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 Th21:
  A c= C |^.. m & B c= C |^.. n implies A ^^ B c= C |^.. (m + n)
proof
  assume that
A1: A c= C |^.. m and
A2: B c= C |^.. n;
  thus for x being object holds x in A ^^ B implies x in C |^.. (m + n)
  proof let x be object;
    assume x in A ^^ B;
    then consider a, b such that
A3: a in A and
A4: b in B and
A5: x = a ^ b by FLANG_1:def 1;
    a ^ b in (C |^.. m) ^^ (C |^.. n) by A1,A2,A3,A4,FLANG_1:def 1;
    hence thesis by A5,Th18;
  end;
end;
