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);
reserve C for Category,
  a,b,a9,b9,c for Object of C,
  f,g,h,f9,g9 for Morphism of C;

theorem Th42:
  hom(a,id c) = id Hom(a,c)
proof
  set A = Hom(a,c);
  now
    A = {} implies A = {};
    hence dom hom(a,id c) = A by FUNCT_2:def 1;
    let x be object;
    assume
A1: x in A;
    then reconsider g = x as Morphism of C;
A2: cod g = c by A1,CAT_1:1;
    thus hom(a,id c).x = id(c)(*)g by A1,Def18
      .= x by A2,CAT_1:21;
  end;
  hence thesis by FUNCT_1:17;
end;
