reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve f for RingMorphismStr;
reserve G,H,G1,G2,G3,G4 for Ring;
reserve F for RingMorphism;
reserve V for Ring_DOMAIN;

theorem
  for a being Object of RingCat(UN), aa being Element of
  RingObjects(UN) st a = aa holds id a = ID aa
 proof let a be Object of RingCat UN, aa be Element of RingObjects UN;
  set C = RingCat UN;
  assume
A1:  a = aa;
   reconsider ii = ID aa as Morphism of C;
   reconsider ia = ii as RingMorphismStr;
A2: dom ii = dom ia by Def19
      .= a by A1;
   cod ii = cod ia by Def20
      .= a by A1;
   then reconsider ii as Morphism of a,a by A2,CAT_1:4;
  for b being Object of C holds
     (Hom(a,b) <> {} implies
       for f being Morphism of a,b holds f(*)ii = f)
   & (Hom(b,a) <> {} implies
     for f being Morphism of b,a holds ii(*)f = f) by A1,Lm11;
  hence id a = ID aa by CAT_1:def 12;
 end;
