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 Th3:
  n1 + n2 = 1 iff n1 = 1 & n2 = 0 or n1 = 0 & n2 = 1
proof
  thus n1 + n2 = 1 implies n1 = 1 & n2 = 0 or n1 = 0 & n2 = 1
  proof
    assume
A1: n1 + n2 = 1;
    now
A2:   n1 = 0 & n2 = 0 implies contradiction by A1;
      assume
A3:   n1 <= 1 & n2 <= 1;
      n1 = 1 & n2 = 1 implies contradiction by A1;
      hence thesis by A3,A2,NAT_1:25;
    end;
    hence thesis by A1,Th1;
  end;
  thus thesis;
end;
