
theorem
for n be non zero Nat holds
  Prod_Measure(L-Meas(n+1)) = Prod_Measure(Prod_Measure(L-Meas n),L-Meas)
proof
    let n be non zero Nat;

    set X = Seg(n+1) --> REAL, S = L-Field (n+1), m = L-Meas (n+1);
    set X1 = SubFin(X,n), S1 = SubFin(S,n), m1 = SubFin(m,n);
    set X11 = ElmFin(X,n+1), S11 = ElmFin(S,n+1), m11 = ElmFin(m,n+1);

A1: Prod_Measure(L-Meas(n+1)) = Prod_Measure(Prod_Measure(m1),m11)
      by MEASUR13:28;

A2: Seg n c= Seg (n+1) by NAT_1:12,FINSEQ_1:5;
    SubFin(X,n) = X|n by NAT_1:12,MEASUR13:def 5; then
    X1 = Seg (n+1) /\ Seg n--> REAL by FUNCOP_1:12; then
A3: X1 = Seg n--> REAL by A2,XBOOLE_1:28;

    SubFin(S,n) = S|n by NAT_1:12,MEASUR13:def 6; then
    S1 = Seg (n+1) /\ Seg n--> L-Field by FUNCOP_1:12; then
A4: S1 = L-Field n by A2,XBOOLE_1:28;
    SubFin(m,n) = m|n by NAT_1:12,MEASUR13:def 9; then
    m1 = Seg (n+1) /\ Seg n--> L-Meas by FUNCOP_1:12; then
A5: m1 = L-Meas n by A2,XBOOLE_1:28;

    ElmFin(X,n+1) = X.(n+1) by MEASUR13:def 1; then
A6: X11 = REAL by FUNCOP_1:7,FINSEQ_1:3;
    ElmFin(S,n+1) = S.(n+1) by MEASUR13:def 7; then
A7: S11 = L-Field by FUNCOP_1:7,FINSEQ_1:3;
    ElmFin(m,n+1) = m.(n+1) by MEASUR13:def 10;
    hence
    Prod_Measure(L-Meas(n+1)) = Prod_Measure(Prod_Measure(L-Meas n),L-Meas)
      by A7,A1,A4,A3,A5,A6,FUNCOP_1:7,FINSEQ_1:3;
end;
