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 Th12:
  A ^^ B = {} iff A = {} or B = {}
proof
  thus A ^^ B = {} implies A = {} or B = {}
  proof
    assume that
A1: A ^^ B = {} and
A2: A <> {} and
A3: B <> {};
    consider a such that
A4: a in A by A2,SUBSET_1:4;
    consider b such that
A5: b in B by A3,SUBSET_1:4;
    a ^ b in A ^^ B by A4,A5,Def1;
    hence contradiction by A1;
  end;
  assume
A6: A = {} or B = {};
  not ex x being object st x in A ^^ B
  proof
    given x being object such that
A7: x in A ^^ B;
    ex a, b st a in A & b in B & x = a ^ b by A7,Def1;
    hence contradiction by A6;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
