reserve e for set;

theorem Th14:
  for C being Category holds the_comps_of C is associative
proof
  let C be Category;
  let i,j,k,l be Object of C;
  let f,g,h be set;
  assume f in (the_hom_sets_of C).(i,j);
  then
A1: f in Hom(i,j) by Def3;
  then reconsider f9 = f as Morphism of i,j by CAT_1:def 5;
  assume g in (the_hom_sets_of C).(j,k);
  then
A2: g in Hom(j,k) by Def3;
  then reconsider g9 = g as Morphism of j,k by CAT_1:def 5;
  assume h in (the_hom_sets_of C).(k,l);
  then
A3: h in Hom(k,l) by Def3;
  then reconsider h9 = h as Morphism of k,l by CAT_1:def 5;
A4: Hom(j,l) <> {} & (the_comps_of C).(j,k,l).(h,g) = h9*g9 by A2,A3,Th13,
CAT_1:24;
  Hom(i,k) <> {} & (the_comps_of C).(i,j,k).(g,f) = g9*f9 by A1,A2,Th13,
CAT_1:24;
  hence
  (the_comps_of C).(i,k,l).(h,(the_comps_of C).(i,j,k).(g,f)) = h9*(g9*f9
  ) by A3,Th13
    .= h9*g9*f9 by A1,A2,A3,CAT_1:25
    .= (the_comps_of C).(i,j,l).((the_comps_of C).(j,k,l).(h,g),f) by A1,A4
,Th13;
end;
