 reserve A, B, C, A1, B1, C1, A2, B2, C2, C3 for Point of TOP-REAL 2,
  lambda, mu, nu, alpha, beta, gamma for Real,
  X, Y, Z for Subset of TOP-REAL 2;

theorem
  Th4: (lambda * A + mu * B) `1 = lambda * A `1 + mu * B `1 &
  (lambda * A + mu * B) `2 = lambda * A `2 + mu * B `2
proof
  (lambda * A + mu * B) `1 = (lambda * A) `1 + (mu * B) `1 by Th1
  .= lambda * A `1 + (mu * B) `1 by Th2;
  hence (lambda * A + mu * B) `1 = lambda * A `1 + mu * B `1 by Th2;
  (lambda * A + mu * B) `2 = (lambda * A) `2 + (mu * B) `2 by Th1
  .= lambda * A `2 + (mu * B) `2 by Th2;
  hence thesis by Th2;
end;
