 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 holds addRel(X,z)~ = addRel(X,-z)
proof
  let X be complex-membered set;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in addRel(X,-z);
      then [a,b] in addRel(X,-z);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Complex;
      [a,b] in addRel(X,-z) by A1;
      then a in X & b in X & b = -z + a by Th11;
      then a in X & b in X & a = z + b;
      hence [y,x] in addRel(X,z) by Th11;
    end;
    assume A2: [y,x] in addRel(X,z);
    then [b,a] in addRel(X,z);
    then a in X & b in X by MMLQUER2:4;
    then reconsider a,b as Complex;
    [b,a] in addRel(X,z) by A2;
    then a in X & b in X & a = z + b by Th11;
    then a in X & b in X & b = -z + a;
    hence [x,y] in addRel(X,-z) by Th11;
  end;
  hence thesis by RELAT_1:def 7;
end;
