 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 rational-membered set holds
    addRel(X,q) = ((curry addrat).q) |_2 X
proof
  let X be rational-membered set;
  set g = (curry addrat).q;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in addRel(X,q);
      then [a,b] in addRel(X,q);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Rational;
      [a,b] in addRel(X,q) by A1;
      then A2: a in X & b in X & b = q + a by Th11;
      q in RAT & a in RAT by RAT_1:def 2;
      then [q,a] in [: RAT, RAT :] by ZFMISC_1:87;
      then A3: [q,a] in dom addrat by FUNCT_2:def 1;
      A4: b = addrat.(q,a) by A2, BINOP_2:def 15
        .= g.a by A3, FUNCT_5:20;
      a in dom g by A3, FUNCT_5:20;
      hence [x,y] in g |_2 X by A2, A4, FUNCT_1:1, MMLQUER2:4;
    end;
    assume [x,y] in g |_2 X;
    then [a,b] in g |_2 X;
    then A5: a in X & b in X & [a,b] in g by MMLQUER2:4;
    then reconsider a,b as Rational;
    q in RAT & a in RAT by RAT_1:def 2;
    then [q,a] in [: RAT, RAT :] by ZFMISC_1:87;
    then A6: [q,a] in dom addrat by FUNCT_2:def 1;
    a in dom g & g.a = b by A5, FUNCT_1:1;
    then b = addrat.(q,a) by A6, FUNCT_5:20
     .= q + a by BINOP_2:def 15;
    hence [x,y] in addRel(X,q) by A5, Th11;
  end;
  hence thesis by RELAT_1:def 2;
end;
