
theorem Th10:
  for C1,C2,C3 being composable with_identities CategoryStr
  st C1 ~= C2 & C2 ~= C3 holds C1 ~= C3
  proof
    let C1,C2,C3 be composable with_identities CategoryStr;
    assume
A1: C1 ~= C2;
    assume
A2: C2 ~= C3;
    per cases;
    suppose
A3:   C2 is non empty & C3 is non empty;
      consider F be Functor of C1,C2 such that
A4:   F is covariant & F is bijective by A1,CAT_7:12;
      consider G be Functor of C2,C3 such that
A5:   G is covariant & G is bijective by A2,CAT_7:12;
      F * G is onto by A4,A5,A3,FUNCT_2:27;
      then G (*) F is bijective by A4,A5,CAT_6:def 27;
      hence thesis by A4,A5,CAT_6:35,CAT_7:12;
    end;
    suppose C2 is empty or C3 is empty;
      then
A6:   C2 is empty & C3 is empty by A2,CAT_7:15;
      then C1 is empty by A1,CAT_7:15;
      hence thesis by A6,CAT_7:13;
    end;
  end;
