 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
  addRel(RAT,q) = the set of all [q1,q1+q] where q1
proof
  set S = the set of all [q1,q1+q] where q1;
  now
    let o be object;
    hereby
      assume A1: o in addRel(RAT,q);
      then consider x,y being object such that
        A2: o = [x,y] by RELAT_1:def 1;
      reconsider x,y as set by TARSKI:1;
      [x,y] in addRel(RAT,q) by A1, A2;
      then x in RAT & y in RAT by MMLQUER2:4;
      then reconsider x, y as Rational;
      y = q + x by A1, A2, Th11;
      hence o in S by A2;
    end;
    assume o in S;
    then consider q1 such that
      A3: o = [q1,q1+q];
    q1 in RAT & q1+q in RAT by RAT_1:def 2;
    hence o in addRel(RAT,q) by A3, Th11;
  end;
  hence thesis by TARSKI:2;
end;
