
theorem Th44:
for n be non zero Nat holds
 Prod_Field(L-Field (n+1)) =
     sigma measurable_rectangles(Prod_Field(L-Field n),L-Field)
proof
    let n be non zero Nat;
    set X = Seg (n+1) --> REAL;
    set X1 = SubFin(X,n);
    set S1 = SubFin(L-Field(n+1),n);
    set S11 = ElmFin(L-Field(n+1),n+1);

A1: Prod_Field(L-Field(n+1))
     = sigma measurable_rectangles(Prod_Field(S1),S11) by MEASUR13:34;

A2: n < n+1 by NAT_1:13; then
A3: Seg(n+1) /\ Seg n = Seg n by FINSEQ_1:5,XBOOLE_1:28;

    SubFin(X,n) = X|n by A2,MEASUR13:def 5; then
A4: SubFin(X,n) = Seg n --> REAL by A3,FUNCOP_1:12;

    SubFin(L-Field(n+1),n) = (L-Field(n+1)) |n by A2,MEASUR13:def 6;  then
A5: SubFin(L-Field(n+1),n) = L-Field n by A3,FUNCOP_1:12;

    ElmFin(X,n+1) = X.(n+1) by MEASUR13:def 1; then
A6: ElmFin(X,n+1) = REAL by FINSEQ_1:4,FUNCOP_1:7;
    ElmFin(L-Field(n+1),n+1) = (L-Field(n+1)).(n+1) by MEASUR13:def 7;
    hence Prod_Field(L-Field(n+1))
     = sigma measurable_rectangles(Prod_Field(L-Field n),L-Field)
       by A1,A4,A5,A6,FINSEQ_1:4,FUNCOP_1:7;
end;
