 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 Th41:
  for X being natural-membered set holds addRel(X,1) = succRel(X)
proof
  let X be natural-membered set;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in addRel(X,1);
      then [a,b] in addRel(X,1);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Nat;
      [a,b] in addRel(X,1) by A1;
      then A2: a in X & b in X & b = 1 + a by Th11;
      then b = Segm(a+1) by ORDINAL1:def 17
        .= succ Segm a by NAT_1:38
        .= succ a by ORDINAL1:def 17;
      hence [x,y] in succRel(X) by A2, Def1;
    end;
    assume [x,y] in succRel(X);
    then [a,b] in succRel(X);
    then A3: a in X & b in X & b = succ a by Def1;
    then reconsider a,b as Nat;
    b = succ Segm a by A3, ORDINAL1:def 17
      .= Segm(a+1) by NAT_1:38
      .= 1 + a by ORDINAL1:def 17;
    hence [x,y] in addRel(X,1) by A3, Th11;
  end;
  hence thesis by RELAT_1:def 2;
end;
