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

    for i be Nat st i in Seg n holds (SubFin(S,n)).i is SigmaField of Y.i
    proof
     let i be Nat;
     assume
A3:   i in Seg n;
A4:  Seg n c= Seg m by A1,FINSEQ_1:5;

     (S|n).i = S.i by A3,FUNCT_1:49; then
A5:  (S|n).i is SigmaField of X.i by A3,A4,Def2;
     Y.i = X.i by A2,A3,FUNCT_1:49;
     hence thesis by A5,A1,Def6;
    end;
    hence SubFin(S,n) is sigmaFieldFamily of Y by Def2;
end;
