 reserve X, Y for set, A for Ordinal;
 reserve z,z1,z2 for Complex;
 reserve r,r1,r2 for Real;
 reserve q,q1,q2 for Rational;
 reserve i,i1,i2 for Integer;
 reserve n,n1,n2 for Nat;

theorem
  for X being complex-membered set st z <> 1 & z <> -1 & not 0 in X
  holds multRel(X,z) is asymmetric
proof
  let X be complex-membered set;
  assume A0: z <> 1 & z <> -1 & not 0 in X;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    assume A1: x in field multRel(X,z) & y in field multRel(X,z) &
      [x,y] in multRel(X,z) & [y,x] in multRel(X,z);
    then A2: a in X & b in X by MMLQUER2:4;
    then reconsider a,b as Complex;
    b = z * a & a = z * b by A1, Th42;
    then b = z*(z*b);
    then b*b" = (z*z*b)*b"
      .= z*z*(b*b");
    then 1 = z*z*(b*b") by A0, A2, XCMPLX_0:def 7;
    then 1 = z*z*1 by A0, A2, XCMPLX_0:def 7;
    hence contradiction by A0, XCMPLX_1:182;
  end;
  hence thesis by RELAT_2:def 13, RELAT_2:def 5;
end;
