
theorem Th19:
  for K be Field, K1 be Subfield of K, x be Element of K,
  x1 be Element of K1 st x = x1 holds -x = -x1
  proof
    let K be Field,
    K1 be Subfield of K,
    x be Element of K,
    x1 be Element of K1;
    set C = the carrier of K;
    set C1 = the carrier of K1;
    set hpd = -x1;
    assume A1: x = x1;
    C1 c= C by EC_PF_1:def 1;
    then reconsider g = hpd as Element of K;
    x + g = x1 + hpd by A1,Th17
    .= 0.K1 by VECTSP_1:19
    .= 0.K by EC_PF_1:def 1;
    hence thesis by VECTSP_1:16;
  end;
