
theorem Th17:
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 ElmFin(M,k) = ElmFin(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;

A3: ElmFin(M,k) = M.k by A1,A2,Def10,XXREAL_0:2;

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

    SubFin(M,n) = M|n by A2,Def9; then
    ElmFin(SubFin(M,n),k) = (M|n).k by A1,Def10;
    hence ElmFin(M,k) = ElmFin(SubFin(M,n),k) by A3,A4,FUNCT_1:49;
end;
