 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(NAT,n) = the set of all [n1,n1+n] where n1
proof
  set S = the set of all [n1,n1+n] where n1;
  now
    let o be object;
    hereby
      assume A1: o in addRel(NAT,n);
      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(NAT,n) by A1, A2;
      then x in NAT & y in NAT by MMLQUER2:4;
      then reconsider x, y as Nat;
      y = n + x by A1, A2, Th11;
      hence o in S by A2;
    end;
    assume o in S;
    then consider n1 such that
      A3: o = [n1,n1+n];
    n1 in NAT & n1+n in NAT by ORDINAL1:def 12;
    hence o in addRel(NAT,n) by A3, Th11;
  end;
  hence thesis by TARSKI:2;
end;
