
theorem
  for F be commutative Skew-Field, x be Element of MultGroup (F),
  x1 be Element of F st x = x1 holds x" = x1"
proof
  let F be commutative Skew-Field, h be Element of MultGroup (F), hp be
  Element of F;
  assume
A1: h = hp;
  set hpd = hp";
  h in the carrier of MultGroup (F);
  then h in NonZero F by UNIROOTS:def 1;
  then not h in {0.F} by XBOOLE_0:def 5;
  then
A2: h <> 0.F by TARSKI:def 1;
  then hp*hpd = 1.F by A1,VECTSP_1:def 10;
  then hpd <> 0.F;
  then not hpd in {0.F} by TARSKI:def 1;
  then hpd in NonZero F by XBOOLE_0:def 5;
  then reconsider g = hpd as Element of MultGroup (F) by UNIROOTS:def 1;
A3: 1_F = 1_MultGroup F by UNIROOTS:17;
  g * h = hpd*hp by A1,UNIROOTS:16
    .= 1_(MultGroup (F)) by A1,A2,A3,VECTSP_1:def 10;
  hence thesis by GROUP_1:def 5;
end;
