 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,0) = id X
proof
  let X be complex-membered set;
  now
    let x,y be object;
    hereby
      assume A1: [x,y] in addRel(X,0);
      reconsider z1=x,z2=y as set by TARSKI:1;
      [z1,z2] in addRel(X,0) by A1;
      then z1 in X & z2 in X by MMLQUER2:4;
      then reconsider z1,z2 as Complex;
      [z1,z2] in addRel(X,0) by A1;
      then z1 in X & z2 = 0 + z1 by Th11;
      hence x in X & x = y;
    end;
    assume A2: x in X & x = y;
    then reconsider z1=x,z2=y as Complex;
    z2 = 0 + z1 by A2;
    hence [x,y] in addRel(X,0) by A2, Th11;
  end;
  hence thesis by RELAT_1:def 10;
end;
