
theorem Th18:
  for C being composable non empty with_identities CategoryStr,
      a,b being Object of C, f being Morphism of a,b
  st Hom(a,b) <> {} holds f * (id- a) = f & (id- b) * f = f
  proof
    let C be composable non empty with_identities CategoryStr;
    let a,b be Object of C;
    let f be Morphism of a,b;
    assume
A1: Hom(a,b) <> {};
A3: id- a = a & id- b = b by CAT_6:def 20;
A4: Hom(a,a) <> {} & Hom(b,b) <> {};
A5: f |> id- a & id- b |> f by A1,A4,Th17;
    thus f * (id- a) = f (*) (id- a) by A4,A1,Def4
    .= f by A3,A5,CAT_6:23;
    thus (id- b) * f = (id- b) (*) f by A4,A1,Def4
    .= f by A3,A5,CAT_6:23;
  end;
