
theorem Th18:
for m,n,k be non zero Nat, X be non-empty m-element FinSequence,
  S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S st
  k <= n & n <= m holds SubFin(M,k) = SubFin(SubFin(M,n),k)
proof
    let m,n,k be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S;
    assume that
A1:  k <= n and
A2:  n <= m;

    SubFin(SubFin(M,n),k) = (SubFin(M,n))|k by A1,Def9; then
    SubFin(SubFin(M,n),k) = (M|n)|k by A2,Def9; then
    SubFin(SubFin(M,n),k) = M|k by A1,FINSEQ_1:82;
    hence SubFin(M,k) = SubFin(SubFin(M,n),k) by Def9,A1,A2,XXREAL_0:2;
end;
