
theorem Th42:
  Prod_Field(L-Field 2) = sigma measurable_rectangles(L-Field,L-Field)
& measurable_rectangles(L-Field,L-Field)
   c= sigma measurable_rectangles(L-Field,L-Field)
& ( the set of all [:A,B:] where A is Element of Borel_Sets,
      B is Element of Borel_Sets )
   c= measurable_rectangles(L-Field,L-Field)
& { [:I,J:] where I,J is Subset of REAL: I is Interval & J is Interval }
   c= ( the set of all [:A,B:]
          where A is Element of Borel_Sets, B is Element of Borel_Sets )
proof
    set X = Seg 2 --> REAL;

A1: len SubFin(X,1) = 1 by CARD_1:def 7;
A2: 1 in Seg 2 & 1 in Seg 1 & 2 in Seg 2;

    consider S1 be SigmaField of CarProduct SubFin(X,1) such that
A3: S1 = (ProdSigmaFldFinSeq(L-Field 2)).1
  & (ProdSigmaFldFinSeq(L-Field 2)).(1+1)
      = sigma measurable_rectangles(S1,ElmFin(L-Field 2,1+1))
        by MEASUR13:def 11;

    SubFin(X,1) = X|1 by MEASUR13:def 5; then
    SubFin(X,1).1 = X.1 by A2,FUNCT_1:49; then
    SubFin(X,1) = <*REAL*> by A1,A2,FUNCOP_1:7,FINSEQ_1:40; then
a4: CarProduct SubFin(X,1) = <*REAL*>.1 by MEASUR13:def 3;

    (ProdSigmaFldFinSeq(L-Field 2)).1 = (L-Field 2).1 by MEASUR13:def 11; then
A5: S1 = L-Field by A3,A2,FUNCOP_1:7;

    ElmFin(X,2) = X.2 by MEASUR13:def 1; then
A6: ElmFin(X,2) = REAL by A2,FUNCOP_1:7;

    ElmFin(L-Field 2,2) = (L-Field 2).2 by MEASUR13:def 7;
    hence Prod_Field(L-Field 2) = sigma measurable_rectangles(L-Field,L-Field)
      by A3,a4,A5,A6,A2,FUNCOP_1:7;

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

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

    now let z be object;
     assume z in XY; then
     consider A,B be Element of Borel_Sets such that
A7:  z=[:A,B:];
     A in L-Field & B in L-Field by MEASUR12:75;
     hence z in measurable_rectangles( L-Field,L-Field) by A7;
    end;
    hence XY c= measurable_rectangles ( L-Field,L-Field);

    set IJ = { [:I,J:] where I,J is Subset of REAL:
                 I is Interval & J is Interval};
    now let z be object;
     assume z in IJ; then
     consider I,J be Subset of REAL such that
A8:  z=[:I,J:] & I is Interval & J is Interval;

     I in Borel_Sets & J in Borel_Sets by A8,MEASUR10:5;
     hence z in XY by A8;
    end;
    hence IJ c= XY;
end;
