
theorem Th22:
  for C being composable with_identities CategoryStr,
      a,b,c being Object of C
  st Hom(a,b) <> {} & Hom(b,c) <> {} holds Hom(a,c) <> {}
  proof
    let C be composable with_identities CategoryStr;
    let a,b,c be Object of C;
    assume
A1: Hom(a,b) <> {};
    set f1 = the Morphism of a,b;
    consider f11,f12 be morphism of C such that
A2: a = f11 & b = f12 & f1 |> f11 & f12 |> f1 by A1,Th16;
    assume
A3: Hom(b,c) <> {};
    set f2 = the Morphism of b,c;
    consider f22,f23 be morphism of C such that
A4: b = f22 & c = f23 & f2 |> f22 & f23 |> f2 by A3,Th16;
A5: C is left_composable & C is right_composable by CAT_6:def 11;
    C is non empty by A2,CAT_6:1;
    then
A6: f2 |> f1 by A2,A4,Th3,CAT_6:22;
    f23 |> f2(*)f1 & f2(*)f1 |> f11 by A2,A4,A6,A5,CAT_6:def 8,def 9;
    then f2(*)f1 in Hom(a,c) by A2,A4;
    hence Hom(a,c) <> {};
  end;
