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 Th5:
  m <= n implies A |^.. n c= A |^.. m
proof
  assume
A1: m <= n;
    let x be object;
    assume x in A |^.. n;
    then consider k such that
A2: n <= k and
A3: x in A |^ k by Th2;
    m <= k by A1,A2,XXREAL_0:2;
    hence x in A |^.. m by A3,Th2;
end;
