
theorem Th21:
  for K be Field, K1 be Subfield of K, x, y be Element of K,
  x1, y1 be Element of K1 st x = x1 & x <> 0.K
  holds x" = x1"
  proof
    let K be Field,
    K1 be Subfield of K,
    x, y be Element of K,
    x1, y1 be Element of K1;
    set C = the carrier of K;
    set C1 = the carrier of K1;
    set hpd = x1";
    assume A1: x = x1 & x <> 0.K;
    then A2: x1 <> 0.K1 by EC_PF_1:def 1;
    C1 c= C by EC_PF_1:def 1;
    then reconsider g = hpd as Element of K;
    A3: x*g = x1*hpd by A1,Th18
    .= 1.K1 by A2,VECTSP_1:def 10;
    x*g = 1.K by A3,EC_PF_1:def 1;
    hence thesis by A1,VECTSP_1:def 10;
  end;
