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 Th15:
  <%>E in A ^^ B iff <%>E in A & <%>E in B
proof
  thus <%>E in A ^^ B implies <%>E in A & <%>E in B
  proof
    assume <%>E in A ^^ B;
    then ex a, b st a in A & b in B & a ^ b = <%>E by Def1;
    hence thesis by AFINSQ_1:30;
  end;
  assume <%>E in A & <%>E in B;
  then {} ^ <%>E in A ^^ B by Def1;
  hence thesis;
end;
