reserve B,C,D,C9,D9 for Category;
reserve E for Subcategory of C;

theorem Th7:
  for f,g being (Morphism of E), f9,g9 being Morphism of C st f =
  f9 & g = g9 & dom g = cod f holds g(*)f = g9(*)f9
proof
  let f,g be (Morphism of E), f9,g9 be Morphism of C such that
A1: f = f9 & g = g9 and
A2: dom g = cod f;
  dom g = dom g9 & cod f = cod f9 by A1,Th5;
  then
A3: g9(*)f9 = (the Comp of C).(g9,f9) by A2,CAT_1:16;
A4: the Comp of E c= the Comp of C by Def4;
  g(*)f = (the Comp of E).(g,f) & [g,f] in dom(the Comp of E)
   by A2,CAT_1:15,16;
  hence thesis by A1,A3,A4,GRFUNC_1:2;
end;
