reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);

theorem Th28:
  id a = id$ @a
proof
     reconsider aa=a as Element of V;
     reconsider ii = id$ aa as Morphism of Ens V;
A1:   cod ii = cod id$ aa by Def10
        .= a;
A2:    dom ii = dom id$ aa by Def9
         .= a;
     then reconsider ii as Morphism of a,a by A1,CAT_1:4;
  for b being Object of Ens V 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)
  proof
    let b be Element of Ens V;
     set p = the Comp of Ens V;
    thus Hom(a,b)<>{} implies for g being Morphism of a,b holds g(*)ii = g
     proof assume
A3:     Hom(a,b)<>{};
      let g be Morphism of a,b;
       reconsider gg = g as Element of Maps V;
A4:     dom gg = dom g by Def9
          .= aa by A3,CAT_1:5;
       then
A5:     cod id$ aa = dom gg;
       dom g = a by A3,CAT_1:5;
       then [g,ii] in dom p by A1,CAT_1:def 6;
      hence g(*)ii = p.(g,ii) by CAT_1:def 1
           .= gg*(id$ aa) by A5,Def11
           .= g by A4,Th14;
     end;
      assume
A6:     Hom(b,a)<>{};
      let g be Morphism of b,a;
       reconsider gg = g as Element of Maps V;
A7:     cod gg = cod g by Def10
          .= aa by A6,CAT_1:5;
       then
A8:     dom id$ aa = cod gg;
       cod g = a by A6,CAT_1:5;
       then [ii,g] in dom p by A2,CAT_1:def 6;
      hence ii(*)g = p.(ii,g) by CAT_1:def 1
           .= (id$ aa)*gg by A8,Def11
           .= g by A7,Th14;
  end;

 hence thesis by CAT_1:def 12;
end;
