reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem
  <%>E in A & m > n implies A |^ n c= A |^ m
proof
  assume that
A1: <%>E in A and
A2: m > n;
  consider k such that
A3: n + k = m by A2,NAT_1:10;
  <%>E in A |^ k by A1,Th30;
  then A |^ n c= A |^ k ^^ A |^ n by Th16;
  hence thesis by A3,Th33;
end;
