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 Th37:
  a in C |^.. m & b in C |^.. n implies a ^ b in C |^.. (m + n)
proof
  assume that
A1: a in C |^.. m and
A2: b in C |^.. n;
A3: (C |^.. m) ^^ (C |^.. n) c= C |^.. (m + n) by Th21;
  a ^ b in (C |^.. m) ^^ (C |^.. n) by A1,A2,FLANG_1:def 1;
  hence thesis by A3;
end;
