reserve e for set;

theorem Th13:
  for C being Category, i,j,k being Object of C st Hom(i,j) <> {}
  & Hom(j,k) <> {} for f being Morphism of i,j, g being Morphism of j,k holds (
  the_comps_of C).(i,j,k).(g,f) = g*f
proof
  let C be Category, i,j,k be Object of C such that
A1: Hom(i,j) <> {} and
A2: Hom(j,k) <> {};
  let f be Morphism of i,j, g be Morphism of j,k;
A3: g in Hom(j,k) by A2,CAT_1:def 5;
  then
A4: g in (the_hom_sets_of C).(j,k) by Def3;
A5: f in Hom(i,j) by A1,CAT_1:def 5;
  then f in (the_hom_sets_of C).(i,j) by Def3;
  then
A6: [g,f] in [:(the_hom_sets_of C).(j,k),(the_hom_sets_of C).(i,j):] by A4,
ZFMISC_1:87;
A7: dom g = j by A3,CAT_1:1
    .= cod f by A5,CAT_1:1;
  thus (the_comps_of C).(i,j,k).(g,f) = ((the Comp of C)| ([:(the_hom_sets_of
  C).(j,k),(the_hom_sets_of C).(i,j):] qua set)) .[g,f] by Def4
    .= (the Comp of C).(g,f) by A6,FUNCT_1:49
    .= g(*)(f qua Morphism of C) by A7,CAT_1:16
    .= g*f by A1,A2,CAT_1:def 13;
end;
