
theorem Th6:
  for C being with_identities composable CategoryStr,
      f being morphism of C st f is identity holds dom f = f & cod f = f
  proof
    let C be with_identities composable CategoryStr;
    let f be morphism of C;
    assume
A1: f is identity;
     per cases;
     suppose
A2:    C is empty;
       then
A3:    the Object of C = {} by SUBSET_1:def 1;
       dom f = the Object of C & cod f = the Object of C
       by A2,CAT_6:def 18,def 19;
       hence thesis by A3,A2,SUBSET_1:def 1;
     end;
     suppose C is non empty;
       hence thesis by A1,CAT_6:24,26,27;
     end;
  end;
