
theorem Th5:
  for C being non empty composable with_identities CategoryStr,
  f1,f2 being morphism of C holds f1 |> f2 iff dom f1 = cod f2
  proof
    let C be non empty composable with_identities CategoryStr;
    let f1,f2 be morphism of C;
A1: dom f1 = (SourceMap C).f1 by CAT_6:def 30;
A2: cod f2 = (TargetMap C).f2 by CAT_6:def 31;
    hereby
      assume f1 |> f2;
      then [f1,f2] in dom the composition of C by CAT_6:def 2;
      then [f1,f2] in dom CompMap C by CAT_6:def 29;
      hence dom f1 = cod f2 by A1,A2,CAT_6:36;
    end;
    assume dom f1 = cod f2;
    then (SourceMap C).f1 = (TargetMap C).f2 by A1,CAT_6:def 31;
    then [f1,f2] in dom CompMap C by CAT_6:36;
    then [f1,f2] in dom the composition of C by CAT_6:def 29;
    hence f1 |> f2 by CAT_6:def 2;
  end;
