
theorem Th12:
for m,n,k be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X st k <= n & n <= m holds
 ElmFin(S,k) = ElmFin(SubFin(S,n),k)
proof
    let m,n,k be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X;
    assume that
A1:  k <= n and
A2:  n <= m;

A3: ElmFin(S,k) = S.k by A1,A2,Def7,XXREAL_0:2;
    1 <= k by NAT_1:14; then
A4: k in Seg n by A1;

    SubFin(S,n) = S|n by A2,Def6; then
    ElmFin(SubFin(S,n),k) = (S|n).k by A1,Def7;
    hence ElmFin(S,k) = ElmFin(SubFin(S,n),k) by A3,A4,FUNCT_1:49;
end;
