
theorem Th39:
for n be non zero Nat, X be non-empty (n+1)-element FinSequence,
 S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S,
 f be PartFunc of CarProduct X,ExtREAL,
 g be PartFunc of [:CarProduct SubFin(X,n),ElmFin(X,n+1):],ExtREAL
 st f = g holds FSqIntg(M,f).1 = f &
       FSqIntg(M,f).2 = Integral2(ElmFin(M,n+1),g)
proof
    let n be non zero Nat, X be non-empty (n+1)-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S,
    f be PartFunc of CarProduct X,ExtREAL,
    g be PartFunc of [:CarProduct SubFin(X,n),ElmFin(X,n+1):],ExtREAL;
    assume
A1:  f = g;
    thus FSqIntg(M,f).1 = f by Def17;
    1 < n+1 by NAT_1:13,14; then
    ex k be non zero Nat,
     g0 be PartFunc of [: CarProduct SubFin(X,k),ElmFin(X,k+1):],ExtREAL
     st k = n+1-1 & g0 = FSqIntg(M,f).1
     & FSqIntg(M,f).(1+1) = Integral2(ElmFin(M,k+1),g0) by Def17;
    hence FSqIntg(M,f).2 = Integral2(ElmFin(M,n+1),g) by A1,Def17;
end;
