
theorem Th5:
  for C being non empty with_identities CategoryStr, f being morphism of C
  ex f1,f2 being morphism of C st f1 is identity & f2 is identity &
  f1 |> f & f |> f2
  proof
    let C be non empty with_identities CategoryStr;
    let f be morphism of C;
    f in Mor C;
    then
A1: f in the carrier of C by CAT_6:def 1;
    then consider f1 be morphism of C such that
A2: f1 |> f & f1 is left_identity by CAT_6:def 6,def 12;
    consider f2 be morphism of C such that
A3: f |> f2 & f2 is right_identity by A1,CAT_6:def 7,def 12;
    take f1,f2;
    f1 is right_identity by A2,CAT_6:9;
    hence f1 is identity by A2,CAT_6:def 14;
    f2 is left_identity by A3,CAT_6:9;
    hence f2 is identity by A3,CAT_6:def 14;
    thus thesis by A2,A3;
  end;
