
theorem Th14:
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 SubFin(S,k) = SubFin(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;

    SubFin(SubFin(S,n),k) = (SubFin(S,n))|k by A1,Def6; then
    SubFin(SubFin(S,n),k) = (S|n)|k by A2,Def6; then
    SubFin(SubFin(S,n),k) = S|k by A1,FINSEQ_1:82;
    hence thesis by A2,A1,Def6,XXREAL_0:2;
end;
