
theorem
for m,n be non zero Nat, X be non-empty m-element FinSequence,
 P be SemialgebraFamily of ProdFinSeq X
 st n <= m holds
  P.n is semialgebra_of_sets of CarProduct SubFin(X,n)
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence,
    P be SemialgebraFamily of ProdFinSeq X;
    assume
A1: n <= m;

    1 <= n by NAT_1:14; then
    n in Seg m by A1; then
    P.n is semialgebra_of_sets of (ProdFinSeq X).n by MEASUR10:def 2;
    hence P.n is semialgebra_of_sets of CarProduct SubFin(X,n) by A1,Th4;
end;
