
theorem Th8:
for m,n,k be non zero Nat, X be non-empty m-element FinSequence
 st k <= n & n <= m holds ElmFin(X,k) = ElmFin(SubFin(X,n),k)
proof
    let m,n,k be non zero Nat, X be non-empty m-element FinSequence;
    assume that
A1:  k <= n and
A2:  n <= m;

A3: ElmFin(X,k) = X.k by A1,A2,Def1,XXREAL_0:2;

    1 <= k by NAT_1:14; then
A4: k in Seg n by A1;

    SubFin(X,n) = X|n by A2,Def5; then
    ElmFin(SubFin(X,n),k) = (X|n).k by A1,Def1;
    hence ElmFin(X,k) = ElmFin(SubFin(X,n),k) by A3,A4,FUNCT_1:49;
end;
