
theorem Th30:
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 SubFin(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;

    set Xn = SubFin(X,n), Sn = SubFin(S,n), Mn = SubFin(M,n);

A3: Xn =X|n & Sn=S|n & Mn=M|n by A1,Def5,Def6,Def9;
A4: Seg n c= Seg m by A1,FINSEQ_1:5;
    for j be Nat st j in Seg n
     ex Xj be non empty set, Sj being SigmaField of Xj,
        Mj be sigma_Measure of Sj st Xj = Xn.j & Sj = Sn.j & Mj = Mn.j
      & Mj is sigma_finite
    proof
     let j be Nat;
     assume
A5:   j in Seg n; then
     consider Xj be non empty set, Sj being SigmaField of Xj,
        Mj be sigma_Measure of Sj such that
A6:   Xj = X.j & Sj = S.j & Mj = M.j & Mj is sigma_finite by A2,A4;
     take Xj,Sj,Mj;
     thus thesis by A3,A5,A6,FUNCT_1:49;
    end;
    hence Mn is sigma_finite;
end;
