 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(INT,i) = the set of all [i1,i1+i] where i1
proof
  set S = the set of all [i1,i1+i] where i1;
  now
    let o be object;
    hereby
      assume A1: o in addRel(INT,i);
      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(INT,i) by A1, A2;
      then x in INT & y in INT by MMLQUER2:4;
      then reconsider x, y as Integer;
      y = i + x by A1, A2, Th11;
      hence o in S by A2;
    end;
    assume o in S;
    then consider i1 such that
      A3: o = [i1,i1+i];
    i1 in INT & i1+i in INT by INT_1:def 2;
    hence o in addRel(INT,i) by A3, Th11;
  end;
  hence thesis by TARSKI:2;
end;
