) by Def6;

          hence ((F # z1) . n) = ((( ProjPMap2 (F,y1)) # x1) . n) by MESFUNC5:def 13;

        end;

        hence (( ProjPMap2 (F,y1)) # x1) is convergent by A18, FUNCT_2:def 8;

        (F # z1) = (( ProjPMap2 (F,y1)) # x1) by A19, FUNCT_2:def 8;

        then ( lim (( ProjPMap2 (F,y1)) # x1)) = (f . (x1,y1)) by A7, A17;

        hence ( lim (( ProjPMap2 (F,y1)) # x1)) = (( ProjPMap2 (f,y1)) . x1) by A17, Def4;

      end;

      

       A21: for y be Element of X2 holds ( lim ( ProjPMap2 (F,y))) = ( ProjPMap2 (f,y)) & (FI1 # y) is convergent & ( lim (FI1 # y)) = ( Integral (M1,( lim ( ProjPMap2 (F,y)))))

      proof

        let y be Element of X2;

        ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom (( ProjPMap2 (F,y)) . 0 )) by MESFUNC8:def 9;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom ( ProjPMap2 ((F . 0 ),y))) by Def6;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( Y-section (( dom (F . 0 )),y)) by Def4;

        then ( dom ( lim ( ProjPMap2 (F,y)))) = ( Y-section (( dom f),y)) by A4;

        then

         A22: ( dom ( lim ( ProjPMap2 (F,y)))) = ( dom ( ProjPMap2 (f,y))) by Def4;

        for x be Element of X1 st x in ( dom ( lim ( ProjPMap2 (F,y)))) holds (( lim ( ProjPMap2 (F,y))) . x) = (( ProjPMap2 (f,y)) . x)

        proof

          let x be Element of X1;

          assume

           A23: x in ( dom ( lim ( ProjPMap2 (F,y))));

          then (( lim ( ProjPMap2 (F,y))) . x) = ( lim (( ProjPMap2 (F,y)) # x)) by MESFUNC8:def 9;

          hence (( lim ( ProjPMap2 (F,y))) . x) = (( ProjPMap2 (f,y)) . x) by A16, A22, A23;

        end;

        hence ( lim ( ProjPMap2 (F,y))) = ( ProjPMap2 (f,y)) by A22, PARTFUN1: 5;

        

         A24: (( ProjPMap2 (F,y)) . 0 ) = ( ProjPMap2 ((F . 0 ),y)) by Def6;

        then ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Y-section (( dom (F . 0 )),y)) by Def4;

        then ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Y-section (A,y)) by A4, A3;

        then

         A25: ( dom (( ProjPMap2 (F,y)) . 0 )) = ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

        (F . 0 ) is nonpositive by A5;

        then

         A26: (( ProjPMap2 (F,y)) . 0 ) is nonpositive by A24, Th33;

        

         A27: for n be Nat holds (( ProjPMap2 (F,y)) . n) is ( Measurable-Y-section (A,y)) -measurable

        proof

          let n be Nat;

          

           A28: ( dom (F . n)) = A by A3, A4;

          (F . n) is A -measurable by A4, MESFUNC2: 34;

          then ( ProjPMap2 ((F . n),y)) is ( Measurable-Y-section (A,y)) -measurable by A28, Th47;

          hence (( ProjPMap2 (F,y)) . n) is ( Measurable-Y-section (A,y)) -measurable by Def6;

        end;

        

         A29: for n,m be Nat st n <= m holds for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds ((( ProjPMap2 (F,y)) . n) . x) >= ((( ProjPMap2 (F,y)) . m) . x)

        proof

          let n,m be Nat;

          assume

           A30: n <= m;

          let x be Element of X1;

          assume

           A31: x in ( Measurable-Y-section (A,y));

          then x in ( dom ( ProjPMap2 ((F . 0 ),y))) by A25, Def6;

          then x in ( Y-section (( dom (F . 0 )),y)) by Def4;

          then x in ( Y-section (( dom f),y)) by A4;

          then

           A32: [x, y] in ( dom f) by Th25;

          

           A33: ( dom (( ProjPMap2 (F,y)) . n)) = ( dom (( ProjPMap2 (F,y)) . 0 )) & ( dom (( ProjPMap2 (F,y)) . m)) = ( dom (( ProjPMap2 (F,y)) . 0 )) by A8, Th58, MESFUNC8:def 2;

          (( ProjPMap2 (F,y)) . n) = ( ProjPMap2 ((F . n),y)) & (( ProjPMap2 (F,y)) . m) = ( ProjPMap2 ((F . m),y)) by Def6;

          then ((( ProjPMap2 (F,y)) . n) . x) = ((F . n) . (x,y)) & ((( ProjPMap2 (F,y)) . m) . x) = ((F . m) . (x,y)) by A25, A31, A33, Th26;

          hence ((( ProjPMap2 (F,y)) . n) . x) >= ((( ProjPMap2 (F,y)) . m) . x) by A6, A30, A32;

        end;

        for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (( ProjPMap2 (F,y)) # x) is convergent

        proof

          let x be Element of X1;

          assume x in ( Measurable-Y-section (A,y));

          then x in ( Y-section (( dom f),y)) by A3, MEASUR11:def 7;

          then x in ( dom ( ProjPMap2 (f,y))) by Def4;

          hence (( ProjPMap2 (F,y)) # x) is convergent by A16;

        end;

        then

        consider J be ExtREAL_sequence such that

         A34: (for n be Nat holds (J . n) = ( Integral (M1,(( ProjPMap2 (F,y)) . n)))) and

         A35: J is convergent and

         A36: ( Integral (M1,( lim ( ProjPMap2 (F,y))))) = ( lim J) by A8, A25, A26, A27, A29, Th58, MESFUN11: 74;

        for n be Element of NAT holds (J . n) = ((FI1 # y) . n)

        proof

          let n be Element of NAT ;

          

           A37: ( dom (FI1 . n)) = X2 by A14;

          ((FI1 # y) . n) = ((FI1 . n) . y) by MESFUNC5:def 13;

          then ((FI1 # y) . n) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by A15, A37;

          then ((FI1 # y) . n) = ( Integral (M1,(( ProjPMap2 (F,y)) . n))) by Def6;

          hence (J . n) = ((FI1 # y) . n) by A34;

        end;

        hence (FI1 # y) is convergent & ( lim (FI1 # y)) = ( Integral (M1,( lim ( ProjPMap2 (F,y))))) by A35, A36, FUNCT_2: 63;

      end;

      ( dom ( lim FI1)) = ( dom (FI1 . 0 )) by MESFUNC8:def 9;

      then

       A38: ( dom ( lim FI1)) = X2 by A14;

      then

      reconsider I1 = ( lim FI1) as Function of X2, ExtREAL by FUNCT_2:def 1;

      take I1;

      for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))))

      proof

        let y be Element of X2;

        (I1 . y) = ( lim (FI1 # y)) by A38, MESFUNC8:def 9;

        then (I1 . y) = ( Integral (M1,( lim ( ProjPMap2 (F,y))))) by A21;

        hence (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A21;

      end;

      hence for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y))));

      thus for V be Element of S2 holds I1 is V -measurable

      proof

        let V be Element of S2;

        now

          let m,n be Nat;

          ( dom (FI1 . m)) = X2 & ( dom (FI1 . n)) = X2 by A14;

          hence ( dom (FI1 . m)) = ( dom (FI1 . n));

        end;

        then

         A39: FI1 is with_the_same_dom by MESFUNC8:def 2;

        

         A40: ( dom (FI1 . 0 )) = XX2 by A14;

        

         A41: for n be Nat holds (FI1 . n) is XX2 -measurable

        proof

          let n be Nat;

          ( dom (F . n)) = A & (F . n) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by A4, A3;

          then

          consider L be Function of X2, ExtREAL such that

           A42: (for y be Element of X2 holds (L . y) = ( Integral (M1,( ProjPMap2 ((F . n),y))))) & (for W be Element of S2 holds L is W -measurable) by A1, A5, Lm5;

          

           A43: ( dom (FI1 . n)) = X2 by A14;

          then

           A44: (FI1 . n) is Function of X2, ExtREAL by FUNCT_2:def 1;

          for y be Element of X2 holds ((FI1 . n) . y) = (L . y)

          proof

            let y be Element of X2;

            ((FI1 . n) . y) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by A15, A43;

            hence ((FI1 . n) . y) = (L . y) by A42;

          end;

          then (FI1 . n) = L by A44, FUNCT_2: 63;

          hence (FI1 . n) is XX2 -measurable by A42;

        end;

        for y be Element of X2 st y in XX2 holds (FI1 # y) is convergent by A21;

        hence I1 is V -measurable by A39, A40, A41, MESFUNC8: 25, MESFUNC1: 30;

      end;

    end;

    

     Lm9: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M2 is sigma_finite & f is nonnegative & A = ( dom f) & f is A -measurable holds ex I2 be Function of X1, ExtREAL st (for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))) & (for V be Element of S1 holds I2 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M2 is sigma_finite and

       A3: f is nonnegative & A = ( dom f) & f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider M = ( product_sigma_Measure (M1,M2)) as sigma_Measure of S by MEASUR11: 8;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL such that

       A4: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

       A5: for n be Nat holds (F . n) is nonnegative and

       A6: for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) <= ((F . m) . z) and

       A7: for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z) by A3, MESFUNC5: 64;

      now

        let m,n be Nat;

        ( dom (F . m)) = ( dom f) by A4;

        hence ( dom (F . m)) = ( dom (F . n)) by A4;

      end;

      then

       A8: F is with_the_same_dom by MESFUNC8:def 2;

      defpred P[ Nat, object] means ex Fx be Function of X1, ExtREAL st $2 = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . $1),x1)))));

      

       A10: for n be Element of NAT holds ex FI2 be Element of ( PFuncs (X1, ExtREAL )) st P[n, FI2]

      proof

        let n be Element of NAT ;

        deffunc F( Element of X1) = ( Integral (M2,( ProjPMap1 ((F . n),$1))));

        consider FI2 be Function such that

         A11: ( dom FI2) = X1 & for x1 be Element of X1 holds (FI2 . x1) = F(x1) from FUNCT_1:sch 4;

        

         A12: for x2 be object st x2 in X1 holds (FI2 . x2) in ExtREAL

        proof

          let x2 be object;

          assume x2 in X1;

          then

          reconsider x1 = x2 as Element of X1;

          (FI2 . x2) = ( Integral (M2,( ProjPMap1 ((F . n),x1)))) by A11;

          hence (FI2 . x2) in ExtREAL ;

        end;

        then FI2 is Function of X1, ExtREAL by A11, FUNCT_2: 3;

        then

        reconsider FI2 as Element of ( PFuncs (X1, ExtREAL )) by PARTFUN1: 45;

        take FI2;

        reconsider Fx = FI2 as Function of X1, ExtREAL by A12, A11, FUNCT_2: 3;

        for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1)))) by A11;

        hence ex Fx be Function of X1, ExtREAL st FI2 = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))) by A11;

      end;

      consider FI2 be Function of NAT , ( PFuncs (X1, ExtREAL )) such that

       A13: for n be Element of NAT holds P[n, (FI2 . n)] from FUNCT_2:sch 3( A10);

      

       A14: for n be Nat holds ( dom (FI2 . n)) = X1

      proof

        let n be Nat;

        n is Element of NAT by ORDINAL1:def 12;

        then ex Fx be Function of X1, ExtREAL st (FI2 . n) = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))) by A13;

        hence ( dom (FI2 . n)) = X1;

      end;

      

       A15: for n be Nat, x1 be Element of X1 st x1 in ( dom (FI2 . n)) holds ((FI2 . n) . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))

      proof

        let n be Nat, x1 be Element of X1;

        assume x1 in ( dom (FI2 . n));

        n is Element of NAT by ORDINAL1:def 12;

        then P[n, (FI2 . n)] by A13;

        hence ((FI2 . n) . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))));

      end;

      

       A16: for x1 be Element of X1, y1 be Element of X2 st y1 in ( dom ( ProjPMap1 (f,x1))) holds (( ProjPMap1 (F,x1)) # y1) is convergent & ( lim (( ProjPMap1 (F,x1)) # y1)) = (( ProjPMap1 (f,x1)) . y1)

      proof

        let x1 be Element of X1, y1 be Element of X2;

        reconsider z1 = [x1, y1] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume y1 in ( dom ( ProjPMap1 (f,x1)));

        then y1 in ( X-section (A,x1)) by A3, Def3;

        then

         A17: z1 in ( dom f) by A3, Th25;

        then

         A18: (F # z1) is convergent by A7;

        

         A19: for n be Element of NAT holds ((F # z1) . n) = ((( ProjPMap1 (F,x1)) # y1) . n)

        proof

          let n be Element of NAT ;

          

           A20: [x1, y1] in ( dom (F . n)) by A4, A17;

          ((F # z1) . n) = ((F . n) . (x1,y1)) by MESFUNC5:def 13;

          then ((F # z1) . n) = (( ProjPMap1 ((F . n),x1)) . y1) by A20, Def3;

          then ((F # z1) . n) = ((( ProjPMap1 (F,x1)) . n) . y1) by Def5;

          hence ((F # z1) . n) = ((( ProjPMap1 (F,x1)) # y1) . n) by MESFUNC5:def 13;

        end;

        hence (( ProjPMap1 (F,x1)) # y1) is convergent by A18, FUNCT_2:def 8;

        (F # z1) = (( ProjPMap1 (F,x1)) # y1) by A19, FUNCT_2:def 8;

        then ( lim (( ProjPMap1 (F,x1)) # y1)) = (f . (x1,y1)) by A7, A17;

        hence ( lim (( ProjPMap1 (F,x1)) # y1)) = (( ProjPMap1 (f,x1)) . y1) by A17, Def3;

      end;

      

       A21: for x be Element of X1 holds ( lim ( ProjPMap1 (F,x))) = ( ProjPMap1 (f,x)) & (FI2 # x) is convergent & ( lim (FI2 # x)) = ( Integral (M2,( lim ( ProjPMap1 (F,x)))))

      proof

        let x be Element of X1;

        ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom (( ProjPMap1 (F,x)) . 0 )) by MESFUNC8:def 9;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom ( ProjPMap1 ((F . 0 ),x))) by Def5;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( X-section (( dom (F . 0 )),x)) by Def3;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( X-section (( dom f),x)) by A4;

        then

         A22: ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom ( ProjPMap1 (f,x))) by Def3;

        for y be Element of X2 st y in ( dom ( lim ( ProjPMap1 (F,x)))) holds (( lim ( ProjPMap1 (F,x))) . y) = (( ProjPMap1 (f,x)) . y)

        proof

          let y be Element of X2;

          assume

           A23: y in ( dom ( lim ( ProjPMap1 (F,x))));

          then (( lim ( ProjPMap1 (F,x))) . y) = ( lim (( ProjPMap1 (F,x)) # y)) by MESFUNC8:def 9;

          hence (( lim ( ProjPMap1 (F,x))) . y) = (( ProjPMap1 (f,x)) . y) by A16, A22, A23;

        end;

        hence ( lim ( ProjPMap1 (F,x))) = ( ProjPMap1 (f,x)) by A22, PARTFUN1: 5;

        

         A24: (( ProjPMap1 (F,x)) . 0 ) = ( ProjPMap1 ((F . 0 ),x)) by Def5;

        then ( dom (( ProjPMap1 (F,x)) . 0 )) = ( X-section (( dom (F . 0 )),x)) by Def3;

        then ( dom (( ProjPMap1 (F,x)) . 0 )) = ( X-section (A,x)) by A4, A3;

        then

         A25: ( dom (( ProjPMap1 (F,x)) . 0 )) = ( Measurable-X-section (A,x)) by MEASUR11:def 6;

        (F . 0 ) is nonnegative by A5;

        then

         A26: (( ProjPMap1 (F,x)) . 0 ) is nonnegative by A24, Th32;

        

         A27: for n be Nat holds (( ProjPMap1 (F,x)) . n) is ( Measurable-X-section (A,x)) -measurable

        proof

          let n be Nat;

          

           A28: ( dom (F . n)) = A by A3, A4;

          (F . n) is A -measurable by A4, MESFUNC2: 34;

          then ( ProjPMap1 ((F . n),x)) is ( Measurable-X-section (A,x)) -measurable by A28, Th47;

          hence (( ProjPMap1 (F,x)) . n) is ( Measurable-X-section (A,x)) -measurable by Def5;

        end;

        

         A29: for n,m be Nat st n <= m holds for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds ((( ProjPMap1 (F,x)) . n) . y) <= ((( ProjPMap1 (F,x)) . m) . y)

        proof

          let n,m be Nat;

          assume

           A30: n <= m;

          let y be Element of X2;

          assume

           A31: y in ( Measurable-X-section (A,x));

          then y in ( dom ( ProjPMap1 ((F . 0 ),x))) by A25, Def5;

          then y in ( X-section (( dom (F . 0 )),x)) by Def3;

          then y in ( X-section (( dom f),x)) by A4;

          then

           A32: [x, y] in ( dom f) by Th25;

          

           A33: ( dom (( ProjPMap1 (F,x)) . n)) = ( dom (( ProjPMap1 (F,x)) . 0 )) & ( dom (( ProjPMap1 (F,x)) . m)) = ( dom (( ProjPMap1 (F,x)) . 0 )) by A8, Th58, MESFUNC8:def 2;

          (( ProjPMap1 (F,x)) . n) = ( ProjPMap1 ((F . n),x)) & (( ProjPMap1 (F,x)) . m) = ( ProjPMap1 ((F . m),x)) by Def5;

          then ((( ProjPMap1 (F,x)) . n) . y) = ((F . n) . (x,y)) & ((( ProjPMap1 (F,x)) . m) . y) = ((F . m) . (x,y)) by A25, A31, A33, Th26;

          hence ((( ProjPMap1 (F,x)) . n) . y) <= ((( ProjPMap1 (F,x)) . m) . y) by A6, A30, A32;

        end;

        for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (( ProjPMap1 (F,x)) # y) is convergent

        proof

          let y be Element of X2;

          assume y in ( Measurable-X-section (A,x));

          then y in ( X-section (( dom f),x)) by A3, MEASUR11:def 6;

          then y in ( dom ( ProjPMap1 (f,x))) by Def3;

          hence (( ProjPMap1 (F,x)) # y) is convergent by A16;

        end;

        then

        consider J be ExtREAL_sequence such that

         A34: (for n be Nat holds (J . n) = ( Integral (M2,(( ProjPMap1 (F,x)) . n)))) and

         A35: J is convergent and

         A36: ( Integral (M2,( lim ( ProjPMap1 (F,x))))) = ( lim J) by A8, A25, A26, A27, A29, Th58, MESFUNC9: 52;

        for n be Element of NAT holds (J . n) = ((FI2 # x) . n)

        proof

          let n be Element of NAT ;

          

           A37: ( dom (FI2 . n)) = X1 by A14;

          ((FI2 # x) . n) = ((FI2 . n) . x) by MESFUNC5:def 13;

          then ((FI2 # x) . n) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by A15, A37;

          then ((FI2 # x) . n) = ( Integral (M2,(( ProjPMap1 (F,x)) . n))) by Def5;

          hence (J . n) = ((FI2 # x) . n) by A34;

        end;

        hence (FI2 # x) is convergent & ( lim (FI2 # x)) = ( Integral (M2,( lim ( ProjPMap1 (F,x))))) by A35, A36, FUNCT_2: 63;

      end;

      ( dom ( lim FI2)) = ( dom (FI2 . 0 )) by MESFUNC8:def 9;

      then

       A38: ( dom ( lim FI2)) = X1 by A14;

      then

      reconsider I2 = ( lim FI2) as Function of X1, ExtREAL by FUNCT_2:def 1;

      take I2;

      for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))

      proof

        let x be Element of X1;

        (I2 . x) = ( lim (FI2 # x)) by A38, MESFUNC8:def 9;

        then (I2 . x) = ( Integral (M2,( lim ( ProjPMap1 (F,x))))) by A21;

        hence (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A21;

      end;

      hence for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))));

      thus for V be Element of S1 holds I2 is V -measurable

      proof

        let V be Element of S1;

        now

          let m,n be Nat;

          ( dom (FI2 . m)) = X1 & ( dom (FI2 . n)) = X1 by A14;

          hence ( dom (FI2 . m)) = ( dom (FI2 . n));

        end;

        then

         A39: FI2 is with_the_same_dom by MESFUNC8:def 2;

        

         A40: ( dom (FI2 . 0 )) = XX1 by A14;

        

         A41: for n be Nat holds (FI2 . n) is XX1 -measurable

        proof

          let n be Nat;

          ( dom (F . n)) = A & (F . n) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by A4, A3;

          then

          consider L be Function of X1, ExtREAL such that

           A42: (for x be Element of X1 holds (L . x) = ( Integral (M2,( ProjPMap1 ((F . n),x))))) & (for W be Element of S1 holds L is W -measurable) by A1, A5, Lm6;

          

           A43: ( dom (FI2 . n)) = X1 by A14;

          then

           A44: (FI2 . n) is Function of X1, ExtREAL by FUNCT_2:def 1;

          for x be Element of X1 holds ((FI2 . n) . x) = (L . x)

          proof

            let x be Element of X1;

            ((FI2 . n) . x) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by A15, A43;

            hence ((FI2 . n) . x) = (L . x) by A42;

          end;

          then (FI2 . n) = L by A44, FUNCT_2: 63;

          hence (FI2 . n) is XX1 -measurable by A42;

        end;

        for x be Element of X1 st x in XX1 holds (FI2 # x) is convergent by A21;

        hence I2 is V -measurable by A39, A40, A41, MESFUNC8: 25, MESFUNC1: 30;

      end;

    end;

    

     Lm10: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M2 is sigma_finite & f is nonpositive & A = ( dom f) & f is A -measurable holds ex I2 be Function of X1, ExtREAL st (for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))) & (for V be Element of S1 holds I2 is V -measurable)

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M2 is sigma_finite and

       A3: f is nonpositive & A = ( dom f) & f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider M = ( product_sigma_Measure (M1,M2)) as sigma_Measure of S by MEASUR11: 8;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL such that

       A4: for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f) and

       A5: for n be Nat holds (F . n) is nonpositive and

       A6: for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) >= ((F . m) . z) and

       A7: for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z) by A3, MESFUN11: 56;

      now

        let m,n be Nat;

        ( dom (F . m)) = ( dom f) by A4;

        hence ( dom (F . m)) = ( dom (F . n)) by A4;

      end;

      then

       A8: F is with_the_same_dom by MESFUNC8:def 2;

      defpred P[ Nat, object] means ex Fx be Function of X1, ExtREAL st $2 = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . $1),x1)))));

      

       A10: for n be Element of NAT holds ex FI2 be Element of ( PFuncs (X1, ExtREAL )) st P[n, FI2]

      proof

        let n be Element of NAT ;

        deffunc F( Element of X1) = ( Integral (M2,( ProjPMap1 ((F . n),$1))));

        consider FI2 be Function such that

         A11: ( dom FI2) = X1 & for x1 be Element of X1 holds (FI2 . x1) = F(x1) from FUNCT_1:sch 4;

        

         A12: for x2 be object st x2 in X1 holds (FI2 . x2) in ExtREAL

        proof

          let x2 be object;

          assume x2 in X1;

          then

          reconsider x1 = x2 as Element of X1;

          (FI2 . x2) = ( Integral (M2,( ProjPMap1 ((F . n),x1)))) by A11;

          hence (FI2 . x2) in ExtREAL ;

        end;

        then FI2 is Function of X1, ExtREAL by A11, FUNCT_2: 3;

        then

        reconsider FI2 as Element of ( PFuncs (X1, ExtREAL )) by PARTFUN1: 45;

        take FI2;

        reconsider Fx = FI2 as Function of X1, ExtREAL by A12, A11, FUNCT_2: 3;

        for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1)))) by A11;

        hence ex Fx be Function of X1, ExtREAL st FI2 = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))) by A11;

      end;

      consider FI2 be Function of NAT , ( PFuncs (X1, ExtREAL )) such that

       A13: for n be Element of NAT holds P[n, (FI2 . n)] from FUNCT_2:sch 3( A10);

      

       A14: for n be Nat holds ( dom (FI2 . n)) = X1

      proof

        let n be Nat;

        n is Element of NAT by ORDINAL1:def 12;

        then ex Fx be Function of X1, ExtREAL st (FI2 . n) = Fx & ( dom Fx) = X1 & (for x1 be Element of X1 st x1 in ( dom Fx) holds (Fx . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))) by A13;

        hence ( dom (FI2 . n)) = X1;

      end;

      

       A15: for n be Nat, x1 be Element of X1 st x1 in ( dom (FI2 . n)) holds ((FI2 . n) . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))))

      proof

        let n be Nat, x1 be Element of X1;

        assume x1 in ( dom (FI2 . n));

        n is Element of NAT by ORDINAL1:def 12;

        then P[n, (FI2 . n)] by A13;

        hence ((FI2 . n) . x1) = ( Integral (M2,( ProjPMap1 ((F . n),x1))));

      end;

      

       A16: for x1 be Element of X1, y1 be Element of X2 st y1 in ( dom ( ProjPMap1 (f,x1))) holds (( ProjPMap1 (F,x1)) # y1) is convergent & ( lim (( ProjPMap1 (F,x1)) # y1)) = (( ProjPMap1 (f,x1)) . y1)

      proof

        let x1 be Element of X1, y1 be Element of X2;

        reconsider z1 = [x1, y1] as Element of [:X1, X2:] by ZFMISC_1:def 2;

        assume y1 in ( dom ( ProjPMap1 (f,x1)));

        then y1 in ( X-section (A,x1)) by A3, Def3;

        then

         A17: z1 in ( dom f) by A3, Th25;

        then

         A18: (F # z1) is convergent by A7;

        

         A19: for n be Element of NAT holds ((F # z1) . n) = ((( ProjPMap1 (F,x1)) # y1) . n)

        proof

          let n be Element of NAT ;

          

           A20: [x1, y1] in ( dom (F . n)) by A4, A17;

          ((F # z1) . n) = ((F . n) . (x1,y1)) by MESFUNC5:def 13;

          then ((F # z1) . n) = (( ProjPMap1 ((F . n),x1)) . y1) by A20, Def3;

          then ((F # z1) . n) = ((( ProjPMap1 (F,x1)) . n) . y1) by Def5;

          hence ((F # z1) . n) = ((( ProjPMap1 (F,x1)) # y1) . n) by MESFUNC5:def 13;

        end;

        hence (( ProjPMap1 (F,x1)) # y1) is convergent by A18, FUNCT_2:def 8;

        (F # z1) = (( ProjPMap1 (F,x1)) # y1) by A19, FUNCT_2:def 8;

        then ( lim (( ProjPMap1 (F,x1)) # y1)) = (f . (x1,y1)) by A7, A17;

        hence ( lim (( ProjPMap1 (F,x1)) # y1)) = (( ProjPMap1 (f,x1)) . y1) by A17, Def3;

      end;

      

       A21: for x be Element of X1 holds ( lim ( ProjPMap1 (F,x))) = ( ProjPMap1 (f,x)) & (FI2 # x) is convergent & ( lim (FI2 # x)) = ( Integral (M2,( lim ( ProjPMap1 (F,x)))))

      proof

        let x be Element of X1;

        ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom (( ProjPMap1 (F,x)) . 0 )) by MESFUNC8:def 9;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom ( ProjPMap1 ((F . 0 ),x))) by Def5;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( X-section (( dom (F . 0 )),x)) by Def3;

        then ( dom ( lim ( ProjPMap1 (F,x)))) = ( X-section (( dom f),x)) by A4;

        then

         A22: ( dom ( lim ( ProjPMap1 (F,x)))) = ( dom ( ProjPMap1 (f,x))) by Def3;

        for y be Element of X2 st y in ( dom ( lim ( ProjPMap1 (F,x)))) holds (( lim ( ProjPMap1 (F,x))) . y) = (( ProjPMap1 (f,x)) . y)

        proof

          let y be Element of X2;

          assume

           A23: y in ( dom ( lim ( ProjPMap1 (F,x))));

          then (( lim ( ProjPMap1 (F,x))) . y) = ( lim (( ProjPMap1 (F,x)) # y)) by MESFUNC8:def 9;

          hence (( lim ( ProjPMap1 (F,x))) . y) = (( ProjPMap1 (f,x)) . y) by A16, A22, A23;

        end;

        hence ( lim ( ProjPMap1 (F,x))) = ( ProjPMap1 (f,x)) by A22, PARTFUN1: 5;

        

         A24: (( ProjPMap1 (F,x)) . 0 ) = ( ProjPMap1 ((F . 0 ),x)) by Def5;

        then ( dom (( ProjPMap1 (F,x)) . 0 )) = ( X-section (( dom (F . 0 )),x)) by Def3;

        then ( dom (( ProjPMap1 (F,x)) . 0 )) = ( X-section (A,x)) by A4, A3;

        then

         A25: ( dom (( ProjPMap1 (F,x)) . 0 )) = ( Measurable-X-section (A,x)) by MEASUR11:def 6;

        (F . 0 ) is nonpositive by A5;

        then

         A26: (( ProjPMap1 (F,x)) . 0 ) is nonpositive by A24, Th33;

        

         A27: for n be Nat holds (( ProjPMap1 (F,x)) . n) is ( Measurable-X-section (A,x)) -measurable

        proof

          let n be Nat;

          

           A28: ( dom (F . n)) = A by A3, A4;

          (F . n) is A -measurable by A4, MESFUNC2: 34;

          then ( ProjPMap1 ((F . n),x)) is ( Measurable-X-section (A,x)) -measurable by A28, Th47;

          hence (( ProjPMap1 (F,x)) . n) is ( Measurable-X-section (A,x)) -measurable by Def5;

        end;

        

         A29: for n,m be Nat st n <= m holds for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds ((( ProjPMap1 (F,x)) . n) . y) >= ((( ProjPMap1 (F,x)) . m) . y)

        proof

          let n,m be Nat;

          assume

           A30: n <= m;

          let y be Element of X2;

          assume

           A31: y in ( Measurable-X-section (A,x));

          then y in ( dom ( ProjPMap1 ((F . 0 ),x))) by A25, Def5;

          then y in ( X-section (( dom (F . 0 )),x)) by Def3;

          then y in ( X-section (( dom f),x)) by A4;

          then

           A32: [x, y] in ( dom f) by Th25;

          

           A33: ( dom (( ProjPMap1 (F,x)) . n)) = ( dom (( ProjPMap1 (F,x)) . 0 )) & ( dom (( ProjPMap1 (F,x)) . m)) = ( dom (( ProjPMap1 (F,x)) . 0 )) by A8, Th58, MESFUNC8:def 2;

          (( ProjPMap1 (F,x)) . n) = ( ProjPMap1 ((F . n),x)) & (( ProjPMap1 (F,x)) . m) = ( ProjPMap1 ((F . m),x)) by Def5;

          then ((( ProjPMap1 (F,x)) . n) . y) = ((F . n) . (x,y)) & ((( ProjPMap1 (F,x)) . m) . y) = ((F . m) . (x,y)) by A25, A31, A33, Th26;

          hence ((( ProjPMap1 (F,x)) . n) . y) >= ((( ProjPMap1 (F,x)) . m) . y) by A6, A30, A32;

        end;

        for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (( ProjPMap1 (F,x)) # y) is convergent

        proof

          let y be Element of X2;

          assume y in ( Measurable-X-section (A,x));

          then y in ( X-section (( dom f),x)) by A3, MEASUR11:def 6;

          then y in ( dom ( ProjPMap1 (f,x))) by Def3;

          hence (( ProjPMap1 (F,x)) # y) is convergent by A16;

        end;

        then

        consider J be ExtREAL_sequence such that

         A34: (for n be Nat holds (J . n) = ( Integral (M2,(( ProjPMap1 (F,x)) . n)))) and

         A35: J is convergent and

         A36: ( Integral (M2,( lim ( ProjPMap1 (F,x))))) = ( lim J) by A8, A25, A26, A27, A29, Th58, MESFUN11: 74;

        for n be Element of NAT holds (J . n) = ((FI2 # x) . n)

        proof

          let n be Element of NAT ;

          

           A37: ( dom (FI2 . n)) = X1 by A14;

          ((FI2 # x) . n) = ((FI2 . n) . x) by MESFUNC5:def 13;

          then ((FI2 # x) . n) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by A15, A37;

          then ((FI2 # x) . n) = ( Integral (M2,(( ProjPMap1 (F,x)) . n))) by Def5;

          hence (J . n) = ((FI2 # x) . n) by A34;

        end;

        hence (FI2 # x) is convergent & ( lim (FI2 # x)) = ( Integral (M2,( lim ( ProjPMap1 (F,x))))) by A35, A36, FUNCT_2: 63;

      end;

      ( dom ( lim FI2)) = ( dom (FI2 . 0 )) by MESFUNC8:def 9;

      then

       A38: ( dom ( lim FI2)) = X1 by A14;

      then

      reconsider I2 = ( lim FI2) as Function of X1, ExtREAL by FUNCT_2:def 1;

      take I2;

      for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))))

      proof

        let x be Element of X1;

        (I2 . x) = ( lim (FI2 # x)) by A38, MESFUNC8:def 9;

        then (I2 . x) = ( Integral (M2,( lim ( ProjPMap1 (F,x))))) by A21;

        hence (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A21;

      end;

      hence for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))));

      thus for V be Element of S1 holds I2 is V -measurable

      proof

        let V be Element of S1;

        now

          let m,n be Nat;

          ( dom (FI2 . m)) = X1 & ( dom (FI2 . n)) = X1 by A14;

          hence ( dom (FI2 . m)) = ( dom (FI2 . n));

        end;

        then

         A39: FI2 is with_the_same_dom by MESFUNC8:def 2;

        

         A40: ( dom (FI2 . 0 )) = XX1 by A14;

        

         A41: for n be Nat holds (FI2 . n) is XX1 -measurable

        proof

          let n be Nat;

          ( dom (F . n)) = A & (F . n) is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by A4, A3;

          then

          consider L be Function of X1, ExtREAL such that

           A42: (for x be Element of X1 holds (L . x) = ( Integral (M2,( ProjPMap1 ((F . n),x))))) & (for W be Element of S1 holds L is W -measurable) by A1, A5, Lm6;

          

           A43: ( dom (FI2 . n)) = X1 by A14;

          then

           A44: (FI2 . n) is Function of X1, ExtREAL by FUNCT_2:def 1;

          for x be Element of X1 holds ((FI2 . n) . x) = (L . x)

          proof

            let x be Element of X1;

            ((FI2 . n) . x) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by A15, A43;

            hence ((FI2 . n) . x) = (L . x) by A42;

          end;

          then (FI2 . n) = L by A44, FUNCT_2: 63;

          hence (FI2 . n) is XX1 -measurable by A42;

        end;

        for x be Element of X1 st x in XX1 holds (FI2 # x) is convergent by A21;

        hence I2 is V -measurable by A39, A40, A41, MESFUNC8: 25, MESFUNC1: 30;

      end;

    end;

    definition

      let X1,X2 be non empty set, S1 be SigmaField of X1, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL ;

      :: MESFUN12:def7

      func Integral1 (M1,f) -> Function of X2, ExtREAL means

      : Def7: for y be Element of X2 holds (it . y) = ( Integral (M1,( ProjPMap2 (f,y))));

      existence

      proof

        deffunc F( Element of X2) = ( Integral (M1,( ProjPMap2 (f,$1))));

        ex IT be Function of X2, ExtREAL st for y be Element of X2 holds (IT . y) = F(y) from FUNCT_2:sch 4;

        hence thesis;

      end;

      uniqueness

      proof

        let I1,I2 be Function of X2, ExtREAL ;

        assume that

         A1: for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) and

         A2: for y be Element of X2 holds (I2 . y) = ( Integral (M1,( ProjPMap2 (f,y))));

        now

          let y be Element of X2;

          (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A1;

          hence (I1 . y) = (I2 . y) by A2;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    definition

      let X1,X2 be non empty set, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL ;

      :: MESFUN12:def8

      func Integral2 (M2,f) -> Function of X1, ExtREAL means

      : Def8: for x be Element of X1 holds (it . x) = ( Integral (M2,( ProjPMap1 (f,x))));

      existence

      proof

        deffunc F( Element of X1) = ( Integral (M2,( ProjPMap1 (f,$1))));

        ex IT be Function of X1, ExtREAL st for x be Element of X1 holds (IT . x) = F(x) from FUNCT_2:sch 4;

        hence thesis;

      end;

      uniqueness

      proof

        let I1,I2 be Function of X1, ExtREAL ;

        assume that

         A1: for x be Element of X1 holds (I1 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) and

         A2: for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x))));

        now

          let x be Element of X1;

          (I1 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A1;

          hence (I1 . x) = (I2 . x) by A2;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    theorem :: MESFUN12:59

    

     Th59: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , E be Element of ( sigma ( measurable_rectangles (S1,S2))), V be Element of S2 st M1 is sigma_finite & (f is nonnegative or f is nonpositive) & E = ( dom f) & f is E -measurable holds ( Integral1 (M1,f)) is V -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))), V be Element of S2;

      assume that

       A1: M1 is sigma_finite and

       A3: f is nonnegative or f is nonpositive and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      consider I1 be Function of X2, ExtREAL such that

       A6: for y be Element of X2 holds (I1 . y) = ( Integral (M1,( ProjPMap2 (f,y)))) and

       A7: for W be Element of S2 holds I1 is W -measurable by A1, A3, A4, A5, Lm7, Lm8;

      I1 = ( Integral1 (M1,f)) by A6, Def7;

      hence ( Integral1 (M1,f)) is V -measurable by A7;

    end;

    theorem :: MESFUN12:60

    

     Th60: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E be Element of ( sigma ( measurable_rectangles (S1,S2))), U be Element of S1 st M2 is sigma_finite & (f is nonnegative or f is nonpositive) & E = ( dom f) & f is E -measurable holds ( Integral2 (M2,f)) is U -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))), U be Element of S1;

      assume that

       A1: M2 is sigma_finite and

       A3: f is nonnegative or f is nonpositive and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      consider I2 be Function of X1, ExtREAL such that

       A6: for x be Element of X1 holds (I2 . x) = ( Integral (M2,( ProjPMap1 (f,x)))) and

       A7: for W be Element of S1 holds I2 is W -measurable by A1, A3, A4, A5, Lm9, Lm10;

      I2 = ( Integral2 (M2,f)) by A6, Def8;

      hence ( Integral2 (M2,f)) is U -measurable by A7;

    end;

    theorem :: MESFUN12:61

    

     Th61: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite holds (( X-vol (E,M1)) . y) = ( Integral (M1,( chi (( Measurable-Y-section (E,y)),X1))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, y be Element of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume M1 is sigma_finite;

      then (( X-vol (A,M1)) . y) = (M1 . ( Measurable-Y-section (A,y))) by MEASUR11:def 14;

      hence (( X-vol (A,M1)) . y) = ( Integral (M1,( chi (( Measurable-Y-section (A,y)),X1)))) by MESFUNC9: 14;

    end;

    theorem :: MESFUN12:62

    

     Th62: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M2 is sigma_finite holds (( Y-vol (E,M2)) . x) = ( Integral (M2,( chi (( Measurable-X-section (E,x)),X2))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, x be Element of X1, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume M2 is sigma_finite;

      then (( Y-vol (A,M2)) . x) = (M2 . ( Measurable-X-section (A,x))) by MEASUR11:def 13;

      hence (( Y-vol (A,M2)) . x) = ( Integral (M2,( chi (( Measurable-X-section (A,x)),X2)))) by MESFUNC9: 14;

    end;

    theorem :: MESFUN12:63

    

     Th63: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), x be Element of X1, y be Element of X2 holds ( ProjPMap1 (( chi (E, [:X1, X2:])),x)) = ( chi (( Measurable-X-section (E,x)),X2)) & ( ProjPMap2 (( chi (E, [:X1, X2:])),y)) = ( chi (( Measurable-Y-section (E,y)),X1))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), x be Element of X1, y be Element of X2;

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      

       A1: x in XX1 implies ( X-section ( [:XX1, XX2:],x)) = XX2 by MEASUR11: 22;

      ( dom ( ProjPMap1 (( chi (A, [:X1, X2:])),x))) = ( X-section (( dom ( chi (A, [:X1, X2:]))),x)) by Def3

      .= ( X-section (XX12,x)) by FUNCT_3:def 3;

      then

       A2: ( dom ( ProjPMap1 (( chi (A, [:X1, X2:])),x))) = ( dom ( chi (( Measurable-X-section (A,x)),X2))) by A1, FUNCT_3:def 3;

      now

        let y be Element of X2;

        assume y in ( dom ( ProjPMap1 (( chi (A, [:X1, X2:])),x)));

        

         A3: [x, y] in [:X1, X2:] by ZFMISC_1:def 2;

        then [x, y] in ( dom ( chi (A, [:X1, X2:]))) by FUNCT_3:def 3;

        then

         A4: (( ProjPMap1 (( chi (A, [:X1, X2:])),x)) . y) = (( chi (A, [:X1, X2:])) . (x,y)) by Def3;

        

         A5: ( Measurable-X-section (A,x)) = ( X-section (A,x)) by MEASUR11:def 6

        .= { y where y be Element of X2 : [x, y] in A } by MEASUR11:def 4;

        per cases ;

          suppose

           A6: [x, y] in A;

          then y in ( Measurable-X-section (A,x)) by A5;

          then (( chi (( Measurable-X-section (A,x)),X2)) . y) = 1 by FUNCT_3:def 3;

          hence (( ProjPMap1 (( chi (A, [:X1, X2:])),x)) . y) = (( chi (( Measurable-X-section (A,x)),X2)) . y) by A4, A6, FUNCT_3:def 3;

        end;

          suppose

           A7: not [x, y] in A;

          now

            assume y in ( Measurable-X-section (A,x));

            then ex y1 be Element of X2 st y1 = y & [x, y1] in A by A5;

            hence contradiction by A7;

          end;

          then (( chi (( Measurable-X-section (A,x)),X2)) . y) = 0 by FUNCT_3:def 3;

          hence (( ProjPMap1 (( chi (A, [:X1, X2:])),x)) . y) = (( chi (( Measurable-X-section (A,x)),X2)) . y) by A3, A4, A7, FUNCT_3:def 3;

        end;

      end;

      hence ( ProjPMap1 (( chi (A, [:X1, X2:])),x)) = ( chi (( Measurable-X-section (A,x)),X2)) by A2, PARTFUN1: 5;

      

       A8: y in XX2 implies ( Y-section ( [:XX1, XX2:],y)) = XX1 by MEASUR11: 22;

      ( dom ( ProjPMap2 (( chi (A, [:X1, X2:])),y))) = ( Y-section (( dom ( chi (A, [:X1, X2:]))),y)) by Def4

      .= ( Y-section (XX12,y)) by FUNCT_3:def 3;

      then

       A9: ( dom ( ProjPMap2 (( chi (A, [:X1, X2:])),y))) = ( dom ( chi (( Measurable-Y-section (A,y)),X1))) by A8, FUNCT_3:def 3;

      now

        let x be Element of X1;

        assume x in ( dom ( ProjPMap2 (( chi (A, [:X1, X2:])),y)));

        

         A10: [x, y] in [:X1, X2:] by ZFMISC_1:def 2;

        then [x, y] in ( dom ( chi (A, [:X1, X2:]))) by FUNCT_3:def 3;

        then

         A11: (( ProjPMap2 (( chi (A, [:X1, X2:])),y)) . x) = (( chi (A, [:X1, X2:])) . (x,y)) by Def4;

        

         A12: ( Measurable-Y-section (A,y)) = ( Y-section (A,y)) by MEASUR11:def 7

        .= { x where x be Element of X1 : [x, y] in A } by MEASUR11:def 5;

        per cases ;

          suppose

           A13: [x, y] in A;

          then x in ( Measurable-Y-section (A,y)) by A12;

          then (( chi (( Measurable-Y-section (A,y)),X1)) . x) = 1 by FUNCT_3:def 3;

          hence (( ProjPMap2 (( chi (A, [:X1, X2:])),y)) . x) = (( chi (( Measurable-Y-section (A,y)),X1)) . x) by A11, A13, FUNCT_3:def 3;

        end;

          suppose

           A14: not [x, y] in A;

          now

            assume x in ( Measurable-Y-section (A,y));

            then ex x1 be Element of X1 st x1 = x & [x1, y] in A by A12;

            hence contradiction by A14;

          end;

          then (( chi (( Measurable-Y-section (A,y)),X1)) . x) = 0 by FUNCT_3:def 3;

          hence (( ProjPMap2 (( chi (A, [:X1, X2:])),y)) . x) = (( chi (( Measurable-Y-section (A,y)),X1)) . x) by A10, A11, A14, FUNCT_3:def 3;

        end;

      end;

      hence thesis by A9, PARTFUN1: 5;

    end;

    theorem :: MESFUN12:64

    

     Th64: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite holds ( X-vol (E,M1)) = ( Integral1 (M1,( chi (E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume

       A1: M1 is sigma_finite;

      now

        let y be Element of X2;

        

         A2: (( X-vol (A,M1)) . y) = ( Integral (M1,( chi (( Measurable-Y-section (A,y)),X1)))) by A1, Th61;

        ( ProjPMap2 (( chi (A, [:X1, X2:])),y)) = ( chi (( Measurable-Y-section (A,y)),X1)) by Th63;

        hence (( X-vol (A,M1)) . y) = (( Integral1 (M1,( chi (A, [:X1, X2:])))) . y) by A2, Def7;

      end;

      hence thesis by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:65

    

     Th65: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M2 is sigma_finite holds ( Y-vol (E,M2)) = ( Integral2 (M2,( chi (E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume

       a1: M2 is sigma_finite;

      now

        let x be Element of X1;

        

         A1: (( Y-vol (A,M2)) . x) = ( Integral (M2,( chi (( Measurable-X-section (A,x)),X2)))) by a1, Th62;

        ( ProjPMap1 (( chi (A, [:X1, X2:])),x)) = ( chi (( Measurable-X-section (A,x)),X2)) by Th63;

        hence (( Y-vol (A,M2)) . x) = (( Integral2 (M2,( chi (A, [:X1, X2:])))) . x) by A1, Def8;

      end;

      hence thesis by FUNCT_2:def 8;

    end;

    definition

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2;

      :: MESFUN12:def9

      func Prod_Measure (M1,M2) -> sigma_Measure of ( sigma ( measurable_rectangles (S1,S2))) equals ( product_sigma_Measure (M1,M2));

      correctness by MEASUR11: 8;

    end

    theorem :: MESFUN12:66

    

     Th66: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))) st E1 = ( dom f) & f is nonnegative & f is E1 -measurable holds ( Integral1 (M1,f)) is nonnegative & ( Integral1 (M1,(f | E2))) is nonnegative & ( Integral2 (M2,f)) is nonnegative & ( Integral2 (M2,(f | E2))) is nonnegative

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A,B be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative and

       A3: f is A -measurable;

      

       A4: (f | B) is nonnegative by A2, MESFUNC5: 15;

      

       A5: ( dom (f | B)) = (A /\ B) by A1, RELAT_1: 61;

      

       A6: f is (A /\ B) -measurable by A3, XBOOLE_1: 17, MESFUNC1: 30;

      

       A7: (( dom f) /\ (A /\ B)) = (A /\ B) by A1, XBOOLE_1: 17, XBOOLE_1: 28;

      (f | (A /\ B)) = ((f | A) /\ (f | B)) by RELAT_1: 79;

      then (f | (A /\ B)) = (f | B) by A1, RELAT_1: 59, XBOOLE_1: 28;

      then

       A8: (f | B) is (A /\ B) -measurable by A6, A7, MESFUNC5: 42;

      now

        let y be object;

        assume y in ( dom ( Integral1 (M1,f)));

        then

        reconsider y1 = y as Element of X2;

        

         A9: ( ProjPMap2 (f,y1)) is ( Measurable-Y-section (A,y1)) -measurable by A1, A3, Th47;

        ( dom ( ProjPMap2 (f,y1))) = ( Y-section (A,y1)) by A1, Def4;

        then

         A10: ( dom ( ProjPMap2 (f,y1))) = ( Measurable-Y-section (A,y1)) by MEASUR11:def 7;

        then ( integral+ (M1,( ProjPMap2 (f,y1)))) >= 0 by A2, A9, Th32, MESFUNC5: 79;

        then ( Integral (M1,( ProjPMap2 (f,y1)))) >= 0 by A2, A9, A10, Th32, MESFUNC5: 88;

        hence (( Integral1 (M1,f)) . y) >= 0 by Def7;

      end;

      hence ( Integral1 (M1,f)) is nonnegative by SUPINF_2: 52;

      now

        let y be object;

        assume y in ( dom ( Integral1 (M1,(f | B))));

        then

        reconsider y1 = y as Element of X2;

        

         A11: ( ProjPMap2 ((f | B),y1)) is ( Measurable-Y-section ((A /\ B),y1)) -measurable by A5, A8, Th47;

        ( dom ( ProjPMap2 ((f | B),y1))) = ( Y-section ((A /\ B),y1)) by A5, Def4;

        then

         A12: ( dom ( ProjPMap2 ((f | B),y1))) = ( Measurable-Y-section ((A /\ B),y1)) by MEASUR11:def 7;

        then ( integral+ (M1,( ProjPMap2 ((f | B),y1)))) >= 0 by A4, A11, Th32, MESFUNC5: 79;

        then ( Integral (M1,( ProjPMap2 ((f | B),y1)))) >= 0 by A4, A11, A12, Th32, MESFUNC5: 88;

        hence (( Integral1 (M1,(f | B))) . y) >= 0 by Def7;

      end;

      hence ( Integral1 (M1,(f | B))) is nonnegative by SUPINF_2: 52;

      now

        let x be object;

        assume x in ( dom ( Integral2 (M2,f)));

        then

        reconsider x1 = x as Element of X1;

        

         A13: ( ProjPMap1 (f,x1)) is ( Measurable-X-section (A,x1)) -measurable by A1, A3, Th47;

        ( dom ( ProjPMap1 (f,x1))) = ( X-section (A,x1)) by A1, Def3;

        then

         A14: ( dom ( ProjPMap1 (f,x1))) = ( Measurable-X-section (A,x1)) by MEASUR11:def 6;

        then ( integral+ (M2,( ProjPMap1 (f,x1)))) >= 0 by A2, A13, Th32, MESFUNC5: 79;

        then ( Integral (M2,( ProjPMap1 (f,x1)))) >= 0 by A2, A13, A14, Th32, MESFUNC5: 88;

        hence (( Integral2 (M2,f)) . x) >= 0 by Def8;

      end;

      hence ( Integral2 (M2,f)) is nonnegative by SUPINF_2: 52;

      now

        let x be object;

        assume x in ( dom ( Integral2 (M2,(f | B))));

        then

        reconsider x1 = x as Element of X1;

        

         A15: ( ProjPMap1 ((f | B),x1)) is ( Measurable-X-section ((A /\ B),x1)) -measurable by A5, A8, Th47;

        ( dom ( ProjPMap1 ((f | B),x1))) = ( X-section ((A /\ B),x1)) by A5, Def3;

        then

         A16: ( dom ( ProjPMap1 ((f | B),x1))) = ( Measurable-X-section ((A /\ B),x1)) by MEASUR11:def 6;

        then ( integral+ (M2,( ProjPMap1 ((f | B),x1)))) >= 0 by A4, A15, Th32, MESFUNC5: 79;

        then ( Integral (M2,( ProjPMap1 ((f | B),x1)))) >= 0 by A4, A15, A16, Th32, MESFUNC5: 88;

        hence (( Integral2 (M2,(f | B))) . x) >= 0 by Def8;

      end;

      hence ( Integral2 (M2,(f | B))) is nonnegative by SUPINF_2: 52;

    end;

    theorem :: MESFUN12:67

    

     Th67: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))) st E1 = ( dom f) & f is nonpositive & f is E1 -measurable holds ( Integral1 (M1,f)) is nonpositive & ( Integral1 (M1,(f | E2))) is nonpositive & ( Integral2 (M2,f)) is nonpositive & ( Integral2 (M2,(f | E2))) is nonpositive

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A,B be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is nonpositive and

       A3: f is A -measurable;

      

       A4: (f | B) is nonpositive by A2, MESFUN11: 1;

      

       A5: ( dom (f | B)) = (A /\ B) by A1, RELAT_1: 61;

      

       A6: f is (A /\ B) -measurable by A3, XBOOLE_1: 17, MESFUNC1: 30;

      

       A7: (( dom f) /\ (A /\ B)) = (A /\ B) by A1, XBOOLE_1: 17, XBOOLE_1: 28;

      (f | (A /\ B)) = ((f | A) /\ (f | B)) by RELAT_1: 79;

      then (f | (A /\ B)) = (f | B) by A1, RELAT_1: 59, XBOOLE_1: 28;

      then

       A8: (f | B) is (A /\ B) -measurable by A6, A7, MESFUNC5: 42;

      now

        let y be set;

        assume y in ( dom ( Integral1 (M1,f)));

        then

        reconsider y1 = y as Element of X2;

        

         A9: ( ProjPMap2 (f,y1)) is ( Measurable-Y-section (A,y1)) -measurable by A1, A3, Th47;

        ( dom ( ProjPMap2 (f,y1))) = ( Y-section (A,y1)) by A1, Def4;

        then ( dom ( ProjPMap2 (f,y1))) = ( Measurable-Y-section (A,y1)) by MEASUR11:def 7;

        then ( Integral (M1,( ProjPMap2 (f,y1)))) <= 0 by A2, A9, Th33, MESFUN11: 61;

        hence (( Integral1 (M1,f)) . y) <= 0 by Def7;

      end;

      hence ( Integral1 (M1,f)) is nonpositive by MESFUNC5: 9;

      now

        let y be set;

        assume y in ( dom ( Integral1 (M1,(f | B))));

        then

        reconsider y1 = y as Element of X2;

        

         A10: ( ProjPMap2 ((f | B),y1)) is ( Measurable-Y-section ((A /\ B),y1)) -measurable by A5, A8, Th47;

        ( dom ( ProjPMap2 ((f | B),y1))) = ( Y-section ((A /\ B),y1)) by A5, Def4;

        then ( dom ( ProjPMap2 ((f | B),y1))) = ( Measurable-Y-section ((A /\ B),y1)) by MEASUR11:def 7;

        then ( Integral (M1,( ProjPMap2 ((f | B),y1)))) <= 0 by A4, A10, Th33, MESFUN11: 61;

        hence (( Integral1 (M1,(f | B))) . y) <= 0 by Def7;

      end;

      hence ( Integral1 (M1,(f | B))) is nonpositive by MESFUNC5: 9;

      now

        let x be set;

        assume x in ( dom ( Integral2 (M2,f)));

        then

        reconsider x1 = x as Element of X1;

        

         A9: ( ProjPMap1 (f,x1)) is ( Measurable-X-section (A,x1)) -measurable by A1, A3, Th47;

        ( dom ( ProjPMap1 (f,x1))) = ( X-section (A,x1)) by A1, Def3;

        then ( dom ( ProjPMap1 (f,x1))) = ( Measurable-X-section (A,x1)) by MEASUR11:def 6;

        then ( Integral (M2,( ProjPMap1 (f,x1)))) <= 0 by A2, A9, Th33, MESFUN11: 61;

        hence (( Integral2 (M2,f)) . x) <= 0 by Def8;

      end;

      hence ( Integral2 (M2,f)) is nonpositive by MESFUNC5: 9;

      now

        let x be set;

        assume x in ( dom ( Integral2 (M2,(f | B))));

        then

        reconsider x1 = x as Element of X1;

        

         A10: ( ProjPMap1 ((f | B),x1)) is ( Measurable-X-section ((A /\ B),x1)) -measurable by A5, A8, Th47;

        ( dom ( ProjPMap1 ((f | B),x1))) = ( X-section ((A /\ B),x1)) by A5, Def3;

        then ( dom ( ProjPMap1 ((f | B),x1))) = ( Measurable-X-section ((A /\ B),x1)) by MEASUR11:def 6;

        then ( Integral (M2,( ProjPMap1 ((f | B),x1)))) <= 0 by A4, A10, Th33, MESFUN11: 61;

        hence (( Integral2 (M2,(f | B))) . x) <= 0 by Def8;

      end;

      hence ( Integral2 (M2,(f | B))) is nonpositive by MESFUNC5: 9;

    end;

    theorem :: MESFUN12:68

    

     Th68: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))), V be Element of S2 st M1 is sigma_finite & (f is nonnegative or f is nonpositive) & E1 = ( dom f) & f is E1 -measurable holds ( Integral1 (M1,(f | E2))) is V -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , E,A be Element of ( sigma ( measurable_rectangles (S1,S2))), V be Element of S2;

      assume that

       A1: M1 is sigma_finite and

       A2: f is nonnegative or f is nonpositive and

       A3: E = ( dom f) and

       A4: f is E -measurable;

      

       A5: ( dom (f | A)) = (E /\ A) by A3, RELAT_1: 61;

      

       A6: (( dom f) /\ (E /\ A)) = (E /\ A) by A3, XBOOLE_1: 17, XBOOLE_1: 28;

      f is (E /\ A) -measurable by A4, XBOOLE_1: 17, MESFUNC1: 30;

      then (f | (E /\ A)) is (E /\ A) -measurable by A6, MESFUNC5: 42;

      then ((f | E) | A) is (E /\ A) -measurable by RELAT_1: 71;

      hence ( Integral1 (M1,(f | A))) is V -measurable by A1, A2, A3, A5, MESFUNC5: 15, MESFUN11: 1, Th59;

    end;

    theorem :: MESFUN12:69

    

     Th69: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))), U be Element of S1 st M2 is sigma_finite & (f is nonnegative or f is nonpositive) & E1 = ( dom f) & f is E1 -measurable holds ( Integral2 (M2,(f | E2))) is U -measurable

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E,A be Element of ( sigma ( measurable_rectangles (S1,S2))), U be Element of S1;

      assume that

       A1: M2 is sigma_finite and

       A2: f is nonnegative or f is nonpositive and

       A3: E = ( dom f) and

       A4: f is E -measurable;

      

       A5: ( dom (f | A)) = (E /\ A) by A3, RELAT_1: 61;

      

       A6: (( dom f) /\ (E /\ A)) = (E /\ A) by A3, XBOOLE_1: 17, XBOOLE_1: 28;

      f is (E /\ A) -measurable by A4, XBOOLE_1: 17, MESFUNC1: 30;

      then (f | (E /\ A)) is (E /\ A) -measurable by A6, MESFUNC5: 42;

      then ((f | E) | A) is (E /\ A) -measurable by RELAT_1: 71;

      hence ( Integral2 (M2,(f | A))) is U -measurable by A1, A2, A3, A5, MESFUNC5: 15, MESFUN11: 1, Th60;

    end;

    theorem :: MESFUN12:70

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , E be Element of ( sigma ( measurable_rectangles (S1,S2))), y be Element of X2 st E = ( dom f) & (f is nonnegative or f is nonpositive) & f is E -measurable & (for x be Element of X1 st x in ( dom ( ProjPMap2 (f,y))) holds (( ProjPMap2 (f,y)) . x) = 0 ) holds (( Integral1 (M1,f)) . y) = 0

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))), y be Element of X2;

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative or f is nonpositive and

       A3: f is A -measurable and

       A4: for x be Element of X1 st x in ( dom ( ProjPMap2 (f,y))) holds (( ProjPMap2 (f,y)) . x) = 0 ;

      

       A5: ( dom ( ProjPMap2 (f,y))) = ( Y-section (A,y)) by A1, Def4

      .= ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

      

       A6: ( ProjPMap2 (f,y)) is ( Measurable-Y-section (A,y)) -measurable by A1, A3, Th47;

      per cases by A2;

        suppose

         A7: f is nonnegative;

        ( integral+ (M1,( ProjPMap2 (f,y)))) = 0 by A1, A3, A4, A5, Th47, MESFUNC5: 87;

        then ( Integral (M1,( ProjPMap2 (f,y)))) = 0 by A5, A6, A7, Th32, MESFUNC5: 88;

        hence (( Integral1 (M1,f)) . y) = 0 by Def7;

      end;

        suppose f is nonpositive;

        then

         A8: ( ProjPMap2 (f,y)) is nonpositive by Th33;

        

         A9: ( dom ( - ( ProjPMap2 (f,y)))) = ( Measurable-Y-section (A,y)) by A5, MESFUNC1:def 7;

        for x be Element of X1 st x in ( dom ( - ( ProjPMap2 (f,y)))) holds (( - ( ProjPMap2 (f,y))) . x) = 0

        proof

          let x be Element of X1;

          assume

           A10: x in ( dom ( - ( ProjPMap2 (f,y))));

          then (( - ( ProjPMap2 (f,y))) . x) = ( - (( ProjPMap2 (f,y)) . x)) by MESFUNC1:def 7;

          then (( - ( ProjPMap2 (f,y))) . x) = ( - 0 ) by A4, A5, A9, A10;

          hence (( - ( ProjPMap2 (f,y))) . x) = 0 ;

        end;

        then ( integral+ (M1,( - ( ProjPMap2 (f,y))))) = 0 by A5, A6, A9, MEASUR11: 63, MESFUNC5: 87;

        then ( - ( integral+ (M1,( - ( ProjPMap2 (f,y)))))) = 0 ;

        then ( Integral (M1,( ProjPMap2 (f,y)))) = 0 by A5, A6, A8, MESFUN11: 57;

        hence (( Integral1 (M1,f)) . y) = 0 by Def7;

      end;

    end;

    theorem :: MESFUN12:71

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E be Element of ( sigma ( measurable_rectangles (S1,S2))), x be Element of X1 st E = ( dom f) & (f is nonnegative or f is nonpositive) & f is E -measurable & (for y be Element of X2 st y in ( dom ( ProjPMap1 (f,x))) holds (( ProjPMap1 (f,x)) . y) = 0 ) holds (( Integral2 (M2,f)) . x) = 0

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))), x be Element of X1;

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative or f is nonpositive and

       A3: f is A -measurable and

       A4: for y be Element of X2 st y in ( dom ( ProjPMap1 (f,x))) holds (( ProjPMap1 (f,x)) . y) = 0 ;

      

       A5: ( dom ( ProjPMap1 (f,x))) = ( X-section (A,x)) by A1, Def3

      .= ( Measurable-X-section (A,x)) by MEASUR11:def 6;

      

       A6: ( ProjPMap1 (f,x)) is ( Measurable-X-section (A,x)) -measurable by A1, A3, Th47;

      per cases by A2;

        suppose

         A7: f is nonnegative;

        ( integral+ (M2,( ProjPMap1 (f,x)))) = 0 by A1, A3, A4, A5, Th47, MESFUNC5: 87;

        then ( Integral (M2,( ProjPMap1 (f,x)))) = 0 by A5, A6, A7, Th32, MESFUNC5: 88;

        hence (( Integral2 (M2,f)) . x) = 0 by Def8;

      end;

        suppose f is nonpositive;

        then

         A8: ( ProjPMap1 (f,x)) is nonpositive by Th33;

        

         A9: ( dom ( - ( ProjPMap1 (f,x)))) = ( Measurable-X-section (A,x)) by A5, MESFUNC1:def 7;

        for y be Element of X2 st y in ( dom ( - ( ProjPMap1 (f,x)))) holds (( - ( ProjPMap1 (f,x))) . y) = 0

        proof

          let y be Element of X2;

          assume

           A10: y in ( dom ( - ( ProjPMap1 (f,x))));

          then (( - ( ProjPMap1 (f,x))) . y) = ( - (( ProjPMap1 (f,x)) . y)) by MESFUNC1:def 7;

          then (( - ( ProjPMap1 (f,x))) . y) = ( - 0 ) by A4, A5, A9, A10;

          hence (( - ( ProjPMap1 (f,x))) . y) = 0 ;

        end;

        then ( integral+ (M2,( - ( ProjPMap1 (f,x))))) = 0 by A5, A6, A9, MEASUR11: 63, MESFUNC5: 87;

        then ( - ( integral+ (M2,( - ( ProjPMap1 (f,x)))))) = 0 ;

        then ( Integral (M2,( ProjPMap1 (f,x)))) = 0 by A5, A6, A8, MESFUN11: 57;

        hence (( Integral2 (M2,f)) . x) = 0 by Def8;

      end;

    end;

    

     Lm11: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E,A,B be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st E = ( dom f) & f is nonnegative & f is E -measurable & A misses B holds ( Integral1 (M1,(f | (A \/ B)))) = (( Integral1 (M1,(f | A))) + ( Integral1 (M1,(f | B)))) & ( Integral2 (M2,(f | (A \/ B)))) = (( Integral2 (M2,(f | A))) + ( Integral2 (M2,(f | B))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E,A,B be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: E = ( dom f) and

       A2: f is nonnegative and

       A3: f is E -measurable and

       A4: A misses B;

      ( Integral1 (M1,(f | A))) is nonnegative by A1, A2, A3, Th66;

      then

      reconsider IA = ( Integral1 (M1,(f | A))) as without-infty Function of X2, ExtREAL ;

      ( Integral1 (M1,(f | B))) is nonnegative by A1, A2, A3, Th66;

      then

      reconsider IB = ( Integral1 (M1,(f | B))) as without-infty Function of X2, ExtREAL ;

      now

        let y be Element of X2;

        

         A5: ( Y-section (A,y)) = ( Measurable-Y-section (A,y)) & ( Y-section (B,y)) = ( Measurable-Y-section (B,y)) & ( Y-section ((A \/ B),y)) = ( Measurable-Y-section ((A \/ B),y)) by MEASUR11:def 7;

        

         A6: ( dom ( ProjPMap2 (f,y))) = ( Y-section (E,y)) by A1, Def4

        .= ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

        

         A7: ( ProjPMap2 (f,y)) is ( Measurable-Y-section (E,y)) -measurable by A1, A3, Th47;

        (A /\ B) = ( {} [:X1, X2:]) by A4;

        then ( Y-section ((A /\ B),y)) = {} by MEASUR11: 24;

        then

         A8: ( Measurable-Y-section (A,y)) misses ( Measurable-Y-section (B,y)) by A5, MEASUR11: 27;

        ( ProjPMap2 ((f | A),y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y))) & ( ProjPMap2 ((f | B),y)) = (( ProjPMap2 (f,y)) | ( Y-section (B,y))) by Th34;

        then

         A9: ( ProjPMap2 ((f | A),y)) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section (A,y))) & ( ProjPMap2 ((f | B),y)) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section (B,y))) by MEASUR11:def 7;

        

         A10: (( Measurable-Y-section (A,y)) \/ ( Measurable-Y-section (B,y))) = ( Measurable-Y-section ((A \/ B),y)) by A5, MEASUR11: 26;

        ((IA + IB) . y) = ((( Integral1 (M1,(f | A))) . y) + (( Integral1 (M1,(f | B))) . y)) by DBLSEQ_3: 7

        .= (( Integral (M1,( ProjPMap2 ((f | A),y)))) + (( Integral1 (M1,(f | B))) . y)) by Def7

        .= (( Integral (M1,( ProjPMap2 ((f | A),y)))) + ( Integral (M1,( ProjPMap2 ((f | B),y))))) by Def7

        .= ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section ((A \/ B),y))))) by A2, A6, A7, A8, A9, A10, Th32, MESFUNC5: 91

        .= ( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section ((A \/ B),y))))) by MEASUR11:def 7

        .= ( Integral (M1,( ProjPMap2 ((f | (A \/ B)),y)))) by Th34;

        hence (( Integral1 (M1,(f | (A \/ B)))) . y) = ((IA + IB) . y) by Def7;

      end;

      hence ( Integral1 (M1,(f | (A \/ B)))) = (( Integral1 (M1,(f | A))) + ( Integral1 (M1,(f | B)))) by FUNCT_2:def 8;

      ( Integral2 (M2,(f | A))) is nonnegative by A1, A2, A3, Th66;

      then

      reconsider JA = ( Integral2 (M2,(f | A))) as without-infty Function of X1, ExtREAL ;

      ( Integral2 (M2,(f | B))) is nonnegative by A1, A2, A3, Th66;

      then

      reconsider JB = ( Integral2 (M2,(f | B))) as without-infty Function of X1, ExtREAL ;

      now

        let x be Element of X1;

        

         A11: ( X-section (A,x)) = ( Measurable-X-section (A,x)) & ( X-section (B,x)) = ( Measurable-X-section (B,x)) & ( X-section ((A \/ B),x)) = ( Measurable-X-section ((A \/ B),x)) by MEASUR11:def 6;

        

         A12: ( dom ( ProjPMap1 (f,x))) = ( X-section (E,x)) by A1, Def3

        .= ( Measurable-X-section (E,x)) by MEASUR11:def 6;

        

         A13: ( ProjPMap1 (f,x)) is ( Measurable-X-section (E,x)) -measurable by A1, A3, Th47;

        (A /\ B) = ( {} [:X1, X2:]) by A4;

        then ( X-section ((A /\ B),x)) = {} by MEASUR11: 24;

        then

         A14: ( Measurable-X-section (A,x)) misses ( Measurable-X-section (B,x)) by A11, MEASUR11: 27;

        ( ProjPMap1 ((f | A),x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) & ( ProjPMap1 ((f | B),x)) = (( ProjPMap1 (f,x)) | ( X-section (B,x))) by Th34;

        then

         A15: ( ProjPMap1 ((f | A),x)) = (( ProjPMap1 (f,x)) | ( Measurable-X-section (A,x))) & ( ProjPMap1 ((f | B),x)) = (( ProjPMap1 (f,x)) | ( Measurable-X-section (B,x))) by MEASUR11:def 6;

        

         A16: (( Measurable-X-section (A,x)) \/ ( Measurable-X-section (B,x))) = ( Measurable-X-section ((A \/ B),x)) by A11, MEASUR11: 26;

        ((JA + JB) . x) = ((( Integral2 (M2,(f | A))) . x) + (( Integral2 (M2,(f | B))) . x)) by DBLSEQ_3: 7

        .= (( Integral (M2,( ProjPMap1 ((f | A),x)))) + (( Integral2 (M2,(f | B))) . x)) by Def8

        .= (( Integral (M2,( ProjPMap1 ((f | A),x)))) + ( Integral (M2,( ProjPMap1 ((f | B),x))))) by Def8

        .= ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section ((A \/ B),x))))) by A2, A12, A13, A14, A15, A16, Th32, MESFUNC5: 91

        .= ( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section ((A \/ B),x))))) by MEASUR11:def 6

        .= ( Integral (M2,( ProjPMap1 ((f | (A \/ B)),x)))) by Th34;

        hence (( Integral2 (M2,(f | (A \/ B)))) . x) = ((JA + JB) . x) by Def8;

      end;

      hence thesis by FUNCT_2:def 8;

    end;

    

     Lm12: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E,A,B be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st E = ( dom f) & f is nonpositive & f is E -measurable & A misses B holds ( Integral1 (M1,(f | (A \/ B)))) = (( Integral1 (M1,(f | A))) + ( Integral1 (M1,(f | B)))) & ( Integral2 (M2,(f | (A \/ B)))) = (( Integral2 (M2,(f | A))) + ( Integral2 (M2,(f | B))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E,A,B be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: E = ( dom f) and

       A2: f is nonpositive and

       A3: f is E -measurable and

       A4: A misses B;

      ( Integral1 (M1,(f | A))) is nonpositive by A1, A2, A3, Th67;

      then

      reconsider IA = ( Integral1 (M1,(f | A))) as without+infty Function of X2, ExtREAL ;

      ( Integral1 (M1,(f | B))) is nonpositive by A1, A2, A3, Th67;

      then

      reconsider IB = ( Integral1 (M1,(f | B))) as without+infty Function of X2, ExtREAL ;

      now

        let y be Element of X2;

        

         A5: ( Y-section (A,y)) = ( Measurable-Y-section (A,y)) & ( Y-section (B,y)) = ( Measurable-Y-section (B,y)) & ( Y-section ((A \/ B),y)) = ( Measurable-Y-section ((A \/ B),y)) by MEASUR11:def 7;

        

         A6: ( dom ( ProjPMap2 (f,y))) = ( Y-section (E,y)) by A1, Def4

        .= ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

        

         A7: ( ProjPMap2 (f,y)) is ( Measurable-Y-section (E,y)) -measurable by A1, A3, Th47;

        (A /\ B) = ( {} [:X1, X2:]) by A4;

        then ( Y-section ((A /\ B),y)) = {} by MEASUR11: 24;

        then

         A8: ( Measurable-Y-section (A,y)) misses ( Measurable-Y-section (B,y)) by A5, MEASUR11: 27;

        ( ProjPMap2 ((f | A),y)) = (( ProjPMap2 (f,y)) | ( Y-section (A,y))) & ( ProjPMap2 ((f | B),y)) = (( ProjPMap2 (f,y)) | ( Y-section (B,y))) by Th34;

        then

         A9: ( ProjPMap2 ((f | A),y)) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section (A,y))) & ( ProjPMap2 ((f | B),y)) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section (B,y))) by MEASUR11:def 7;

        

         A10: (( Measurable-Y-section (A,y)) \/ ( Measurable-Y-section (B,y))) = ( Measurable-Y-section ((A \/ B),y)) by A5, MEASUR11: 26;

        ((IA + IB) . y) = ((( Integral1 (M1,(f | A))) . y) + (( Integral1 (M1,(f | B))) . y)) by DBLSEQ_3: 7

        .= (( Integral (M1,( ProjPMap2 ((f | A),y)))) + (( Integral1 (M1,(f | B))) . y)) by Def7

        .= (( Integral (M1,( ProjPMap2 ((f | A),y)))) + ( Integral (M1,( ProjPMap2 ((f | B),y))))) by Def7

        .= ( Integral (M1,(( ProjPMap2 (f,y)) | ( Measurable-Y-section ((A \/ B),y))))) by A2, A6, A7, A8, A9, A10, Th33, MESFUN11: 62

        .= ( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section ((A \/ B),y))))) by MEASUR11:def 7

        .= ( Integral (M1,( ProjPMap2 ((f | (A \/ B)),y)))) by Th34;

        hence (( Integral1 (M1,(f | (A \/ B)))) . y) = ((IA + IB) . y) by Def7;

      end;

      hence ( Integral1 (M1,(f | (A \/ B)))) = (( Integral1 (M1,(f | A))) + ( Integral1 (M1,(f | B)))) by FUNCT_2:def 8;

      ( Integral2 (M2,(f | A))) is nonpositive by A1, A2, A3, Th67;

      then

      reconsider JA = ( Integral2 (M2,(f | A))) as without+infty Function of X1, ExtREAL ;

      ( Integral2 (M2,(f | B))) is nonpositive by A1, A2, A3, Th67;

      then

      reconsider JB = ( Integral2 (M2,(f | B))) as without+infty Function of X1, ExtREAL ;

      now

        let x be Element of X1;

        

         A5: ( X-section (A,x)) = ( Measurable-X-section (A,x)) & ( X-section (B,x)) = ( Measurable-X-section (B,x)) & ( X-section ((A \/ B),x)) = ( Measurable-X-section ((A \/ B),x)) by MEASUR11:def 6;

        

         A6: ( dom ( ProjPMap1 (f,x))) = ( X-section (E,x)) by A1, Def3

        .= ( Measurable-X-section (E,x)) by MEASUR11:def 6;

        

         A7: ( ProjPMap1 (f,x)) is ( Measurable-X-section (E,x)) -measurable by A1, A3, Th47;

        (A /\ B) = ( {} [:X1, X2:]) by A4;

        then ( X-section ((A /\ B),x)) = {} by MEASUR11: 24;

        then

         A8: ( Measurable-X-section (A,x)) misses ( Measurable-X-section (B,x)) by A5, MEASUR11: 27;

        ( ProjPMap1 ((f | A),x)) = (( ProjPMap1 (f,x)) | ( X-section (A,x))) & ( ProjPMap1 ((f | B),x)) = (( ProjPMap1 (f,x)) | ( X-section (B,x))) by Th34;

        then

         A9: ( ProjPMap1 ((f | A),x)) = (( ProjPMap1 (f,x)) | ( Measurable-X-section (A,x))) & ( ProjPMap1 ((f | B),x)) = (( ProjPMap1 (f,x)) | ( Measurable-X-section (B,x))) by MEASUR11:def 6;

        

         A10: (( Measurable-X-section (A,x)) \/ ( Measurable-X-section (B,x))) = ( Measurable-X-section ((A \/ B),x)) by A5, MEASUR11: 26;

        ((JA + JB) . x) = ((( Integral2 (M2,(f | A))) . x) + (( Integral2 (M2,(f | B))) . x)) by DBLSEQ_3: 7

        .= (( Integral (M2,( ProjPMap1 ((f | A),x)))) + (( Integral2 (M2,(f | B))) . x)) by Def8

        .= (( Integral (M2,( ProjPMap1 ((f | A),x)))) + ( Integral (M2,( ProjPMap1 ((f | B),x))))) by Def8

        .= ( Integral (M2,(( ProjPMap1 (f,x)) | ( Measurable-X-section ((A \/ B),x))))) by A2, A6, A7, A8, A9, A10, Th33, MESFUN11: 62

        .= ( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section ((A \/ B),x))))) by MEASUR11:def 6

        .= ( Integral (M2,( ProjPMap1 ((f | (A \/ B)),x)))) by Th34;

        hence (( Integral2 (M2,(f | (A \/ B)))) . x) = ((JA + JB) . x) by Def8;

      end;

      hence thesis by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:72

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E,E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st E = ( dom f) & (f is nonnegative or f is nonpositive) & f is E -measurable & E1 misses E2 holds ( Integral1 (M1,(f | (E1 \/ E2)))) = (( Integral1 (M1,(f | E1))) + ( Integral1 (M1,(f | E2)))) & ( Integral2 (M2,(f | (E1 \/ E2)))) = (( Integral2 (M2,(f | E1))) + ( Integral2 (M2,(f | E2)))) by Lm11, Lm12;

    theorem :: MESFUN12:73

    

     Th73: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , E be Element of ( sigma ( measurable_rectangles (S1,S2))) st E = ( dom f) & f is E -measurable holds ( Integral1 (M1,( - f))) = ( - ( Integral1 (M1,f))) & ( Integral2 (M2,( - f))) = ( - ( Integral2 (M2,f)))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is A -measurable;

      

       A3: ( dom ( - ( Integral1 (M1,f)))) = X2 & ( dom ( - ( Integral2 (M2,f)))) = X1 by FUNCT_2:def 1;

      now

        let y be Element of X2;

        ( ProjPMap2 (( - f),y)) = ( ProjPMap2 ((( - 1) (#) f),y)) by MESFUNC2: 9

        .= (( - 1) (#) ( ProjPMap2 (f,y))) by Th29

        .= ( - ( ProjPMap2 (f,y))) by MESFUNC2: 9;

        then

         A4: (( Integral1 (M1,( - f))) . y) = ( Integral (M1,( - ( ProjPMap2 (f,y))))) by Def7;

        ( dom ( ProjPMap2 (f,y))) = ( Y-section (A,y)) by A1, Def4;

        then

         A5: ( dom ( ProjPMap2 (f,y))) = ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

        (( - ( Integral1 (M1,f))) . y) = ( - (( Integral1 (M1,f)) . y)) by A3, MESFUNC1:def 7;

        then (( - ( Integral1 (M1,f))) . y) = ( - ( Integral (M1,( ProjPMap2 (f,y))))) by Def7;

        hence (( Integral1 (M1,( - f))) . y) = (( - ( Integral1 (M1,f))) . y) by A1, A2, A4, A5, Th47, MESFUN11: 52;

      end;

      hence ( Integral1 (M1,( - f))) = ( - ( Integral1 (M1,f))) by FUNCT_2:def 8;

      now

        let x be Element of X1;

        ( ProjPMap1 (( - f),x)) = ( ProjPMap1 ((( - 1) (#) f),x)) by MESFUNC2: 9

        .= (( - 1) (#) ( ProjPMap1 (f,x))) by Th29

        .= ( - ( ProjPMap1 (f,x))) by MESFUNC2: 9;

        then

         A6: (( Integral2 (M2,( - f))) . x) = ( Integral (M2,( - ( ProjPMap1 (f,x))))) by Def8;

        ( dom ( ProjPMap1 (f,x))) = ( X-section (A,x)) by A1, Def3;

        then

         A7: ( dom ( ProjPMap1 (f,x))) = ( Measurable-X-section (A,x)) by MEASUR11:def 6;

        (( - ( Integral2 (M2,f))) . x) = ( - (( Integral2 (M2,f)) . x)) by A3, MESFUNC1:def 7;

        then (( - ( Integral2 (M2,f))) . x) = ( - ( Integral (M2,( ProjPMap1 (f,x))))) by Def8;

        hence (( Integral2 (M2,( - f))) . x) = (( - ( Integral2 (M2,f))) . x) by A1, A2, A6, A7, Th47, MESFUN11: 52;

      end;

      hence ( Integral2 (M2,( - f))) = ( - ( Integral2 (M2,f))) by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:74

    

     Th74: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))) st E1 = ( dom f) & f is nonnegative & f is E1 -measurable & E2 = ( dom g) & g is nonnegative & g is E2 -measurable holds ( Integral1 (M1,(f + g))) = (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g)))))) & ( Integral2 (M2,(f + g))) = (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g))))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , A,B be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonnegative and

       A6: g is B -measurable;

      

       A7: ( dom (f + g)) = (A /\ B) by A1, A2, A4, A5, MESFUNC5: 22;

      set f1 = (f | (A /\ B)), g1 = (g | (A /\ B));

      

       A8: ( dom f1) = (A /\ B) & ( dom g1) = (A /\ B) by A1, A4, XBOOLE_1: 17, RELAT_1: 62;

      

       A9: (( dom f) /\ (A /\ B)) = (A /\ B) & (( dom g) /\ (A /\ B)) = (A /\ B) by A1, A4, XBOOLE_1: 17, XBOOLE_1: 28;

      

       A10: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      then

       A11: f1 is (A /\ B) -measurable & g1 is (A /\ B) -measurable by A9, MESFUNC5: 42;

      

       A12: f1 is nonnegative & g1 is nonnegative by A2, A5, MESFUNC5: 15;

      then

       A13: ( Integral1 (M1,f1)) is nonnegative & ( Integral1 (M1,g1)) is nonnegative & ( Integral2 (M2,f1)) is nonnegative & ( Integral2 (M2,g1)) is nonnegative by A8, A11, Th66;

      then

      reconsider IF1 = ( Integral1 (M1,f1)), IG1 = ( Integral1 (M1,g1)) as without-infty Function of X2, ExtREAL ;

      reconsider IF2 = ( Integral2 (M2,f1)), IG2 = ( Integral2 (M2,g1)) as without-infty Function of X1, ExtREAL by A13;

      

       A14: (IF1 + IG1) = (( Integral1 (M1,f1)) + ( Integral1 (M1,g1))) & (IF2 + IG2) = (( Integral2 (M2,f1)) + ( Integral2 (M2,g1)));

      

       A21: (f + g) is nonnegative by A2, A5, MESFUNC5: 22;

      for y be Element of X2 holds ((( Integral1 (M1,f1)) + ( Integral1 (M1,g1))) . y) = (( Integral1 (M1,(f + g))) . y)

      proof

        let y be Element of X2;

        ( dom ( ProjPMap2 (f1,y))) = ( Y-section ((A /\ B),y)) & ( dom ( ProjPMap2 (g1,y))) = ( Y-section ((A /\ B),y)) by A8, Def4;

        then

         A15: ( dom ( ProjPMap2 (f1,y))) = ( Measurable-Y-section ((A /\ B),y)) & ( dom ( ProjPMap2 (g1,y))) = ( Measurable-Y-section ((A /\ B),y)) by MEASUR11:def 7;

        ( ProjPMap2 (f1,y)) is ( Measurable-Y-section ((A /\ B),y)) -measurable & ( ProjPMap2 (g1,y)) is ( Measurable-Y-section ((A /\ B),y)) -measurable by A8, A11, Th47;

        then

         A16: ( Integral (M1,( ProjPMap2 (f1,y)))) = ( integral+ (M1,( ProjPMap2 (f1,y)))) & ( Integral (M1,( ProjPMap2 (g1,y)))) = ( integral+ (M1,( ProjPMap2 (g1,y)))) by A12, A15, Th32, MESFUNC5: 88;

        

         A17: ( ProjPMap2 ((f + g),y)) = (( ProjPMap2 (f,y)) + ( ProjPMap2 (g,y))) by Th44;

        ( ProjPMap2 (f1,y)) = (( ProjPMap2 (f,y)) | ( Y-section ((A /\ B),y))) & ( ProjPMap2 (g1,y)) = (( ProjPMap2 (g,y)) | ( Y-section ((A /\ B),y))) by Th34;

        then

         A18: ( ProjPMap2 (f1,y)) = (( ProjPMap2 (f,y)) | ( Measurable-Y-section ((A /\ B),y))) & ( ProjPMap2 (g1,y)) = (( ProjPMap2 (g,y)) | ( Measurable-Y-section ((A /\ B),y))) by MEASUR11:def 7;

        ( dom ( ProjPMap2 (f,y))) = ( Y-section (A,y)) & ( dom ( ProjPMap2 (g,y))) = ( Y-section (B,y)) by A1, A4, Def4;

        then

         A19: ( dom ( ProjPMap2 (f,y))) = ( Measurable-Y-section (A,y)) & ( dom ( ProjPMap2 (g,y))) = ( Measurable-Y-section (B,y)) by MEASUR11:def 7;

        ( dom ( ProjPMap2 ((f + g),y))) = ( Y-section ((A /\ B),y)) by A7, Def4;

        then

         A20: ( Measurable-Y-section ((A /\ B),y)) = ( dom ( ProjPMap2 ((f + g),y))) by MEASUR11:def 7;

        (f + g) is (A /\ B) -measurable by A2, A5, A10, MESFUNC5: 31;

        then

         A22: ( ProjPMap2 ((f + g),y)) is ( Measurable-Y-section ((A /\ B),y)) -measurable by A7, Th47;

        

         A23: ((( Integral1 (M1,f1)) + ( Integral1 (M1,g1))) . y) = ((( Integral1 (M1,f1)) . y) + (( Integral1 (M1,g1)) . y)) by A13, DBLSEQ_3: 7

        .= (( Integral (M1,( ProjPMap2 (f1,y)))) + (( Integral1 (M1,g1)) . y)) by Def7

        .= (( integral+ (M1,( ProjPMap2 (f1,y)))) + ( integral+ (M1,( ProjPMap2 (g1,y))))) by A16, Def7;

        ( ProjPMap2 (f,y)) is nonnegative & ( ProjPMap2 (g,y)) is nonnegative & ( ProjPMap2 (f,y)) is ( Measurable-Y-section (A,y)) -measurable & ( ProjPMap2 (g,y)) is ( Measurable-Y-section (B,y)) -measurable by A1, A3, A4, A6, A2, A5, Th32, Th47;

        then ex C be Element of S1 st C = ( dom (( ProjPMap2 (f,y)) + ( ProjPMap2 (g,y)))) & ( integral+ (M1,(( ProjPMap2 (f,y)) + ( ProjPMap2 (g,y))))) = (( integral+ (M1,(( ProjPMap2 (f,y)) | C))) + ( integral+ (M1,(( ProjPMap2 (g,y)) | C)))) by A19, MESFUNC5: 78;

        then ((( Integral1 (M1,f1)) + ( Integral1 (M1,g1))) . y) = ( Integral (M1,( ProjPMap2 ((f + g),y)))) by A17, A18, A20, A23, A21, A22, Th32, MESFUNC5: 88;

        hence (( Integral1 (M1,(f + g))) . y) = ((( Integral1 (M1,f1)) + ( Integral1 (M1,g1))) . y) by Def7;

      end;

      hence ( Integral1 (M1,(f + g))) = (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g)))))) by A7, A14, FUNCT_2: 63;

      for x be Element of X1 holds ((( Integral2 (M2,f1)) + ( Integral2 (M2,g1))) . x) = (( Integral2 (M2,(f + g))) . x)

      proof

        let x be Element of X1;

        ( dom ( ProjPMap1 (f1,x))) = ( X-section ((A /\ B),x)) & ( dom ( ProjPMap1 (g1,x))) = ( X-section ((A /\ B),x)) by A8, Def3;

        then

         B15: ( dom ( ProjPMap1 (f1,x))) = ( Measurable-X-section ((A /\ B),x)) & ( dom ( ProjPMap1 (g1,x))) = ( Measurable-X-section ((A /\ B),x)) by MEASUR11:def 6;

        ( ProjPMap1 (f1,x)) is ( Measurable-X-section ((A /\ B),x)) -measurable & ( ProjPMap1 (g1,x)) is ( Measurable-X-section ((A /\ B),x)) -measurable by A8, A11, Th47;

        then

         B16: ( Integral (M2,( ProjPMap1 (f1,x)))) = ( integral+ (M2,( ProjPMap1 (f1,x)))) & ( Integral (M2,( ProjPMap1 (g1,x)))) = ( integral+ (M2,( ProjPMap1 (g1,x)))) by A12, B15, Th32, MESFUNC5: 88;

        

         B17: ( ProjPMap1 ((f + g),x)) = (( ProjPMap1 (f,x)) + ( ProjPMap1 (g,x))) by Th44;

        ( ProjPMap1 (f1,x)) = (( ProjPMap1 (f,x)) | ( X-section ((A /\ B),x))) & ( ProjPMap1 (g1,x)) = (( ProjPMap1 (g,x)) | ( X-section ((A /\ B),x))) by Th34;

        then

         B18: ( ProjPMap1 (f1,x)) = (( ProjPMap1 (f,x)) | ( Measurable-X-section ((A /\ B),x))) & ( ProjPMap1 (g1,x)) = (( ProjPMap1 (g,x)) | ( Measurable-X-section ((A /\ B),x))) by MEASUR11:def 6;

        ( dom ( ProjPMap1 (f,x))) = ( X-section (A,x)) & ( dom ( ProjPMap1 (g,x))) = ( X-section (B,x)) by A1, A4, Def3;

        then

         B19: ( dom ( ProjPMap1 (f,x))) = ( Measurable-X-section (A,x)) & ( dom ( ProjPMap1 (g,x))) = ( Measurable-X-section (B,x)) by MEASUR11:def 6;

        ( dom ( ProjPMap1 ((f + g),x))) = ( X-section ((A /\ B),x)) by A7, Def3;

        then

         B20: ( Measurable-X-section ((A /\ B),x)) = ( dom ( ProjPMap1 ((f + g),x))) by MEASUR11:def 6;

        (f + g) is (A /\ B) -measurable by A2, A5, A10, MESFUNC5: 31;

        then

         B22: ( ProjPMap1 ((f + g),x)) is ( Measurable-X-section ((A /\ B),x)) -measurable by A7, Th47;

        

         B23: ((( Integral2 (M2,f1)) + ( Integral2 (M2,g1))) . x) = ((( Integral2 (M2,f1)) . x) + (( Integral2 (M2,g1)) . x)) by A13, DBLSEQ_3: 7

        .= (( Integral (M2,( ProjPMap1 (f1,x)))) + (( Integral2 (M2,g1)) . x)) by Def8

        .= (( integral+ (M2,( ProjPMap1 (f1,x)))) + ( integral+ (M2,( ProjPMap1 (g1,x))))) by B16, Def8;

        ( ProjPMap1 (f,x)) is nonnegative & ( ProjPMap1 (g,x)) is nonnegative & ( ProjPMap1 (f,x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap1 (g,x)) is ( Measurable-X-section (B,x)) -measurable by A1, A3, A4, A6, A2, A5, Th32, Th47;

        then ex C be Element of S2 st C = ( dom (( ProjPMap1 (f,x)) + ( ProjPMap1 (g,x)))) & ( integral+ (M2,(( ProjPMap1 (f,x)) + ( ProjPMap1 (g,x))))) = (( integral+ (M2,(( ProjPMap1 (f,x)) | C))) + ( integral+ (M2,(( ProjPMap1 (g,x)) | C)))) by B19, MESFUNC5: 78;

        then ((( Integral2 (M2,f1)) + ( Integral2 (M2,g1))) . x) = ( Integral (M2,( ProjPMap1 ((f + g),x)))) by B17, B18, B20, B23, A21, B22, Th32, MESFUNC5: 88;

        hence (( Integral2 (M2,(f + g))) . x) = ((( Integral2 (M2,f1)) + ( Integral2 (M2,g1))) . x) by Def8;

      end;

      hence ( Integral2 (M2,(f + g))) = (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g)))))) by A7, A14, FUNCT_2: 63;

    end;

    theorem :: MESFUN12:75

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))) st E1 = ( dom f) & f is nonpositive & f is E1 -measurable & E2 = ( dom g) & g is nonpositive & g is E2 -measurable holds ( Integral1 (M1,(f + g))) = (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g)))))) & ( Integral2 (M2,(f + g))) = (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g))))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , A,B be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is nonpositive and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonpositive and

       A6: g is B -measurable;

      reconsider f1 = ( - f) as nonnegative PartFunc of [:X1, X2:], ExtREAL by A2;

      reconsider g1 = ( - g) as nonnegative PartFunc of [:X1, X2:], ExtREAL by A5;

      

       A7: (f1 + g1) = ( - (f + g)) by MEASUR11: 64;

      

       A8: ( dom f1) = A & ( dom g1) = B by A1, A4, MESFUNC1:def 7;

      then

       A9: ( dom (f1 + g1)) = (A /\ B) by MESFUNC5: 22;

      then

       A10: ( dom (f + g)) = (A /\ B) by A7, MESFUNC1:def 7;

      then

       A11: ( dom (f | ( dom (f + g)))) = (A /\ B) & ( dom (g | ( dom (f + g)))) = (A /\ B) by A1, A4, XBOOLE_1: 17, RELAT_1: 62;

      

       A12: (( dom f) /\ (A /\ B)) = (A /\ B) & (( dom g) /\ (A /\ B)) = (A /\ B) by A1, A4, XBOOLE_1: 17, XBOOLE_1: 28;

      

       A13: (f1 | ( dom (f1 + g1))) = ( - (f | ( dom (f + g)))) & (g1 | ( dom (f1 + g1))) = ( - (g | ( dom (f + g)))) by A9, A10, MESFUN11: 3;

      

       A14: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      then (f | ( dom (f + g))) is (A /\ B) -measurable & (g | ( dom (f + g))) is (A /\ B) -measurable by A10, A12, MESFUNC5: 42;

      then

       A15: ( Integral1 (M1,(f1 | ( dom (f1 + g1))))) = ( - ( Integral1 (M1,(f | ( dom (f + g)))))) & ( Integral1 (M1,(g1 | ( dom (f1 + g1))))) = ( - ( Integral1 (M1,(g | ( dom (f + g)))))) & ( Integral2 (M2,(f1 | ( dom (f1 + g1))))) = ( - ( Integral2 (M2,(f | ( dom (f + g)))))) & ( Integral2 (M2,(g1 | ( dom (f1 + g1))))) = ( - ( Integral2 (M2,(g | ( dom (f + g)))))) by A11, A13, Th73;

      (f + g) is (A /\ B) -measurable by A2, A5, A10, A14, MEASUR11: 65;

      then

       A16: ( Integral1 (M1,(f1 + g1))) = ( - ( Integral1 (M1,(f + g)))) & ( Integral2 (M2,(f1 + g1))) = ( - ( Integral2 (M2,(f + g)))) by A7, A10, Th73;

      

       A17: f1 is A -measurable & g1 is B -measurable by A1, A3, A4, A6, MEASUR11: 63;

      then ( Integral1 (M1,(f1 + g1))) = (( Integral1 (M1,(f1 | ( dom (f1 + g1))))) + ( Integral1 (M1,(g1 | ( dom (f1 + g1)))))) by A8, Th74;

      then ( - ( Integral1 (M1,(f + g)))) = ( - (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g))))))) by A15, A16, MEASUR11: 64;

      then ( Integral1 (M1,(f + g))) = ( - ( - (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g)))))))) by DBLSEQ_3: 2;

      hence ( Integral1 (M1,(f + g))) = (( Integral1 (M1,(f | ( dom (f + g))))) + ( Integral1 (M1,(g | ( dom (f + g)))))) by DBLSEQ_3: 2;

      ( Integral2 (M2,(f1 + g1))) = (( Integral2 (M2,(f1 | ( dom (f1 + g1))))) + ( Integral2 (M2,(g1 | ( dom (f1 + g1)))))) by A8, A17, Th74;

      then ( - ( Integral2 (M2,(f + g)))) = ( - (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g))))))) by A15, A16, MEASUR11: 64;

      then ( Integral2 (M2,(f + g))) = ( - ( - (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g)))))))) by DBLSEQ_3: 2;

      hence ( Integral2 (M2,(f + g))) = (( Integral2 (M2,(f | ( dom (f + g))))) + ( Integral2 (M2,(g | ( dom (f + g)))))) by DBLSEQ_3: 2;

    end;

    theorem :: MESFUN12:76

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , E1,E2 be Element of ( sigma ( measurable_rectangles (S1,S2))) st E1 = ( dom f) & f is nonnegative & f is E1 -measurable & E2 = ( dom g) & g is nonpositive & g is E2 -measurable holds ( Integral1 (M1,(f - g))) = (( Integral1 (M1,(f | ( dom (f - g))))) - ( Integral1 (M1,(g | ( dom (f - g)))))) & ( Integral1 (M1,(g - f))) = (( Integral1 (M1,(g | ( dom (g - f))))) - ( Integral1 (M1,(f | ( dom (g - f)))))) & ( Integral2 (M2,(f - g))) = (( Integral2 (M2,(f | ( dom (f - g))))) - ( Integral2 (M2,(g | ( dom (f - g)))))) & ( Integral2 (M2,(g - f))) = (( Integral2 (M2,(g | ( dom (g - f))))) - ( Integral2 (M2,(f | ( dom (g - f))))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f,g be PartFunc of [:X1, X2:], ExtREAL , A,B be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: A = ( dom f) and

       A2: f is nonnegative and

       A3: f is A -measurable and

       A4: B = ( dom g) and

       A5: g is nonpositive and

       A6: g is B -measurable;

      reconsider g1 = ( - g) as nonnegative PartFunc of [:X1, X2:], ExtREAL by A5;

      

       A7: B = ( dom g1) by A4, MESFUNC1:def 7;

      

       A8: g1 is B -measurable by A4, A6, MEASUR11: 63;

      

       A9: f is (A /\ B) -measurable & g is (A /\ B) -measurable by A3, A6, XBOOLE_1: 17, MESFUNC1: 30;

      

       A10: ( dom (f - g)) = (A /\ B) by A1, A2, A4, A5, MESFUNC5: 17;

      then

       A11: (A /\ B) = ( dom (g | ( dom (f - g)))) by A4, XBOOLE_1: 17, RELAT_1: 62;

      then (A /\ B) = (( dom g) /\ ( dom (f - g))) by RELAT_1: 61;

      then

       A12: (g | ( dom (f - g))) is (A /\ B) -measurable by A9, A10, MESFUNC5: 42;

      

       A13: (f + g1) = (f - g) by MESFUNC2: 8;

      

      then

       A14: ( Integral1 (M1,(f - g))) = (( Integral1 (M1,(f | ( dom (f - g))))) + ( Integral1 (M1,(g1 | ( dom (f - g)))))) by A1, A2, A3, A7, A8, Th74

      .= (( Integral1 (M1,(f | ( dom (f - g))))) + ( Integral1 (M1,( - (g | ( dom (f - g))))))) by MESFUN11: 3

      .= (( Integral1 (M1,(f | ( dom (f - g))))) + ( - ( Integral1 (M1,(g | ( dom (f - g))))))) by A11, A12, Th73;

      hence ( Integral1 (M1,(f - g))) = (( Integral1 (M1,(f | ( dom (f - g))))) - ( Integral1 (M1,(g | ( dom (f - g)))))) by MESFUNC2: 8;

      

       A15: (f - g) is (A /\ B) -measurable by A2, A5, A9, A10, MEASUR11: 67;

      

       A16: (g - f) = ( - (f - g)) by MEASUR11: 64;

      then

       A17: ( dom (g - f)) = (A /\ B) by A10, MESFUNC1:def 7;

      ( Integral1 (M1,(g - f))) = ( - ( Integral1 (M1,(f - g)))) by A10, A16, A15, Th73;

      hence ( Integral1 (M1,(g - f))) = (( Integral1 (M1,(g | ( dom (g - f))))) - ( Integral1 (M1,(f | ( dom (g - f)))))) by A10, A14, A17, MEASUR11: 64;

      

       A18: ( Integral2 (M2,(f - g))) = (( Integral2 (M2,(f | ( dom (f - g))))) + ( Integral2 (M2,(g1 | ( dom (f - g)))))) by A1, A2, A3, A7, A8, A13, Th74

      .= (( Integral2 (M2,(f | ( dom (f - g))))) + ( Integral2 (M2,( - (g | ( dom (f - g))))))) by MESFUN11: 3

      .= (( Integral2 (M2,(f | ( dom (f - g))))) + ( - ( Integral2 (M2,(g | ( dom (f - g))))))) by A11, A12, Th73;

      hence ( Integral2 (M2,(f - g))) = (( Integral2 (M2,(f | ( dom (f - g))))) - ( Integral2 (M2,(g | ( dom (f - g)))))) by MESFUNC2: 8;

      ( Integral2 (M2,(g - f))) = ( - ( Integral2 (M2,(f - g)))) by A10, A16, A15, Th73;

      hence ( Integral2 (M2,(g - f))) = (( Integral2 (M2,(g | ( dom (g - f))))) - ( Integral2 (M2,(f | ( dom (g - f)))))) by A10, A18, A17, MEASUR11: 64;

    end;

    theorem :: MESFUN12:77

    

     Th77: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite & M2 is sigma_finite holds ( Integral (M1,( Y-vol (E,M2)))) = ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) & ( Integral (M2,( X-vol (E,M1)))) = ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite;

      ( Integral (M2,( X-vol (E,M1)))) = (( product_sigma_Measure (M1,M2)) . E) & ( Integral (M1,( Y-vol (E,M2)))) = (( product_sigma_Measure (M1,M2)) . E) by A1, A2, MEASUR11: 118, MEASUR11: 117;

      hence thesis by MESFUNC9: 14;

    end;

    theorem :: MESFUN12:78

    

     Th78: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL , r be Real st E = ( dom f) & (f is nonnegative or f is nonpositive) & f is E -measurable holds ( Integral1 (M1,(r (#) f))) = (r (#) ( Integral1 (M1,f))) & ( Integral2 (M2,(r (#) f))) = (r (#) ( Integral2 (M2,f)))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL , r be Real;

      assume that

       A1: E = ( dom f) and

       A2: (f is nonnegative or f is nonpositive) and

       A3: f is E -measurable;

      

       A4: ( dom (r (#) ( Integral1 (M1,f)))) = X2 & ( dom (r (#) ( Integral2 (M2,f)))) = X1 by FUNCT_2:def 1;

      now

        let y be Element of X2;

        ( dom ( ProjPMap2 (f,y))) = ( Y-section (E,y)) by A1, Def4;

        then

         A5: ( dom ( ProjPMap2 (f,y))) = ( Measurable-Y-section (E,y)) by MEASUR11:def 7;

        

         A6: ( ProjPMap2 (f,y)) is nonnegative or ( ProjPMap2 (f,y)) is nonpositive by A2, Th32, Th33;

        (( Integral1 (M1,(r (#) f))) . y) = ( Integral (M1,( ProjPMap2 ((r (#) f),y)))) by Def7

        .= ( Integral (M1,(r (#) ( ProjPMap2 (f,y))))) by Th29

        .= (r * ( Integral (M1,( ProjPMap2 (f,y))))) by A5, A6, A1, A3, Th47, Lm1, Lm2

        .= (r * (( Integral1 (M1,f)) . y)) by Def7;

        hence (( Integral1 (M1,(r (#) f))) . y) = ((r (#) ( Integral1 (M1,f))) . y) by A4, MESFUNC1:def 6;

      end;

      hence ( Integral1 (M1,(r (#) f))) = (r (#) ( Integral1 (M1,f))) by FUNCT_2:def 8;

      now

        let x be Element of X1;

        ( dom ( ProjPMap1 (f,x))) = ( X-section (E,x)) by A1, Def3;

        then

         B5: ( dom ( ProjPMap1 (f,x))) = ( Measurable-X-section (E,x)) by MEASUR11:def 6;

        

         B6: ( ProjPMap1 (f,x)) is nonnegative or ( ProjPMap1 (f,x)) is nonpositive by A2, Th32, Th33;

        (( Integral2 (M2,(r (#) f))) . x) = ( Integral (M2,( ProjPMap1 ((r (#) f),x)))) by Def8

        .= ( Integral (M2,(r (#) ( ProjPMap1 (f,x))))) by Th29

        .= (r * ( Integral (M2,( ProjPMap1 (f,x))))) by B6, B5, A1, A3, Th47, Lm1, Lm2

        .= (r * (( Integral2 (M2,f)) . x)) by Def8;

        hence (( Integral2 (M2,(r (#) f))) . x) = ((r (#) ( Integral2 (M2,f))) . x) by A4, MESFUNC1:def 6;

      end;

      hence ( Integral2 (M2,(r (#) f))) = (r (#) ( Integral2 (M2,f))) by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:79

    

     Th79: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) holds ( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (E, [:X1, X2:])))) & ( Integral2 (M2,(( chi (E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      now

        let y be Element of X2;

        

         A1: ( ProjPMap2 ((( chi (E, [:X1, X2:])) | E),y)) = (( ProjPMap2 (( chi (E, [:X1, X2:])),y)) | ( Y-section (E,y))) by Th34

        .= (( chi (( Y-section (E,y)),X1)) | ( Y-section (E,y))) by Th48

        .= (( chi (( Measurable-Y-section (E,y)),X1)) | ( Y-section (E,y))) by MEASUR11:def 7

        .= (( chi (( Measurable-Y-section (E,y)),X1)) | ( Measurable-Y-section (E,y))) by MEASUR11:def 7;

        (( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))) . y) = ( Integral (M1,( ProjPMap2 ((( chi (E, [:X1, X2:])) | E),y)))) by Def7

        .= (M1 . ( Measurable-Y-section (E,y))) by A1, MESFUNC9: 14

        .= ( Integral (M1,( chi (( Measurable-Y-section (E,y)),X1)))) by MESFUNC9: 14

        .= ( Integral (M1,( chi (( Y-section (E,y)),X1)))) by MEASUR11:def 7

        .= ( Integral (M1,( ProjPMap2 (( chi (E, [:X1, X2:])),y)))) by Th48;

        hence (( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))) . y) = (( Integral1 (M1,( chi (E, [:X1, X2:])))) . y) by Def7;

      end;

      hence ( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (E, [:X1, X2:])))) by FUNCT_2:def 8;

      now

        let x be Element of X1;

        

         A2: ( ProjPMap1 ((( chi (E, [:X1, X2:])) | E),x)) = (( ProjPMap1 (( chi (E, [:X1, X2:])),x)) | ( X-section (E,x))) by Th34

        .= (( chi (( X-section (E,x)),X2)) | ( X-section (E,x))) by Th48

        .= (( chi (( Measurable-X-section (E,x)),X2)) | ( X-section (E,x))) by MEASUR11:def 6

        .= (( chi (( Measurable-X-section (E,x)),X2)) | ( Measurable-X-section (E,x))) by MEASUR11:def 6;

        (( Integral2 (M2,(( chi (E, [:X1, X2:])) | E))) . x) = ( Integral (M2,( ProjPMap1 ((( chi (E, [:X1, X2:])) | E),x)))) by Def8

        .= (M2 . ( Measurable-X-section (E,x))) by A2, MESFUNC9: 14

        .= ( Integral (M2,( chi (( Measurable-X-section (E,x)),X2)))) by MESFUNC9: 14

        .= ( Integral (M2,( chi (( X-section (E,x)),X2)))) by MEASUR11:def 6

        .= ( Integral (M2,( ProjPMap1 (( chi (E, [:X1, X2:])),x)))) by Th48;

        hence (( Integral2 (M2,(( chi (E, [:X1, X2:])) | E))) . x) = (( Integral2 (M2,( chi (E, [:X1, X2:])))) . x) by Def8;

      end;

      hence ( Integral2 (M2,(( chi (E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (E, [:X1, X2:])))) by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:80

    

     Th80: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) holds ( Integral1 (M1,(( Xchi (E, [:X1, X2:])) | E))) = ( Integral1 (M1,( Xchi (E, [:X1, X2:])))) & ( Integral2 (M2,(( Xchi (E, [:X1, X2:])) | E))) = ( Integral2 (M2,( Xchi (E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      now

        let y be Element of X2;

        set XC1 = ( Xchi (( Measurable-Y-section (E,y)),X1));

        

         A1: ( ProjPMap2 ((( Xchi (E, [:X1, X2:])) | E),y)) = (( ProjPMap2 (( Xchi (E, [:X1, X2:])),y)) | ( Y-section (E,y))) by Th34

        .= (( Xchi (( Y-section (E,y)),X1)) | ( Y-section (E,y))) by Th35

        .= (( Xchi (( Measurable-Y-section (E,y)),X1)) | ( Y-section (E,y))) by MEASUR11:def 7

        .= (XC1 | ( Measurable-Y-section (E,y))) by MEASUR11:def 7

        .= (( chi ( +infty ,( Measurable-Y-section (E,y)),X1)) | ( Measurable-Y-section (E,y))) by Th2;

        (( Integral1 (M1,(( Xchi (E, [:X1, X2:])) | E))) . y) = ( Integral (M1,( ProjPMap2 ((( Xchi (E, [:X1, X2:])) | E),y)))) by Def7

        .= ( +infty * (M1 . ( Measurable-Y-section (E,y)))) by A1, Th50

        .= ( Integral (M1,( chi ( +infty ,( Measurable-Y-section (E,y)),X1)))) by Th49

        .= ( Integral (M1,( Xchi (( Measurable-Y-section (E,y)),X1)))) by Th2

        .= ( Integral (M1,( Xchi (( Y-section (E,y)),X1)))) by MEASUR11:def 7

        .= ( Integral (M1,( ProjPMap2 (( Xchi (E, [:X1, X2:])),y)))) by Th35;

        hence (( Integral1 (M1,(( Xchi (E, [:X1, X2:])) | E))) . y) = (( Integral1 (M1,( Xchi (E, [:X1, X2:])))) . y) by Def7;

      end;

      hence ( Integral1 (M1,(( Xchi (E, [:X1, X2:])) | E))) = ( Integral1 (M1,( Xchi (E, [:X1, X2:])))) by FUNCT_2:def 8;

      now

        let x be Element of X1;

        set XC2 = ( Xchi (( Measurable-X-section (E,x)),X2));

        

         A1: ( ProjPMap1 ((( Xchi (E, [:X1, X2:])) | E),x)) = (( ProjPMap1 (( Xchi (E, [:X1, X2:])),x)) | ( X-section (E,x))) by Th34

        .= (( Xchi (( X-section (E,x)),X2)) | ( X-section (E,x))) by Th35

        .= (( Xchi (( Measurable-X-section (E,x)),X2)) | ( X-section (E,x))) by MEASUR11:def 6

        .= (XC2 | ( Measurable-X-section (E,x))) by MEASUR11:def 6

        .= (( chi ( +infty ,( Measurable-X-section (E,x)),X2)) | ( Measurable-X-section (E,x))) by Th2;

        (( Integral2 (M2,(( Xchi (E, [:X1, X2:])) | E))) . x) = ( Integral (M2,( ProjPMap1 ((( Xchi (E, [:X1, X2:])) | E),x)))) by Def8

        .= ( +infty * (M2 . ( Measurable-X-section (E,x)))) by A1, Th50

        .= ( Integral (M2,( chi ( +infty ,( Measurable-X-section (E,x)),X2)))) by Th49

        .= ( Integral (M2,( Xchi (( Measurable-X-section (E,x)),X2)))) by Th2

        .= ( Integral (M2,( Xchi (( X-section (E,x)),X2)))) by MEASUR11:def 6

        .= ( Integral (M2,( ProjPMap1 (( Xchi (E, [:X1, X2:])),x)))) by Th35;

        hence (( Integral2 (M2,(( Xchi (E, [:X1, X2:])) | E))) . x) = (( Integral2 (M2,( Xchi (E, [:X1, X2:])))) . x) by Def8;

      end;

      hence ( Integral2 (M2,(( Xchi (E, [:X1, X2:])) | E))) = ( Integral2 (M2,( Xchi (E, [:X1, X2:])))) by FUNCT_2:def 8;

    end;

    theorem :: MESFUN12:81

    

     Th81: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), er be ExtReal holds ( Integral1 (M1,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (er,E, [:X1, X2:])))) & ( Integral2 (M2,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (er,E, [:X1, X2:]))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), er be ExtReal;

      reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      reconsider C = (( chi (E, [:X1, X2:])) | E) as PartFunc of [:X1, X2:], ExtREAL ;

      per cases by XXREAL_0: 14;

        suppose er in REAL ;

        then

        reconsider r = er as Real;

        

         A1: ( chi (r,E, [:X1, X2:])) = (r (#) ( chi (E, [:X1, X2:]))) by Th1;

        

         A2: ( chi (E, [:X1, X2:])) is XX12 -measurable by MESFUNC2: 29;

        

         A3: ( dom ( chi (E, [:X1, X2:]))) = XX12 by FUNCT_2:def 1;

        

         A4: ( dom (( chi (E, [:X1, X2:])) | E)) = (( dom ( chi (E, [:X1, X2:]))) /\ E) by RELAT_1: 61

        .= ( [:X1, X2:] /\ E) by FUNCT_2:def 1

        .= E by XBOOLE_1: 28;

        

         A5: (( chi (E, [:X1, X2:])) | E) is nonnegative by MESFUNC5: 15;

        E = (( dom ( chi (E, [:X1, X2:]))) /\ E) by A3, XBOOLE_1: 28;

        then

         A6: (( chi (E, [:X1, X2:])) | E) is E -measurable by MESFUNC2: 29, MESFUNC5: 42;

        ( Integral1 (M1,(( chi (r,E, [:X1, X2:])) | E))) = ( Integral1 (M1,(r (#) C))) by A1, MESFUN11: 2

        .= (r (#) ( Integral1 (M1,C))) by A4, A5, A6, Th78

        .= (r (#) ( Integral1 (M1,( chi (E, [:X1, X2:]))))) by Th79

        .= ( Integral1 (M1,(r (#) ( chi (E, [:X1, X2:]))))) by A2, A3, Th78;

        hence ( Integral1 (M1,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (er,E, [:X1, X2:])))) by Th1;

        ( Integral2 (M2,(( chi (r,E, [:X1, X2:])) | E))) = ( Integral2 (M2,(r (#) C))) by A1, MESFUN11: 2

        .= (r (#) ( Integral2 (M2,C))) by A4, A5, A6, Th78

        .= (r (#) ( Integral2 (M2,( chi (E, [:X1, X2:]))))) by Th79

        .= ( Integral2 (M2,(r (#) ( chi (E, [:X1, X2:]))))) by A2, A3, Th78;

        hence ( Integral2 (M2,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (er,E, [:X1, X2:])))) by Th1;

      end;

        suppose er = +infty ;

        then ( chi (er,E, [:X1, X2:])) = ( Xchi (E, [:X1, X2:])) by Th2;

        hence ( Integral1 (M1,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (er,E, [:X1, X2:])))) & ( Integral2 (M2,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (er,E, [:X1, X2:])))) by Th80;

      end;

        suppose

         d0: er = -infty ;

        reconsider XX12 = [:X1, X2:] as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

        reconsider XE = (( Xchi (E, [:X1, X2:])) | E) as PartFunc of [:X1, X2:], ExtREAL ;

        

         d3: ( Xchi (E, [:X1, X2:])) is XX12 -measurable by MEASUR10: 32;

        

         e2: XE is nonnegative by MESFUNC5: 15;

        

         d4: ( dom ( Xchi (E, [:X1, X2:]))) = XX12 by FUNCT_2:def 1;

        then

         e1: ( dom XE) = E by RELAT_1: 62;

        then E = (( dom ( Xchi (E, [:X1, X2:]))) /\ E) by RELAT_1: 61;

        then

         e3: XE is E -measurable by MESFUNC5: 42;

        

         d1: ( chi (er,E, [:X1, X2:])) = ( - ( Xchi (E, [:X1, X2:]))) by d0, Th2

        .= (( - 1) (#) ( Xchi (E, [:X1, X2:]))) by MESFUNC2: 9;

        ( Integral1 (M1,( chi (er,E, [:X1, X2:])))) = (( - 1) (#) ( Integral1 (M1,( Xchi (E, [:X1, X2:]))))) by d1, d3, d4, Th78

        .= (( - 1) (#) ( Integral1 (M1,(( Xchi (E, [:X1, X2:])) | E)))) by Th80

        .= ( Integral1 (M1,(( - 1) (#) XE))) by e1, e2, e3, Th78

        .= ( Integral1 (M1,(( chi (er,E, [:X1, X2:])) | E))) by d1, MESFUN11: 2;

        hence ( Integral1 (M1,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral1 (M1,( chi (er,E, [:X1, X2:]))));

        ( Integral2 (M2,( chi (er,E, [:X1, X2:])))) = (( - 1) (#) ( Integral2 (M2,( Xchi (E, [:X1, X2:]))))) by d1, d3, d4, Th78

        .= (( - 1) (#) ( Integral2 (M2,(( Xchi (E, [:X1, X2:])) | E)))) by Th80

        .= ( Integral2 (M2,(( - 1) (#) XE))) by e1, e2, e3, Th78

        .= ( Integral2 (M2,(( chi (er,E, [:X1, X2:])) | E))) by d1, MESFUN11: 2;

        hence ( Integral2 (M2,(( chi (er,E, [:X1, X2:])) | E))) = ( Integral2 (M2,( chi (er,E, [:X1, X2:]))));

      end;

    end;

    theorem :: MESFUN12:82

    

     Th82: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite & M2 is sigma_finite holds ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) = ( Integral (M2,( Integral1 (M1,( chi (E, [:X1, X2:])))))) & ( Integral (( Prod_Measure (M1,M2)),(( chi (E, [:X1, X2:])) | E))) = ( Integral (M2,( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))))) & ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) = ( Integral (M1,( Integral2 (M2,( chi (E, [:X1, X2:])))))) & ( Integral (( Prod_Measure (M1,M2)),(( chi (E, [:X1, X2:])) | E))) = ( Integral (M1,( Integral2 (M2,(( chi (E, [:X1, X2:])) | E)))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite;

      ( X-vol (E,M1)) = ( Integral1 (M1,( chi (E, [:X1, X2:])))) by A1, Th64;

      hence

       A4: ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) = ( Integral (M2,( Integral1 (M1,( chi (E, [:X1, X2:])))))) by A1, A2, Th77;

      

       A5: ( Integral (( Prod_Measure (M1,M2)),(( chi (E, [:X1, X2:])) | E))) = (( Prod_Measure (M1,M2)) . E) by MESFUNC9: 14

      .= ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) by MESFUNC9: 14;

      hence ( Integral (( Prod_Measure (M1,M2)),(( chi (E, [:X1, X2:])) | E))) = ( Integral (M2,( Integral1 (M1,(( chi (E, [:X1, X2:])) | E))))) by A4, Th79;

      ( Y-vol (E,M2)) = ( Integral2 (M2,( chi (E, [:X1, X2:])))) by A2, Th65;

      hence ( Integral (( Prod_Measure (M1,M2)),( chi (E, [:X1, X2:])))) = ( Integral (M1,( Integral2 (M2,( chi (E, [:X1, X2:])))))) by A1, A2, Th77;

      hence ( Integral (( Prod_Measure (M1,M2)),(( chi (E, [:X1, X2:])) | E))) = ( Integral (M1,( Integral2 (M2,(( chi (E, [:X1, X2:])) | E))))) by A5, Th79;

    end;

    theorem :: MESFUN12:83

    

     Th83: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real st M1 is sigma_finite & M2 is sigma_finite holds ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) = ( Integral (M2,( Integral1 (M1,( chi (r,E, [:X1, X2:])))))) & ( Integral (( Prod_Measure (M1,M2)),(( chi (r,E, [:X1, X2:])) | E))) = ( Integral (M2,( Integral1 (M1,(( chi (r,E, [:X1, X2:])) | E))))) & ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) = ( Integral (M1,( Integral2 (M2,( chi (r,E, [:X1, X2:])))))) & ( Integral (( Prod_Measure (M1,M2)),(( chi (r,E, [:X1, X2:])) | E))) = ( Integral (M1,( Integral2 (M2,(( chi (r,E, [:X1, X2:])) | E)))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, E be Element of ( sigma ( measurable_rectangles (S1,S2))), r be Real;

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      set M = ( Prod_Measure (M1,M2));

      reconsider XX12 = [:X1, X2:] as Element of S by MEASURE1: 7;

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      

       A3: ( chi (r,E, [:X1, X2:])) = (r (#) ( chi (E, [:X1, X2:]))) by Th1;

      

       A4: ( chi (E, [:X1, X2:])) is_simple_func_in S by Th12;

      

       A5: ( chi (E, [:X1, X2:])) is XX12 -measurable by Th12, MESFUNC2: 34;

      

       A6: ( dom ( chi (E, [:X1, X2:]))) = XX12 by FUNCT_2:def 1;

      

       A7: ( Integral1 (M1,( chi (E, [:X1, X2:])))) = ( X-vol (E,M1)) by A1, Th64;

      

       A8: ( X-vol (E,M1)) is XX2 -measurable by A1, MEASUR11:def 14;

      

       A9: ( dom ( Integral1 (M1,( chi (E, [:X1, X2:]))))) = XX2 by FUNCT_2:def 1;

      

       A10: ( Integral (M,( chi (r,E, [:X1, X2:])))) = ( Integral (M,(r (#) ( chi (E, [:X1, X2:]))))) by Th1

      .= (r * ( integral' (M,( chi (E, [:X1, X2:]))))) by Th12, MESFUN11: 59

      .= (r * ( Integral (M,( chi (E, [:X1, X2:]))))) by A4, MESFUNC5: 89;

      

      then

       A14: ( Integral (M,( chi (r,E, [:X1, X2:])))) = (r * ( Integral (M2,( Integral1 (M1,( chi (E, [:X1, X2:]))))))) by A1, A2, Th82

      .= ( Integral (M2,(r (#) ( X-vol (E,M1))))) by A7, A8, A9, Lm1;

      hence ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) = ( Integral (M2,( Integral1 (M1,( chi (r,E, [:X1, X2:])))))) by A3, A5, A6, A7, Th78;

      reconsider C = (( chi (E, [:X1, X2:])) | E) as PartFunc of [:X1, X2:], ExtREAL ;

      

       A11: ( dom C) = E by A6, RELAT_1: 62;

      

       A12: (( chi (r,E, [:X1, X2:])) | E) = ((r (#) ( chi (E, [:X1, X2:]))) | E) by Th1

      .= (r (#) C) by MESFUN11: 2;

      

       A13: ( Integral (M2,( Integral1 (M1,(( chi (r,E, [:X1, X2:])) | E))))) = ( Integral (M2,( Integral1 (M1,( chi (r,E, [:X1, X2:])))))) by Th81

      .= ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) by A3, A5, A6, A7, A14, Th78;

      C is E -measurable by A4, MESFUNC2: 34, MESFUNC5: 34;

      

      then

       A15: ( Integral (M,(( chi (r,E, [:X1, X2:])) | E))) = (r * ( Integral (M,C))) by A11, A12, Lm1, MESFUNC5: 15

      .= (r * (( Prod_Measure (M1,M2)) . E)) by MESFUNC9: 14

      .= (r * ( Integral (M,( chi (E, [:X1, X2:]))))) by MESFUNC9: 14

      .= ( Integral (M,(r (#) ( chi (E, [:X1, X2:]))))) by A4, A6, Lm1, MESFUNC2: 34;

      hence ( Integral (( Prod_Measure (M1,M2)),(( chi (r,E, [:X1, X2:])) | E))) = ( Integral (M2,( Integral1 (M1,(( chi (r,E, [:X1, X2:])) | E))))) by A13, Th1;

      

       B7: ( Integral2 (M2,( chi (E, [:X1, X2:])))) = ( Y-vol (E,M2)) by A2, Th65;

      

       B8: ( Y-vol (E,M2)) is XX1 -measurable by A2, MEASUR11:def 13;

      

       B9: ( dom ( Integral2 (M2,( chi (E, [:X1, X2:]))))) = XX1 by FUNCT_2:def 1;

      

       B14: ( Integral (M,( chi (r,E, [:X1, X2:])))) = (r * ( Integral (M1,( Integral2 (M2,( chi (E, [:X1, X2:]))))))) by A1, A2, A10, Th82

      .= ( Integral (M1,(r (#) ( Y-vol (E,M2))))) by B7, B8, B9, Lm1;

      hence ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) = ( Integral (M1,( Integral2 (M2,( chi (r,E, [:X1, X2:])))))) by A3, A5, A6, B7, Th78;

      ( Integral (M1,( Integral2 (M2,(( chi (r,E, [:X1, X2:])) | E))))) = ( Integral (M1,( Integral2 (M2,( chi (r,E, [:X1, X2:])))))) by Th81

      .= ( Integral (( Prod_Measure (M1,M2)),( chi (r,E, [:X1, X2:])))) by A3, A5, A6, B7, B14, Th78;

      hence ( Integral (( Prod_Measure (M1,M2)),(( chi (r,E, [:X1, X2:])) | E))) = ( Integral (M1,( Integral2 (M2,(( chi (r,E, [:X1, X2:])) | E))))) by A15, Th1;

    end;

    

     Lm13: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be non empty PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) & (f is nonnegative or f is nonpositive) & A = ( dom f) holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) & ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be non empty PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite and

       A3: f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) and

       A4: (f is nonnegative or f is nonpositive) and

       A5: A = ( dom f);

      

       A6: f is A -measurable by A3, MESFUNC2: 34;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      set M = ( Prod_Measure (M1,M2));

      reconsider XX12 = [:X1, X2:] as Element of S by MEASURE1: 7;

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      consider E be non empty Finite_Sep_Sequence of S, a be FinSequence of ExtREAL , r be FinSequence of REAL such that

       A7: (E,a) are_Re-presentation_of f & for n be Nat holds (a . n) = (r . n) & (f | (E . n)) = (( chi ((r . n),(E . n), [:X1, X2:])) | (E . n)) & ((E . n) = {} implies (r . n) = 0 ) by A3, Th5;

      defpred P[ Nat] means ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | $1)))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | $1))))))));

      

       A8: P[ 0 ]

      proof

        reconsider E0 = {} as Element of S by MEASURE1: 7;

        reconsider E01 = {} as Element of S1 by MEASURE1: 7;

        (M . E0) = 0 by VALUED_0:def 19;

        then

         A9: ( Integral (M,(f | ( union ( rng (E | 0 )))))) = 0 by A3, A5, MESFUNC2: 34, ZFMISC_1: 2, MESFUNC5: 94;

        

         A10: for y be Element of X2 st y in ( dom ( Integral1 (M1,(f | ( union ( rng (E | 0 ))))))) holds (( Integral1 (M1,(f | ( union ( rng (E | 0 )))))) . y) = 0

        proof

          let y be Element of X2;

          assume y in ( dom ( Integral1 (M1,(f | ( union ( rng (E | 0 )))))));

          (( Integral1 (M1,(f | ( union ( rng (E | 0 )))))) . y) = ( Integral (M1,( ProjPMap2 ((f | ( union ( rng (E | 0 )))),y)))) by Def7;

          then

           A11: (( Integral1 (M1,(f | ( union ( rng (E | 0 )))))) . y) = ( Integral (M1,(( ProjPMap2 (f,y)) | ( Y-section (E0,y))))) by Th34, ZFMISC_1: 2;

          

           A12: (M1 . E01) = 0 by VALUED_0:def 19;

          ( dom ( ProjPMap2 (f,y))) = ( Y-section (( dom f),y)) by Def4;

          then

           A13: ( dom ( ProjPMap2 (f,y))) = ( Measurable-Y-section (A,y)) by A5, MEASUR11:def 7;

          E0 = ( {} [:X1, X2:]);

          then (( Integral1 (M1,(f | ( union ( rng (E | 0 )))))) . y) = ( Integral (M1,(( ProjPMap2 (f,y)) | E01))) by A11, MEASUR11: 24;

          hence (( Integral1 (M1,(f | ( union ( rng (E | 0 )))))) . y) = 0 by A5, A6, A12, A13, Th47, MESFUNC5: 94;

        end;

        ( dom ( Integral1 (M1,(f | ( union ( rng (E | 0 ))))))) = XX2 by FUNCT_2:def 1;

        hence thesis by A9, A10, Th57;

      end;

      

       A14: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A15: P[n];

        per cases ;

          suppose n >= ( len E);

          then (E | n) = E & (E | (n + 1)) = E by FINSEQ_1: 58, NAT_1: 12;

          hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | (n + 1))))))))) by A15;

        end;

          suppose n < ( len E);

          ( Union (E | n)) is Element of S;

          then

          reconsider En = ( union ( rng (E | n))) as Element of S by CARD_3:def 4;

          reconsider En1 = (E . (n + 1)) as Element of S;

          

           A16: En misses En1 & ( union ( rng (E | (n + 1)))) = (En \/ En1) by NAT_1: 16, MEASUR11: 1, MEASUR11: 3;

          set CH = ( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:]));

          

           A17: ( Integral (M,(CH | (E . (n + 1))))) = ( Integral (M2,( Integral1 (M1,(CH | (E . (n + 1))))))) by A1, A2, Th83;

          

           A18: ( dom ( Integral1 (M1,(f | En)))) = XX2 & ( dom ( Integral1 (M1,(f | En1)))) = XX2 by FUNCT_2:def 1;

          

           A19: ( Integral1 (M1,(f | En))) is XX2 -measurable & ( Integral1 (M1,(f | En1))) is XX2 -measurable by A1, A4, A5, A6, Th68;

          

           A20: (( Integral1 (M1,(f | En))) | XX2) = ( Integral1 (M1,(f | En))) & (( Integral1 (M1,(f | En1))) | XX2) = ( Integral1 (M1,(f | En1)));

          ( Integral (M,(f | En1))) = ( Integral (M,(( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])) | (E . (n + 1))))) by A7;

          then

           A21: ( Integral (M,(f | En1))) = ( Integral (M2,( Integral1 (M1,(f | En1))))) by A7, A17;

          per cases by A4;

            suppose

             A22: f is nonnegative;

            then

             A23: ( Integral1 (M1,(f | En))) is nonnegative & ( Integral1 (M1,(f | En1))) is nonnegative by A5, A6, Th66;

            then

            reconsider I1 = ( Integral1 (M1,(f | En))), I2 = ( Integral1 (M1,(f | En1))) as without-infty Function of X2, ExtREAL ;

            (I1 + I2) = (( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1))));

            then

             A24: ( dom (( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1))))) = XX2 by FUNCT_2:def 1;

            ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = (( Integral (M,(f | En))) + ( Integral (M,(f | En1)))) by A3, A5, A22, A16, MESFUNC2: 34, MESFUNC5: 91;

            then ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M2,(( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1)))))) by A15, A18, A19, A20, A21, A23, A24, Th21;

            hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | (n + 1))))))))) by A5, A6, A16, A22, Lm11;

          end;

            suppose

             A25: f is nonpositive;

            then

             A26: ( Integral1 (M1,(f | En))) is nonpositive & ( Integral1 (M1,(f | En1))) is nonpositive by A5, A6, Th67;

            then

            reconsider I1 = ( Integral1 (M1,(f | En))), I2 = ( Integral1 (M1,(f | En1))) as without+infty Function of X2, ExtREAL ;

            (I1 + I2) = (( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1))));

            then

             A27: ( dom (( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1))))) = XX2 by FUNCT_2:def 1;

            ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = (( Integral (M,(f | En))) + ( Integral (M,(f | En1)))) by A3, A5, A16, A25, MESFUNC2: 34, MESFUN11: 62;

            then ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M2,(( Integral1 (M1,(f | En))) + ( Integral1 (M1,(f | En1)))))) by A15, A18, A19, A20, A21, A26, A27, Th22;

            hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | (n + 1))))))))) by A5, A6, A16, A25, Lm12;

          end;

        end;

      end;

      

       A28: ( union ( rng E)) = ( dom f) by A7, MESFUNC3:def 1;

      for n be Nat holds P[n] from NAT_1:sch 2( A8, A14);

      then ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | ( len E))))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | ( len E)))))))));

      then ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng E))))) = ( Integral (M2,( Integral1 (M1,(f | ( union ( rng (E | ( len E))))))))) by FINSEQ_1: 58;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) by A28, FINSEQ_1: 58;

      defpred P[ Nat] means ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | $1)))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | $1))))))));

      

       A8: P[ 0 ]

      proof

        reconsider E0 = {} as Element of S by MEASURE1: 7;

        reconsider E01 = {} as Element of S2 by MEASURE1: 7;

        (M . E0) = 0 by VALUED_0:def 19;

        then

         A9: ( Integral (M,(f | ( union ( rng (E | 0 )))))) = 0 by A3, A5, MESFUNC2: 34, ZFMISC_1: 2, MESFUNC5: 94;

        

         A10: for x be Element of X1 st x in ( dom ( Integral2 (M2,(f | ( union ( rng (E | 0 ))))))) holds (( Integral2 (M2,(f | ( union ( rng (E | 0 )))))) . x) = 0

        proof

          let x be Element of X1;

          assume x in ( dom ( Integral2 (M2,(f | ( union ( rng (E | 0 )))))));

          (( Integral2 (M2,(f | ( union ( rng (E | 0 )))))) . x) = ( Integral (M2,( ProjPMap1 ((f | ( union ( rng (E | 0 )))),x)))) by Def8;

          then

           A11: (( Integral2 (M2,(f | ( union ( rng (E | 0 )))))) . x) = ( Integral (M2,(( ProjPMap1 (f,x)) | ( X-section (E0,x))))) by Th34, ZFMISC_1: 2;

          

           A12: (M2 . E01) = 0 by VALUED_0:def 19;

          ( dom ( ProjPMap1 (f,x))) = ( X-section (( dom f),x)) by Def3;

          then

           A13: ( dom ( ProjPMap1 (f,x))) = ( Measurable-X-section (A,x)) by A5, MEASUR11:def 6;

          E0 = ( {} [:X1, X2:]);

          then (( Integral2 (M2,(f | ( union ( rng (E | 0 )))))) . x) = ( Integral (M2,(( ProjPMap1 (f,x)) | E01))) by A11, MEASUR11: 24;

          hence (( Integral2 (M2,(f | ( union ( rng (E | 0 )))))) . x) = 0 by A5, A6, A12, A13, Th47, MESFUNC5: 94;

        end;

        ( dom ( Integral2 (M2,(f | ( union ( rng (E | 0 ))))))) = XX1 by FUNCT_2:def 1;

        hence thesis by A9, A10, Th57;

      end;

      

       A14: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A15: P[n];

        per cases ;

          suppose n >= ( len E);

          then (E | n) = E & (E | (n + 1)) = E by FINSEQ_1: 58, NAT_1: 12;

          hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | (n + 1))))))))) by A15;

        end;

          suppose n < ( len E);

          ( Union (E | n)) is Element of S;

          then

          reconsider En = ( union ( rng (E | n))) as Element of S by CARD_3:def 4;

          reconsider En1 = (E . (n + 1)) as Element of S;

          

           A16: En misses En1 & ( union ( rng (E | (n + 1)))) = (En \/ En1) by NAT_1: 16, MEASUR11: 1, MEASUR11: 3;

          set CH = ( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:]));

          

           A17: ( Integral (M,(CH | (E . (n + 1))))) = ( Integral (M1,( Integral2 (M2,(CH | (E . (n + 1))))))) by A1, A2, Th83;

          

           A18: ( dom ( Integral2 (M2,(f | En)))) = XX1 & ( dom ( Integral2 (M2,(f | En1)))) = XX1 by FUNCT_2:def 1;

          

           A19: ( Integral2 (M2,(f | En))) is XX1 -measurable & ( Integral2 (M2,(f | En1))) is XX1 -measurable by A2, A4, A5, A6, Th69;

          

           A20: (( Integral2 (M2,(f | En))) | XX1) = ( Integral2 (M2,(f | En))) & (( Integral2 (M2,(f | En1))) | XX1) = ( Integral2 (M2,(f | En1)));

          ( Integral (M,(f | En1))) = ( Integral (M,(( chi ((r . (n + 1)),(E . (n + 1)), [:X1, X2:])) | (E . (n + 1))))) by A7;

          then

           A21: ( Integral (M,(f | En1))) = ( Integral (M1,( Integral2 (M2,(f | En1))))) by A7, A17;

          per cases by A4;

            suppose

             A22: f is nonnegative;

            then

             A23: ( Integral2 (M2,(f | En))) is nonnegative & ( Integral2 (M2,(f | En1))) is nonnegative by A5, A6, Th66;

            then

            reconsider I1 = ( Integral2 (M2,(f | En))), I2 = ( Integral2 (M2,(f | En1))) as without-infty Function of X1, ExtREAL ;

            (I1 + I2) = (( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1))));

            then

             A24: ( dom (( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1))))) = XX1 by FUNCT_2:def 1;

            ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = (( Integral (M,(f | En))) + ( Integral (M,(f | En1)))) by A3, A5, A22, A16, MESFUNC2: 34, MESFUNC5: 91;

            then ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M1,(( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1)))))) by A15, A18, A19, A20, A21, A23, A24, Th21;

            hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | (n + 1))))))))) by A5, A6, A16, A22, Lm11;

          end;

            suppose

             A25: f is nonpositive;

            then

             A26: ( Integral2 (M2,(f | En))) is nonpositive & ( Integral2 (M2,(f | En1))) is nonpositive by A5, A6, Th67;

            then

            reconsider I1 = ( Integral2 (M2,(f | En))), I2 = ( Integral2 (M2,(f | En1))) as without+infty Function of X1, ExtREAL ;

            (I1 + I2) = (( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1))));

            then

             A27: ( dom (( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1))))) = XX1 by FUNCT_2:def 1;

            ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = (( Integral (M,(f | En))) + ( Integral (M,(f | En1)))) by A3, A5, A16, A25, MESFUNC2: 34, MESFUN11: 62;

            then ( Integral (M,(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M1,(( Integral2 (M2,(f | En))) + ( Integral2 (M2,(f | En1)))))) by A15, A18, A19, A20, A21, A26, A27, Th22;

            hence ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | (n + 1))))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | (n + 1))))))))) by A5, A6, A16, A25, Lm12;

          end;

        end;

      end;

      

       A28: ( union ( rng E)) = ( dom f) by A7, MESFUNC3:def 1;

      for n be Nat holds P[n] from NAT_1:sch 2( A8, A14);

      then ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng (E | ( len E))))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | ( len E)))))))));

      then ( Integral (( Prod_Measure (M1,M2)),(f | ( union ( rng E))))) = ( Integral (M1,( Integral2 (M2,(f | ( union ( rng (E | ( len E))))))))) by FINSEQ_1: 58;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))) by A28, FINSEQ_1: 58;

    end;

    

     Lm14: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be empty PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))) holds (M1 is sigma_finite implies ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f))))) & (M2 is sigma_finite implies ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be empty PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      reconsider EMP = {} as Element of ( sigma ( measurable_rectangles (S1,S2))) by MEASURE1: 7;

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      

       A2: f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) by Th19;

      

       A3: f is EMP -measurable by Th19, MESFUNC2: 34;

      

       A5: for x be object st x in ( dom f) holds 0 <= (f . x);

      then

       A6: f is nonnegative by SUPINF_2: 52;

      

       A4: ( dom f) = EMP;

      then ( integral' (( Prod_Measure (M1,M2)),f)) = 0 by MESFUNC5:def 14;

      then

       A7: ( Integral (( Prod_Measure (M1,M2)),f)) = 0 by A2, A6, MESFUNC5: 89;

      

       A8: ( dom ( Integral1 (M1,f))) = XX2 & ( dom ( Integral2 (M2,f))) = XX1 by FUNCT_2:def 1;

      

       A10: ( Integral1 (M1,f)) is nonnegative & ( Integral2 (M2,f)) is nonnegative by A3, A4, A6, Th66;

      hereby

        assume M1 is sigma_finite;

        then

         A9: ( Integral1 (M1,f)) is XX2 -measurable by A3, A5, Th59, SUPINF_2: 52;

        for y be Element of X2 st y in ( dom ( Integral1 (M1,f))) holds (( Integral1 (M1,f)) . y) = 0

        proof

          let y be Element of X2;

          assume y in ( dom ( Integral1 (M1,f)));

          

           A11: ( ProjPMap2 (f,y)) is_simple_func_in S1 & ( ProjPMap2 (f,y)) is nonnegative by A6, A2, Th31, Th32;

          ( dom f) = ( {} [:X1, X2:]);

          

          then ( dom ( ProjPMap2 (f,y))) = ( Y-section (( {} [:X1, X2:]),y)) by Def4

          .= {} by MEASUR11: 24;

          then ( integral' (M1,( ProjPMap2 (f,y)))) = 0 by MESFUNC5:def 14;

          then ( Integral (M1,( ProjPMap2 (f,y)))) = 0 by A11, MESFUNC5: 89;

          hence (( Integral1 (M1,f)) . y) = 0 by Def7;

        end;

        then ( integral+ (M2,( Integral1 (M1,f)))) = 0 by A8, A9, MESFUNC5: 87;

        hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) by A7, A8, A9, A10, MESFUNC5: 88;

      end;

      assume M2 is sigma_finite;

      then

       B9: ( Integral2 (M2,f)) is XX1 -measurable by A3, A5, Th60, SUPINF_2: 52;

      for x be Element of X1 st x in ( dom ( Integral2 (M2,f))) holds (( Integral2 (M2,f)) . x) = 0

      proof

        let x be Element of X1;

        assume x in ( dom ( Integral2 (M2,f)));

        

         B11: ( ProjPMap1 (f,x)) is_simple_func_in S2 & ( ProjPMap1 (f,x)) is nonnegative by A6, A2, Th31, Th32;

        ( dom f) = ( {} [:X1, X2:]);

        

        then ( dom ( ProjPMap1 (f,x))) = ( X-section (( {} [:X1, X2:]),x)) by Def3

        .= {} by MEASUR11: 24;

        then ( integral' (M2,( ProjPMap1 (f,x)))) = 0 by MESFUNC5:def 14;

        then ( Integral (M2,( ProjPMap1 (f,x)))) = 0 by B11, MESFUNC5: 89;

        hence (( Integral2 (M2,f)) . x) = 0 by Def8;

      end;

      then ( integral+ (M1,( Integral2 (M2,f)))) = 0 by A8, B9, MESFUNC5: 87;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))) by A7, A8, B9, A10, MESFUNC5: 88;

    end;

    

     Lm15: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2))) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) & (f is nonnegative or f is nonpositive) & A = ( dom f) holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) & ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, f be PartFunc of [:X1, X2:], ExtREAL , A be Element of ( sigma ( measurable_rectangles (S1,S2)));

      assume

       a1: M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in ( sigma ( measurable_rectangles (S1,S2))) & (f is nonnegative or f is nonpositive) & A = ( dom f);

      per cases ;

        suppose f is non empty;

        hence thesis by a1, Lm13;

      end;

        suppose f is empty;

        hence thesis by a1, Lm14;

      end;

    end;

    

     Lm16: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonnegative & A = ( dom f) & f is A -measurable holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite and

       A3: f is nonnegative and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider EX1 = {} as Element of S1 by MEASURE1: 7;

      ( Integral (( Prod_Measure (M1,M2)),f)) = ( integral+ (( Prod_Measure (M1,M2)),f)) by A3, A4, A5, MESFUNC5: 88;

      then

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL , K be ExtREAL_sequence such that

       A6: (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) <= ((F . m) . z)) & (for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z)) & for n be Nat holds (K . n) = ( integral' (( Prod_Measure (M1,M2)),(F . n))) and K is convergent and

       A7: ( Integral (( Prod_Measure (M1,M2)),f)) = ( lim K) by A3, A4, A5, MESFUNC5:def 15;

      ( dom (F . 0 )) = ( dom f) by A6;

      then

       A8: ( dom ( lim F)) = ( dom f) by MESFUNC8:def 9;

      for z be Element of [:X1, X2:] st z in ( dom ( lim F)) holds (( lim F) . z) = (f . z)

      proof

        let z be Element of [:X1, X2:];

        assume

         A9: z in ( dom ( lim F));

        

        hence (( lim F) . z) = ( lim (F # z)) by MESFUNC8:def 9

        .= (f . z) by A9, A8, A6;

      end;

      then

       A10: ( lim F) = f by A8, PARTFUN1: 5;

      deffunc G( Nat) = ( Integral1 (M1,(F . $1)));

      consider G be Functional_Sequence of X2, ExtREAL such that

       A11: for n be Nat holds (G . n) = G(n) from SEQFUNC:sch 1;

      

       A12: for n be Nat, y be Element of X2 holds ( dom ( ProjPMap2 ((F . n),y))) = ( Measurable-Y-section (A,y)) & ( ProjPMap2 ((F . n),y)) is ( Measurable-Y-section (A,y)) -measurable & ( ProjPMap2 ((F . n),y)) is nonnegative

      proof

        let n be Nat, y be Element of X2;

        

         A13: ( dom (F . n)) = A by A4, A6;

        then ( dom ( ProjPMap2 ((F . n),y))) = ( Y-section (A,y)) by Def4;

        hence ( dom ( ProjPMap2 ((F . n),y))) = ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

        (F . n) is A -measurable by A6, MESFUNC2: 34;

        hence ( ProjPMap2 ((F . n),y)) is ( Measurable-Y-section (A,y)) -measurable by A13, Th47;

        (F . n) is nonnegative by A6;

        hence ( ProjPMap2 ((F . n),y)) is nonnegative by Th32;

      end;

      

       A14: for n be Nat holds ( dom (G . n)) = XX2 & (G . n) is XX2 -measurable & (G . n) is nonnegative

      proof

        let n be Nat;

        

         A15: (G . n) = ( Integral1 (M1,(F . n))) by A11;

        hence ( dom (G . n)) = XX2 by FUNCT_2:def 1;

        ( dom (F . n)) = A & (F . n) is A -measurable by A4, A6, MESFUNC2: 34;

        hence (G . n) is XX2 -measurable by A1, A15, A6, Th59;

        now

          let y be object;

          assume y in ( dom (G . n));

          then

          reconsider y1 = y as Element of X2;

          ((G . n) . y) = ( Integral (M1,( ProjPMap2 ((F . n),y1)))) & ( ProjPMap2 ((F . n),y1)) is ( Measurable-Y-section (A,y1)) -measurable & ( dom ( ProjPMap2 ((F . n),y1))) = ( Measurable-Y-section (A,y1)) by A12, A15, Def7;

          hence ((G . n) . y) >= 0 by A12, MESFUNC5: 90;

        end;

        hence (G . n) is nonnegative by SUPINF_2: 52;

      end;

      

       A16: for y be Element of X2, n,m be Nat st n <= m holds for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (( ProjPMap2 ((F . n),y)) . x) <= (( ProjPMap2 ((F . m),y)) . x)

      proof

        let y be Element of X2, n,m be Nat;

        assume

         A17: n <= m;

        hereby

          let x be Element of X1;

          assume x in ( Measurable-Y-section (A,y));

          then x in ( Y-section (A,y)) by MEASUR11:def 7;

          then x in ( Y-section (( dom (F . n)),y)) by A4, A6;

          then x in { x where x be Element of X1 : [x, y] in ( dom (F . n)) } by MEASUR11:def 5;

          then

           A18: ex x1 be Element of X1 st x1 = x & [x1, y] in ( dom (F . n));

          then

           A19: [x, y] in ( dom f) by A6;

          then [x, y] in ( dom (F . m)) by A6;

          then (( ProjPMap2 ((F . n),y)) . x) = ((F . n) . (x,y)) & (( ProjPMap2 ((F . m),y)) . x) = ((F . m) . (x,y)) by A18, Def4;

          hence (( ProjPMap2 ((F . n),y)) . x) <= (( ProjPMap2 ((F . m),y)) . x) by A6, A17, A19;

        end;

      end;

      

       A20: for n,m be Nat st n <= m holds for y be Element of X2 st y in XX2 holds ((G . n) . y) <= ((G . m) . y)

      proof

        let n,m be Nat;

        assume

         A21: n <= m;

        hereby

          let y be Element of X2;

          assume y in XX2;

          

           A22: ( dom ( ProjPMap2 ((F . n),y))) = ( Measurable-Y-section (A,y)) & ( dom ( ProjPMap2 ((F . m),y))) = ( Measurable-Y-section (A,y)) & ( ProjPMap2 ((F . n),y)) is ( Measurable-Y-section (A,y)) -measurable & ( ProjPMap2 ((F . m),y)) is ( Measurable-Y-section (A,y)) -measurable & ( ProjPMap2 ((F . n),y)) is nonnegative & ( ProjPMap2 ((F . m),y)) is nonnegative by A12;

          for x be Element of X1 st x in ( dom ( ProjPMap2 ((F . n),y))) holds (( ProjPMap2 ((F . n),y)) . x) <= (( ProjPMap2 ((F . m),y)) . x) by A16, A21, A22;

          then ( integral+ (M1,( ProjPMap2 ((F . n),y)))) <= ( integral+ (M1,( ProjPMap2 ((F . m),y)))) by A22, MESFUNC5: 85;

          then ( Integral (M1,( ProjPMap2 ((F . n),y)))) <= ( integral+ (M1,( ProjPMap2 ((F . m),y)))) by A22, MESFUNC5: 88;

          then

           A23: ( Integral (M1,( ProjPMap2 ((F . n),y)))) <= ( Integral (M1,( ProjPMap2 ((F . m),y)))) by A22, MESFUNC5: 88;

          ((G . n) . y) = (( Integral1 (M1,(F . n))) . y) by A11;

          then

           A24: ((G . n) . y) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by Def7;

          ((G . m) . y) = (( Integral1 (M1,(F . m))) . y) by A11;

          hence ((G . n) . y) <= ((G . m) . y) by A23, A24, Def7;

        end;

      end;

      

       A25: for y be Element of X2 st y in XX2 holds (G # y) is convergent & ( lim (G # y)) = ( Integral (M1,( ProjPMap2 (f,y))))

      proof

        let y be Element of X2;

        assume y in XX2;

        defpred P2[ Element of NAT , object] means $2 = ( ProjPMap2 ((F . $1),y));

        

         A26: for n be Element of NAT holds ex f be Element of ( PFuncs (X1, ExtREAL )) st P2[n, f]

        proof

          let n be Element of NAT ;

          reconsider f = ( ProjPMap2 ((F . n),y)) as Element of ( PFuncs (X1, ExtREAL )) by PARTFUN1: 45;

          take f;

          thus thesis;

        end;

        consider FX be sequence of ( PFuncs (X1, ExtREAL )) such that

         A27: for n be Element of NAT holds P2[n, (FX . n)] from FUNCT_2:sch 3( A26);

        

         A28: for n be Nat holds ( dom (FX . n)) = ( Measurable-Y-section (A,y))

        proof

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          then (FX . n) = ( ProjPMap2 ((F . n),y)) by A27;

          then ( dom (FX . n)) = ( Y-section (( dom (F . n)),y)) by Def4;

          then ( dom (FX . n)) = ( Y-section (A,y)) by A4, A6;

          hence ( dom (FX . n)) = ( Measurable-Y-section (A,y)) by MEASUR11:def 7;

        end;

        for m,n be Nat holds ( dom (FX . m)) = ( dom (FX . n))

        proof

          let m,n be Nat;

          ( dom (FX . m)) = ( Measurable-Y-section (A,y)) by A28;

          hence ( dom (FX . m)) = ( dom (FX . n)) by A28;

        end;

        then

        reconsider FX as with_the_same_dom Functional_Sequence of X1, ExtREAL by MESFUNC8:def 2;

        

         A29: ( dom (FX . 0 )) = ( Measurable-Y-section (A,y)) by A28;

        

         A30: for n be Nat holds (FX . n) is ( Measurable-Y-section (A,y)) -measurable

        proof

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          then

           A31: (FX . n) = ( ProjPMap2 ((F . n),y)) by A27;

          (F . n) is_simple_func_in S by A6;

          hence (FX . n) is ( Measurable-Y-section (A,y)) -measurable by A31, Th31, MESFUNC2: 34;

        end;

        ( ProjPMap2 ((F . 0 ),y)) is nonnegative by A12;

        then

         A32: (FX . 0 ) is nonnegative by A27;

        

         A33: for n,m be Nat st n <= m holds for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds ((FX . n) . x) <= ((FX . m) . x)

        proof

          let n,m be Nat;

          assume

           A34: n <= m;

          n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;

          then (FX . n) = ( ProjPMap2 ((F . n),y)) & (FX . m) = ( ProjPMap2 ((F . m),y)) by A27;

          hence for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds ((FX . n) . x) <= ((FX . m) . x) by A16, A34;

        end;

        

         A36: ( dom ( ProjPMap2 (f,y))) = ( Y-section (A,y)) by A4, Def4;

        

         A37: for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (FX # x) is convergent & (( ProjPMap2 (f,y)) . x) = ( lim (FX # x))

        proof

          let x be Element of X1;

          reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

          assume x in ( Measurable-Y-section (A,y));

          then x in ( Y-section (A,y)) by MEASUR11:def 7;

          then

           A38: [x, y] in ( dom f) by A4, Th25;

          then

           A39: (F # z) is convergent & ( lim (F # z)) = (f . z) by A6;

          

           A40: for n be Element of NAT holds ((FX # x) . n) = ((F # z) . n)

          proof

            let n be Element of NAT ;

            

             A41: [x, y] in ( dom (F . n)) by A38, A6;

            ((FX # x) . n) = ((FX . n) . x) by MESFUNC5:def 13;

            then ((FX # x) . n) = (( ProjPMap2 ((F . n),y)) . x) by A27;

            then ((FX # x) . n) = ((F . n) . (x,y)) by A41, Def4;

            hence ((FX # x) . n) = ((F # z) . n) by MESFUNC5:def 13;

          end;

          hence (FX # x) is convergent by A39, FUNCT_2: 63;

          (( ProjPMap2 (f,y)) . x) = (f . (x,y)) by A38, Def4;

          hence (( ProjPMap2 (f,y)) . x) = ( lim (FX # x)) by A39, A40, FUNCT_2: 63;

        end;

        then for x be Element of X1 st x in ( Measurable-Y-section (A,y)) holds (FX # x) is convergent;

        then

        consider I be ExtREAL_sequence such that

         A42: (for n be Nat holds (I . n) = ( Integral (M1,(FX . n)))) & I is convergent & ( Integral (M1,( lim FX))) = ( lim I) by A29, A30, A32, A33, MESFUNC9: 52;

        

         A43: for n be Element of NAT holds ((G # y) . n) = (I . n)

        proof

          let n be Element of NAT ;

          ((G # y) . n) = ((G . n) . y) by MESFUNC5:def 13;

          then ((G # y) . n) = (( Integral1 (M1,(F . n))) . y) by A11;

          then ((G # y) . n) = ( Integral (M1,( ProjPMap2 ((F . n),y)))) by Def7;

          then ((G # y) . n) = ( Integral (M1,(FX . n))) by A27;

          hence ((G # y) . n) = (I . n) by A42;

        end;

        hence (G # y) is convergent by A42, FUNCT_2:def 8;

        

         A44: (G # y) = I by A43, FUNCT_2:def 8;

        

         A45: ( dom ( lim FX)) = ( Measurable-Y-section (A,y)) by A29, MESFUNC8:def 9;

        for x be Element of X1 st x in ( dom ( lim FX)) holds (( lim FX) . x) = (( ProjPMap2 (f,y)) . x)

        proof

          let x be Element of X1;

          assume

           A46: x in ( dom ( lim FX));

          then (( lim FX) . x) = ( lim (FX # x)) by MESFUNC8:def 9;

          hence (( lim FX) . x) = (( ProjPMap2 (f,y)) . x) by A37, A45, A46;

        end;

        hence ( lim (G # y)) = ( Integral (M1,( ProjPMap2 (f,y)))) by A44, A45, A36, A42, PARTFUN1: 5, MEASUR11:def 7;

      end;

      then

       A47: for y be Element of X2 st y in XX2 holds (G # y) is convergent;

      now

        let n,m be Nat;

        ( dom (G . n)) = XX2 & ( dom (G . m)) = XX2 by A14;

        hence ( dom (G . n)) = ( dom (G . m));

      end;

      then

       A48: G is with_the_same_dom by MESFUNC8:def 2;

      (G . 0 ) = ( Integral1 (M1,(F . 0 ))) by A11;

      then XX2 = ( dom (G . 0 )) by FUNCT_2:def 1;

      then

      consider J be ExtREAL_sequence such that

       A49: (for n be Nat holds (J . n) = ( Integral (M2,(G . n)))) & J is convergent & ( Integral (M2,( lim G))) = ( lim J) by A14, A20, A47, A48, MESFUNC9: 52;

      ( dom ( lim G)) = ( dom (G . 0 )) by MESFUNC8:def 9;

      then ( dom ( lim G)) = ( dom ( Integral1 (M1,(F . 0 )))) by A11;

      then ( dom ( lim G)) = XX2 by FUNCT_2:def 1;

      then

       A50: ( dom ( lim G)) = ( dom ( Integral1 (M1,( lim F)))) by FUNCT_2:def 1;

      for y be Element of X2 st y in ( dom ( lim G)) holds (( lim G) . y) = (( Integral1 (M1,( lim F))) . y)

      proof

        let y be Element of X2;

        assume y in ( dom ( lim G));

        then (( lim G) . y) = ( lim (G # y)) by MESFUNC8:def 9;

        then (( lim G) . y) = ( Integral (M1,( ProjPMap2 (f,y)))) by A25;

        hence (( lim G) . y) = (( Integral1 (M1,( lim F))) . y) by A10, Def7;

      end;

      then

       A51: ( lim G) = ( Integral1 (M1,( lim F))) by A50, PARTFUN1: 5;

      for n be Element of NAT holds (K . n) = (J . n)

      proof

        let n be Element of NAT ;

        

         A52: A = ( dom (F . n)) by A4, A6;

        

         A53: (F . n) is nonnegative & (F . n) is_simple_func_in S by A6;

        (K . n) = ( integral' (( Prod_Measure (M1,M2)),(F . n))) by A6;

        then (K . n) = ( Integral (( Prod_Measure (M1,M2)),(F . n))) by A53, MESFUNC5: 89;

        then (K . n) = ( Integral (M2,( Integral1 (M1,(F . n))))) by A1, A2, A52, A53, Lm15;

        then (K . n) = ( Integral (M2,(G . n))) by A11;

        hence (K . n) = (J . n) by A49;

      end;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) by A7, A10, A49, A51, FUNCT_2:def 8;

    end;

    

     Lm17: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonnegative & A = ( dom f) & f is A -measurable holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite and

       A3: f is nonnegative and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      set S = ( sigma ( measurable_rectangles (S1,S2)));

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider EX1 = {} as Element of S1 by MEASURE1: 7;

      ( Integral (( Prod_Measure (M1,M2)),f)) = ( integral+ (( Prod_Measure (M1,M2)),f)) by A3, A4, A5, MESFUNC5: 88;

      then

      consider F be Functional_Sequence of [:X1, X2:], ExtREAL , K be ExtREAL_sequence such that

       A6: (for n be Nat holds (F . n) is_simple_func_in S & ( dom (F . n)) = ( dom f)) & (for n be Nat holds (F . n) is nonnegative) & (for n,m be Nat st n <= m holds for z be Element of [:X1, X2:] st z in ( dom f) holds ((F . n) . z) <= ((F . m) . z)) & (for z be Element of [:X1, X2:] st z in ( dom f) holds (F # z) is convergent & ( lim (F # z)) = (f . z)) & for n be Nat holds (K . n) = ( integral' (( Prod_Measure (M1,M2)),(F . n))) and K is convergent and

       A7: ( Integral (( Prod_Measure (M1,M2)),f)) = ( lim K) by A3, A4, A5, MESFUNC5:def 15;

      ( dom (F . 0 )) = ( dom f) by A6;

      then

       A8: ( dom ( lim F)) = ( dom f) by MESFUNC8:def 9;

      for z be Element of [:X1, X2:] st z in ( dom ( lim F)) holds (( lim F) . z) = (f . z)

      proof

        let z be Element of [:X1, X2:];

        assume

         A9: z in ( dom ( lim F));

        

        hence (( lim F) . z) = ( lim (F # z)) by MESFUNC8:def 9

        .= (f . z) by A9, A8, A6;

      end;

      then

       A10: ( lim F) = f by A8, PARTFUN1: 5;

      deffunc G( Nat) = ( Integral2 (M2,(F . $1)));

      consider G be Functional_Sequence of X1, ExtREAL such that

       A11: for n be Nat holds (G . n) = G(n) from SEQFUNC:sch 1;

      

       A12: for n be Nat, x be Element of X1 holds ( dom ( ProjPMap1 ((F . n),x))) = ( Measurable-X-section (A,x)) & ( ProjPMap1 ((F . n),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap1 ((F . n),x)) is nonnegative

      proof

        let n be Nat, x be Element of X1;

        

         A13: ( dom (F . n)) = A by A4, A6;

        then ( dom ( ProjPMap1 ((F . n),x))) = ( X-section (A,x)) by Def3;

        hence ( dom ( ProjPMap1 ((F . n),x))) = ( Measurable-X-section (A,x)) by MEASUR11:def 6;

        (F . n) is A -measurable by A6, MESFUNC2: 34;

        hence ( ProjPMap1 ((F . n),x)) is ( Measurable-X-section (A,x)) -measurable by A13, Th47;

        (F . n) is nonnegative by A6;

        hence ( ProjPMap1 ((F . n),x)) is nonnegative by Th32;

      end;

      

       A14: for n be Nat holds ( dom (G . n)) = XX1 & (G . n) is XX1 -measurable & (G . n) is nonnegative

      proof

        let n be Nat;

        

         A15: (G . n) = ( Integral2 (M2,(F . n))) by A11;

        hence ( dom (G . n)) = XX1 by FUNCT_2:def 1;

        ( dom (F . n)) = A & (F . n) is A -measurable by A4, A6, MESFUNC2: 34;

        hence (G . n) is XX1 -measurable by A2, A15, A6, Th60;

        now

          let x be object;

          assume x in ( dom (G . n));

          then

          reconsider x1 = x as Element of X1;

          ((G . n) . x) = ( Integral (M2,( ProjPMap1 ((F . n),x1)))) & ( ProjPMap1 ((F . n),x1)) is ( Measurable-X-section (A,x1)) -measurable & ( dom ( ProjPMap1 ((F . n),x1))) = ( Measurable-X-section (A,x1)) by A12, A15, Def8;

          hence ((G . n) . x) >= 0 by A12, MESFUNC5: 90;

        end;

        hence (G . n) is nonnegative by SUPINF_2: 52;

      end;

      

       A16: for x be Element of X1, n,m be Nat st n <= m holds for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (( ProjPMap1 ((F . n),x)) . y) <= (( ProjPMap1 ((F . m),x)) . y)

      proof

        let x be Element of X1, n,m be Nat;

        assume

         A17: n <= m;

        hereby

          let y be Element of X2;

          assume y in ( Measurable-X-section (A,x));

          then y in ( X-section (A,x)) by MEASUR11:def 6;

          then y in ( X-section (( dom (F . n)),x)) by A4, A6;

          then y in { y where y be Element of X2 : [x, y] in ( dom (F . n)) } by MEASUR11:def 4;

          then

           A18: ex y1 be Element of X2 st y1 = y & [x, y1] in ( dom (F . n));

          then

           A19: [x, y] in ( dom f) by A6;

          then [x, y] in ( dom (F . m)) by A6;

          then (( ProjPMap1 ((F . n),x)) . y) = ((F . n) . (x,y)) & (( ProjPMap1 ((F . m),x)) . y) = ((F . m) . (x,y)) by A18, Def3;

          hence (( ProjPMap1 ((F . n),x)) . y) <= (( ProjPMap1 ((F . m),x)) . y) by A6, A17, A19;

        end;

      end;

      

       A20: for n,m be Nat st n <= m holds for x be Element of X1 st x in XX1 holds ((G . n) . x) <= ((G . m) . x)

      proof

        let n,m be Nat;

        assume

         A21: n <= m;

        hereby

          let x be Element of X1;

          assume x in XX1;

          

           A22: ( dom ( ProjPMap1 ((F . n),x))) = ( Measurable-X-section (A,x)) & ( dom ( ProjPMap1 ((F . m),x))) = ( Measurable-X-section (A,x)) & ( ProjPMap1 ((F . n),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap1 ((F . m),x)) is ( Measurable-X-section (A,x)) -measurable & ( ProjPMap1 ((F . n),x)) is nonnegative & ( ProjPMap1 ((F . m),x)) is nonnegative by A12;

          for y be Element of X2 st y in ( dom ( ProjPMap1 ((F . n),x))) holds (( ProjPMap1 ((F . n),x)) . y) <= (( ProjPMap1 ((F . m),x)) . y) by A16, A21, A22;

          then ( integral+ (M2,( ProjPMap1 ((F . n),x)))) <= ( integral+ (M2,( ProjPMap1 ((F . m),x)))) by A22, MESFUNC5: 85;

          then ( Integral (M2,( ProjPMap1 ((F . n),x)))) <= ( integral+ (M2,( ProjPMap1 ((F . m),x)))) by A22, MESFUNC5: 88;

          then

           A23: ( Integral (M2,( ProjPMap1 ((F . n),x)))) <= ( Integral (M2,( ProjPMap1 ((F . m),x)))) by A22, MESFUNC5: 88;

          ((G . n) . x) = (( Integral2 (M2,(F . n))) . x) by A11;

          then

           A24: ((G . n) . x) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by Def8;

          ((G . m) . x) = (( Integral2 (M2,(F . m))) . x) by A11;

          hence ((G . n) . x) <= ((G . m) . x) by A23, A24, Def8;

        end;

      end;

      

       A25: for x be Element of X1 st x in XX1 holds (G # x) is convergent & ( lim (G # x)) = ( Integral (M2,( ProjPMap1 (f,x))))

      proof

        let x be Element of X1;

        assume x in XX1;

        defpred P2[ Element of NAT , object] means $2 = ( ProjPMap1 ((F . $1),x));

        

         A26: for n be Element of NAT holds ex f be Element of ( PFuncs (X2, ExtREAL )) st P2[n, f]

        proof

          let n be Element of NAT ;

          reconsider f = ( ProjPMap1 ((F . n),x)) as Element of ( PFuncs (X2, ExtREAL )) by PARTFUN1: 45;

          take f;

          thus thesis;

        end;

        consider FX be sequence of ( PFuncs (X2, ExtREAL )) such that

         A27: for n be Element of NAT holds P2[n, (FX . n)] from FUNCT_2:sch 3( A26);

        

         A28: for n be Nat holds ( dom (FX . n)) = ( Measurable-X-section (A,x))

        proof

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          then (FX . n) = ( ProjPMap1 ((F . n),x)) by A27;

          then ( dom (FX . n)) = ( X-section (( dom (F . n)),x)) by Def3;

          then ( dom (FX . n)) = ( X-section (A,x)) by A4, A6;

          hence ( dom (FX . n)) = ( Measurable-X-section (A,x)) by MEASUR11:def 6;

        end;

        for m,n be Nat holds ( dom (FX . m)) = ( dom (FX . n))

        proof

          let m,n be Nat;

          ( dom (FX . m)) = ( Measurable-X-section (A,x)) by A28;

          hence ( dom (FX . m)) = ( dom (FX . n)) by A28;

        end;

        then

        reconsider FX as with_the_same_dom Functional_Sequence of X2, ExtREAL by MESFUNC8:def 2;

        

         A29: ( dom (FX . 0 )) = ( Measurable-X-section (A,x)) by A28;

        

         A30: for n be Nat holds (FX . n) is ( Measurable-X-section (A,x)) -measurable

        proof

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          then

           A31: (FX . n) = ( ProjPMap1 ((F . n),x)) by A27;

          (F . n) is_simple_func_in S by A6;

          hence (FX . n) is ( Measurable-X-section (A,x)) -measurable by A31, Th31, MESFUNC2: 34;

        end;

        ( ProjPMap1 ((F . 0 ),x)) is nonnegative by A12;

        then

         A32: (FX . 0 ) is nonnegative by A27;

        

         A33: for n,m be Nat st n <= m holds for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds ((FX . n) . y) <= ((FX . m) . y)

        proof

          let n,m be Nat;

          assume

           A34: n <= m;

          n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;

          then (FX . n) = ( ProjPMap1 ((F . n),x)) & (FX . m) = ( ProjPMap1 ((F . m),x)) by A27;

          hence for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds ((FX . n) . y) <= ((FX . m) . y) by A16, A34;

        end;

        

         A36: ( dom ( ProjPMap1 (f,x))) = ( X-section (A,x)) by A4, Def3;

        

         A37: for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (FX # y) is convergent & (( ProjPMap1 (f,x)) . y) = ( lim (FX # y))

        proof

          let y be Element of X2;

          reconsider z = [x, y] as Element of [:X1, X2:] by ZFMISC_1:def 2;

          assume y in ( Measurable-X-section (A,x));

          then y in ( X-section (A,x)) by MEASUR11:def 6;

          then

           A38: [x, y] in ( dom f) by A4, Th25;

          then

           A39: (F # z) is convergent & ( lim (F # z)) = (f . z) by A6;

          

           A40: for n be Element of NAT holds ((FX # y) . n) = ((F # z) . n)

          proof

            let n be Element of NAT ;

            

             A41: [x, y] in ( dom (F . n)) by A38, A6;

            ((FX # y) . n) = ((FX . n) . y) by MESFUNC5:def 13;

            then ((FX # y) . n) = (( ProjPMap1 ((F . n),x)) . y) by A27;

            then ((FX # y) . n) = ((F . n) . (x,y)) by A41, Def3;

            hence ((FX # y) . n) = ((F # z) . n) by MESFUNC5:def 13;

          end;

          hence (FX # y) is convergent by A39, FUNCT_2: 63;

          (( ProjPMap1 (f,x)) . y) = (f . (x,y)) by A38, Def3;

          hence (( ProjPMap1 (f,x)) . y) = ( lim (FX # y)) by A39, A40, FUNCT_2: 63;

        end;

        then for y be Element of X2 st y in ( Measurable-X-section (A,x)) holds (FX # y) is convergent;

        then

        consider I be ExtREAL_sequence such that

         A42: (for n be Nat holds (I . n) = ( Integral (M2,(FX . n)))) & I is convergent & ( Integral (M2,( lim FX))) = ( lim I) by A29, A30, A32, A33, MESFUNC9: 52;

        

         A43: for n be Element of NAT holds ((G # x) . n) = (I . n)

        proof

          let n be Element of NAT ;

          ((G # x) . n) = ((G . n) . x) by MESFUNC5:def 13;

          then ((G # x) . n) = (( Integral2 (M2,(F . n))) . x) by A11;

          then ((G # x) . n) = ( Integral (M2,( ProjPMap1 ((F . n),x)))) by Def8;

          then ((G # x) . n) = ( Integral (M2,(FX . n))) by A27;

          hence ((G # x) . n) = (I . n) by A42;

        end;

        hence (G # x) is convergent by A42, FUNCT_2:def 8;

        

         A44: (G # x) = I by A43, FUNCT_2:def 8;

        

         A45: ( dom ( lim FX)) = ( Measurable-X-section (A,x)) by A29, MESFUNC8:def 9;

        for y be Element of X2 st y in ( dom ( lim FX)) holds (( lim FX) . y) = (( ProjPMap1 (f,x)) . y)

        proof

          let y be Element of X2;

          assume

           A46: y in ( dom ( lim FX));

          then (( lim FX) . y) = ( lim (FX # y)) by MESFUNC8:def 9;

          hence (( lim FX) . y) = (( ProjPMap1 (f,x)) . y) by A37, A45, A46;

        end;

        hence ( lim (G # x)) = ( Integral (M2,( ProjPMap1 (f,x)))) by A44, A45, A36, A42, PARTFUN1: 5, MEASUR11:def 6;

      end;

      then

       A47: for x be Element of X1 st x in XX1 holds (G # x) is convergent;

      now

        let n,m be Nat;

        ( dom (G . n)) = XX1 & ( dom (G . m)) = XX1 by A14;

        hence ( dom (G . n)) = ( dom (G . m));

      end;

      then

       A48: G is with_the_same_dom by MESFUNC8:def 2;

      (G . 0 ) = ( Integral2 (M2,(F . 0 ))) by A11;

      then XX1 = ( dom (G . 0 )) by FUNCT_2:def 1;

      then

      consider J be ExtREAL_sequence such that

       A49: (for n be Nat holds (J . n) = ( Integral (M1,(G . n)))) & J is convergent & ( Integral (M1,( lim G))) = ( lim J) by A14, A20, A47, A48, MESFUNC9: 52;

      ( dom ( lim G)) = ( dom (G . 0 )) by MESFUNC8:def 9;

      then ( dom ( lim G)) = ( dom ( Integral2 (M2,(F . 0 )))) by A11;

      then ( dom ( lim G)) = XX1 by FUNCT_2:def 1;

      then

       A50: ( dom ( lim G)) = ( dom ( Integral2 (M2,( lim F)))) by FUNCT_2:def 1;

      for x be Element of X1 st x in ( dom ( lim G)) holds (( lim G) . x) = (( Integral2 (M2,( lim F))) . x)

      proof

        let x be Element of X1;

        assume x in ( dom ( lim G));

        then (( lim G) . x) = ( lim (G # x)) by MESFUNC8:def 9;

        then (( lim G) . x) = ( Integral (M2,( ProjPMap1 (f,x)))) by A25;

        hence (( lim G) . x) = (( Integral2 (M2,( lim F))) . x) by A10, Def8;

      end;

      then

       A51: ( lim G) = ( Integral2 (M2,( lim F))) by A50, PARTFUN1: 5;

      for n be Element of NAT holds (K . n) = (J . n)

      proof

        let n be Element of NAT ;

        

         A52: A = ( dom (F . n)) by A4, A6;

        

         A53: (F . n) is nonnegative & (F . n) is_simple_func_in S by A6;

        (K . n) = ( integral' (( Prod_Measure (M1,M2)),(F . n))) by A6;

        then (K . n) = ( Integral (( Prod_Measure (M1,M2)),(F . n))) by A53, MESFUNC5: 89;

        then (K . n) = ( Integral (M1,( Integral2 (M2,(F . n))))) by A1, A2, A52, A53, Lm15;

        then (K . n) = ( Integral (M1,(G . n))) by A11;

        hence (K . n) = (J . n) by A49;

      end;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))) by A7, A10, A49, A51, FUNCT_2:def 8;

    end;

    

     Lm18: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = ( dom f) & f is A -measurable holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite and

       A3: f is nonpositive and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      reconsider XX2 = X2 as Element of S2 by MEASURE1: 7;

      reconsider g = ( - f) as nonnegative PartFunc of [:X1, X2:], ExtREAL by A3;

      

       A6: g = (( - 1) (#) f) by MESFUNC2: 9;

      ( - ( Integral1 (M1,f))) = (( - 1) (#) ( Integral1 (M1,f))) & ( dom ( Integral1 (M1,f))) = XX2 & ( Integral1 (M1,f)) is nonpositive & ( Integral1 (M1,f)) is XX2 -measurable by A1, A3, A4, A5, Th67, Th59, MESFUNC2: 9, FUNCT_2:def 1;

      then

       A7: ( Integral (M2,( - ( Integral1 (M1,f))))) = (( - 1) * ( Integral (M2,( Integral1 (M1,f))))) by Lm2;

      A = ( dom g) & g is A -measurable by A4, A5, MESFUNC1:def 7, MEASUR11: 63;

      then ( Integral (( Prod_Measure (M1,M2)),g)) = ( Integral (M2,( Integral1 (M1,g)))) by A1, A2, Lm16;

      then (( - 1) * ( Integral (( Prod_Measure (M1,M2)),f))) = ( Integral (M2,( Integral1 (M1,g)))) by A3, A4, A5, A6, Lm2;

      then (( - 1) * ( Integral (( Prod_Measure (M1,M2)),f))) = ( Integral (M2,( - ( Integral1 (M1,f))))) by A4, A5, Th73;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) by A7, XXREAL_3: 68;

    end;

    

     Lm19: for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = ( dom f) & f is A -measurable holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f))))

    proof

      let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL ;

      assume that

       A1: M1 is sigma_finite and

       A2: M2 is sigma_finite and

       A3: f is nonpositive and

       A4: A = ( dom f) and

       A5: f is A -measurable;

      reconsider XX1 = X1 as Element of S1 by MEASURE1: 7;

      reconsider g = ( - f) as nonnegative PartFunc of [:X1, X2:], ExtREAL by A3;

      

       A6: g = (( - 1) (#) f) by MESFUNC2: 9;

      ( - ( Integral2 (M2,f))) = (( - 1) (#) ( Integral2 (M2,f))) & ( dom ( Integral2 (M2,f))) = XX1 & ( Integral2 (M2,f)) is nonpositive & ( Integral2 (M2,f)) is XX1 -measurable by A2, A3, A4, A5, Th67, Th60, MESFUNC2: 9, FUNCT_2:def 1;

      then

       A7: ( Integral (M1,( - ( Integral2 (M2,f))))) = (( - 1) * ( Integral (M1,( Integral2 (M2,f))))) by Lm2;

      A = ( dom g) & g is A -measurable by A4, A5, MESFUNC1:def 7, MEASUR11: 63;

      then ( Integral (( Prod_Measure (M1,M2)),g)) = ( Integral (M1,( Integral2 (M2,g)))) by A1, A2, Lm17;

      then (( - 1) * ( Integral (( Prod_Measure (M1,M2)),f))) = ( Integral (M1,( Integral2 (M2,g)))) by A3, A4, A5, A6, Lm2;

      then (( - 1) * ( Integral (( Prod_Measure (M1,M2)),f))) = ( Integral (M1,( - ( Integral2 (M2,f))))) by A4, A5, Th73;

      hence ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))) by A7, XXREAL_3: 68;

    end;

    theorem :: MESFUN12:84

    for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2, M1 be sigma_Measure of S1, M2 be sigma_Measure of S2, A be Element of ( sigma ( measurable_rectangles (S1,S2))), f be PartFunc of [:X1, X2:], ExtREAL st M1 is sigma_finite & M2 is sigma_finite & (f is nonnegative or f is nonpositive) & A = ( dom f) & f is A -measurable holds ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M2,( Integral1 (M1,f)))) & ( Integral (( Prod_Measure (M1,M2)),f)) = ( Integral (M1,( Integral2 (M2,f)))) by Lm16, Lm18, Lm17, Lm19;