
theorem Th8:
for m,n1,n2,k be non zero Nat, X be non-empty m-element FinSequence
 st k <= n1 & n1 <= n2 & n2 <= m holds
  SubFin(SubFin(X,n1),k) = SubFin(SubFin(X,n2),k)
& ElmFin(SubFin(X,n1),k) = ElmFin(SubFin(X,n2),k)
proof
    let m,n1,n2,k be non zero Nat, X be non-empty m-element FinSequence;
    assume that
A1: k <= n1 and
A2: n1 <= n2 and
A3: n2 <= m;

A4: SubFin(X,n1) = X|n1 & SubFin(X,n2) = X|n2
      by A2,A3,XXREAL_0:2,MEASUR13:def 5; then
A5: SubFin(SubFin(X,n2),k) = (X|n2)|k by A1,A2,XXREAL_0:2,MEASUR13:def 5;

A6: k <= n2 by A1,A2,XXREAL_0:2; then
    X|n1|k = X|k & X|n2|k = X|k by A1,FINSEQ_1:5,RELAT_1:74;
    hence SubFin(SubFin(X,n1),k) = SubFin(SubFin(X,n2),k)
      by A5,A1,A4,MEASUR13:def 5;

    1 <= k by NAT_1:14; then
A7: k in Seg n1 & k in Seg n2 by A1,A6;
    ElmFin(SubFin(X,n1),k) = (X|n1).k & ElmFin(SubFin(X,n2),k) = (X|n2).k
      by A1,A4,A2,XXREAL_0:2,MEASUR13:def 1; then
    ElmFin(SubFin(X,n1),k) = X.k & ElmFin(SubFin(X,n2),k) = X.k
      by A7,FUNCT_1:49;
    hence ElmFin(SubFin(X,n1),k) = ElmFin(SubFin(X,n2),k);
end;
