
theorem Th17:
  for C being composable with_identities CategoryStr,
      a,b,c being Object of C,
      f1 being Morphism of a,b, f2 being Morphism of b,c
  st Hom(a,b) <> {} & Hom(b,c) <> {} holds f2 |> f1
  proof
    let C be composable with_identities CategoryStr;
    let a,b,c be Object of C;
    let f1 be Morphism of a,b;
    let f2 be Morphism of b,c;
    assume Hom(a,b) <> {};
    then consider f11,f12 be morphism of C such that
A1: a = f11 & b = f12 & f1 |> f11 & f12 |> f1 by Th16;
    assume Hom(b,c) <> {};
    then consider f21,f22 be morphism of C such that
A2: b = f21 & c = f22 & f2 |> f21 & f22 |> f2 by Th16;
    C is non empty by A1,CAT_6:1;
    hence f2 |> f1 by A1,A2,Th3,CAT_6:22;
  end;
