reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th30:
  for C being Category, o being Object of C, f being Element of Hom o
  for a being Object of C-SliceCat o st a = f holds id a = [[a,a], id dom f]
proof
  let C be Category, o be Object of C, f be Element of Hom o;
  let a be Object of C-SliceCat o;
  assume
A1: a = f;
  consider b,c being Element of Hom o, g being Morphism of C such that
A2: id a = [[b,c], g] and
A3: dom c = cod g and b = c(*)g by Def11;
A4: cod id dom f = dom f;
  f = f(*)id dom f by CAT_1:22;
  then reconsider h = [[f,f], id dom f] as Morphism of C-SliceCat o by A4,Def11
;
A5: (id a)`11 = b by A2,MCART_1:85;
A6: (id a)`12 = c by A2,MCART_1:85;
A7: dom id a = b by A5,Th2;
A8: cod id a = c by A6,Th2;
A9: b = a by A7;
A10: c = a by A8;
  dom h = h`11 by Th2
    .= a by A1,MCART_1:85;
  then h = h(*)id a by CAT_1:22
    .= [[f,f], (id dom f)(*)g] by A1,A2,A9,A10,Def11
    .= [[f,f], g] by A1,A3,A10,CAT_1:21;
  hence thesis by A1,A2,A7,A10;
end;
