
theorem Th40:
  for C being non empty category, f being morphism of C st f is identity holds
  for g being morphism of OrdC 2 holds (MORPHISM f).g = f
  proof
    let C be non empty category;
    let f be morphism of C;
    assume
A1: f is identity;
    let g be morphism of OrdC 2;
    consider f1 be morphism of OrdC(2) such that
A2: f1 is not identity & Ob OrdC 2 = {dom f1, cod f1} &
    Mor OrdC 2 = {dom f1, cod f1, f1} &
    dom f1, cod f1, f1 are_mutually_distinct by Th39;
    per cases by A2,ENUMSET1:def 1;
    suppose g = dom f1;
      hence (MORPHISM f).g = (MORPHISM f).(dom f1) by CAT_6:def 21
      .= dom ((MORPHISM f).f1) by CAT_6:32
      .= dom f by A2,Def16
      .= f by A1,Th6;
    end;
    suppose g = cod f1;
      hence (MORPHISM f).g = (MORPHISM f).(cod f1) by CAT_6:def 21
      .= cod ((MORPHISM f).f1) by CAT_6:32
      .= cod f by A2,Def16
      .= f by A1,Th6;
    end;
    suppose g = f1; hence thesis by A2,Def16; end;
  end;
