 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 <> 0
  holds multRel(X,z)~ = multRel(X,z")
proof
  let X be complex-membered set;
  assume A1: z <> 0;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A2: [x,y] in multRel(X,z");
      then [a,b] in multRel(X,z");
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Complex;
      [a,b] in multRel(X,z") by A2;
      then a in X & b in X & b = z" * a by Th42;
      then a in X & b in X & z" * z * a = z * b;
      then a in X & b in X & 1 * a = z * b by A1, XCMPLX_0:def 7;
      hence [y,x] in multRel(X,z) by Th42;
    end;
    assume A3: [y,x] in multRel(X,z);
    then [b,a] in multRel(X,z);
    then a in X & b in X by MMLQUER2:4;
    then reconsider a,b as Complex;
    [b,a] in multRel(X,z) by A3;
    then a in X & b in X & a = z * b by Th42;
    then a in X & b in X & z" * z * b = z" * a;
    then a in X & b in X & 1 * b = z" * a by A1, XCMPLX_0:def 7;
    hence [x,y] in multRel(X,z") by Th42;
  end;
  hence thesis by RELAT_1:def 7;
end;
