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 Th45:
  a in C* & b in C* implies a ^ b in C*
proof
  assume that
A1: a in C* and
A2: b in C*;
  consider k such that
A3: a in C |^ k by A1,Th41;
  consider l such that
A4: b in C |^ l by A2,Th41;
  a ^ b in C |^ (k + l) by A3,A4,Th40;
  hence thesis by Th41;
end;
