theorem Th17:
  for c being Complex
  for f being complex-valued FinSequence holds (c(#)f)^2 = (c^2) (#) (f^2)
  proof
    let c be Complex;
    let f be complex-valued FinSequence;
A1: dom ((c(#)f)^2) = dom (c(#)f) by VALUED_1:11
    .= dom f by VALUED_1:def 5;
A2: dom ((c^2) (#) (f^2)) = dom (f^2) by VALUED_1:def 5
    .= dom f by VALUED_1:11;
    now
      let x be object;
      assume x in dom ((c(#)f)^2);
      thus ((c(#)f)^2).x = ((c(#)f).x)^2 by VALUED_1:11
      .= (c*(f.x))^2 by VALUED_1:6
      .= c^2*(f.x)^2
      .= (c^2) * (f^2).x by VALUED_1:11
      .= ((c^2) (#) (f^2)).x by VALUED_1:6;
    end;
    hence thesis by A1,A2;
  end;
