
theorem Th31:
for m,n be non zero Nat, X be non-empty m-element FinSequence,
  S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S
  st n <= m & M is sigma_finite holds ElmFin(M,n) is sigma_finite
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S;
    assume that
A1:  n <= m and
A2:  M is sigma_finite;
A3: Seg n c= Seg m by A1,FINSEQ_1:5;
A4: n in Seg n by FINSEQ_1:3;
A5: ElmFin(X,n) = X.n & ElmFin(S,n) = S.n & ElmFin(M,n) = M.n
      by A1,Def1,Def7,Def10;

    ex Xi be non empty set, Fi being SigmaField of Xi,
     Mi be sigma_Measure of Fi st Xi = X.n & Fi = S.n & Mi=M.n
   & Mi is sigma_finite by A2,A3,A4;
    hence ElmFin(M,n) is sigma_finite by A5;
end;
