
theorem
  Prod_Field(L-Field 3)
   = sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
& measurable_rectangles(sigma measurable_rectangles(L-Field,L-Field),L-Field)
   c= sigma measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field)
& ( the set of all [:A,B,C:] where A is Element of Borel_Sets,
         B is Element of Borel_Sets, C is Element of Borel_Sets)
   c= measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field)
& { [:I,J,K:] where I,J,K is Subset of REAL:
       I is Interval & J is Interval & K is Interval }
   c= ( the set of all [:A,B,C:] where A is Element of Borel_Sets,
           B is Element of Borel_Sets, C is Element of Borel_Sets )
proof
    set X = Seg 3 --> REAL;
A1: len SubFin(X,1) = 1 by CARD_1:def 7;
A2: 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3;
A3: 1 in Seg 2 & 2 in Seg 2;
A4: 1 in Seg 1;

    consider S2 be SigmaField of CarProduct SubFin(X,2) such that
A5: S2 = (ProdSigmaFldFinSeq(L-Field 3)).2
  & (ProdSigmaFldFinSeq(L-Field 3)).(2+1)
     = sigma measurable_rectangles(S2,ElmFin(L-Field 3,2+1))
       by MEASUR13:def 11;
    consider S1 be SigmaField of CarProduct SubFin(X,1) such that
A6: S1 = (ProdSigmaFldFinSeq(L-Field 3)).1
  & (ProdSigmaFldFinSeq(L-Field 3)).(1+1)
     = sigma measurable_rectangles(S1,ElmFin(L-Field 3,1+1))
       by MEASUR13:def 11;

    SubFin(X,1) = X|1 by MEASUR13:def 5; then
    SubFin(X,1).1 = X.1 by A4,FUNCT_1:49; then
    SubFin(X,1) = <*REAL*> by A1,A2,FUNCOP_1:7,FINSEQ_1:40; then
a7: CarProduct SubFin(X,1) = <*REAL*>.1 by MEASUR13:def 3;

    SubFin(X,2) = X|2 by MEASUR13:def 5; then
    SubFin(X,2).1 = X.1 & SubFin(X,2).2 = X.2 by A3,FUNCT_1:49; then
A8: SubFin(X,2).1 = REAL & SubFin(X,2).2 = REAL by A2,FUNCOP_1:7; then
    (ProdFinSeq SubFin(X,2)).1 = REAL by MEASUR13:def 3; then
A9: (ProdFinSeq SubFin(X,2)).(1+1) = [:REAL,REAL:] by A8,MEASUR13:def 3;

    (ProdSigmaFldFinSeq(L-Field 3)).1 = (L-Field 3).1 by MEASUR13:def 11; then
A10: S1 = L-Field by A6,A2,FUNCOP_1:7;

    ElmFin(X,2) = X.2 by MEASUR13:def 1; then
A11: ElmFin(X,2) = REAL by A2,FUNCOP_1:7;
    ElmFin(L-Field 3,2) = (L-Field 3).2 by MEASUR13:def 7; then
A12: ElmFin(L-Field 3,2) = L-Field by A2,FUNCOP_1:7;

    ElmFin(X,3) = X.3 by MEASUR13:def 1; then
A13:ElmFin(X,3) = REAL by A2,FUNCOP_1:7;
    ElmFin(L-Field 3,3) = (L-Field 3).3 by MEASUR13:def 7;
    hence Prod_Field(L-Field 3)
     = sigma measurable_rectangles(
        sigma measurable_rectangles(L-Field,L-Field),L-Field)
          by A9,A11,A12,a7,A6,A10,A5,A13,A2,FUNCOP_1:7;

    thus measurable_rectangles(
     sigma measurable_rectangles(L-Field,L-Field),L-Field)
      c= sigma measurable_rectangles(
          sigma measurable_rectangles(L-Field,L-Field),L-Field)
            by PROB_1:def 9;

    set XYZ = the set of all [:A,B,C:] where A is Element of Borel_Sets,
                B is Element of Borel_Sets, C is Element of Borel_Sets;

    now let z be object;
     assume z in XYZ; then
     consider A,B,C be Element of Borel_Sets such that
A14: z = [:A,B,C:];
A15: A in L-Field & B in L-Field & C in L-Fieldby MEASUR12:75; then
A16: [:A,B:] in measurable_rectangles(L-Field,L-Field);

     measurable_rectangles(L-Field,L-Field)
      c= sigma measurable_rectangles(L-Field,L-Field) by PROB_1:def 9; then
     [:[:A,B:],C:] in measurable_rectangles(
          sigma measurable_rectangles(L-Field,L-Field),L-Field) by A15,A16;
     hence z in measurable_rectangles(
                 sigma measurable_rectangles(L-Field,L-Field),L-Field)
        by A14,ZFMISC_1:def 3;
    end;
    hence XYZ c= measurable_rectangles(
                  sigma measurable_rectangles(L-Field,L-Field),L-Field);

    set IJK = { [:I,J,K:] where I,J,K is Subset of REAL: I is Interval
                 & J is Interval & K is Interval};
    now let z be object;
     assume z in IJK; then
     consider I,J,K be Subset of REAL such that
A17: z = [:I,J,K:] & I is Interval & J is Interval & K is Interval;
     I in Borel_Sets & J in Borel_Sets & K in Borel_Sets by A17,MEASUR10:5;
     hence z in XYZ by A17;
    end;
    hence IJK c= XYZ;
end;
