reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th29:
  A c= B |^.. k & n > 0 implies A |^.. n c= B |^.. k
proof
  assume that
A1: A c= B |^.. k and
A2: n > 0;
  let x be object;
  assume x in A |^.. n;
  then consider m such that
A3: m >= n and
A4: x in A |^ m by Th2;
  A |^ m c= B |^.. k by A1,A2,A3,Th28;
  hence thesis by A4;
end;
