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

theorem Th23:
  for f,f9 being Morphism of C for g,g9 being Morphism of D st dom
  f9 = cod f & dom g9 = cod g holds [f9,g9](*)[f,g] = [f9(*)f,g9(*)g]
proof
  let f,f9 be Morphism of C;
  let g,g9 be Morphism of D;
  assume that
A1: dom f9 = cod f and
A2: dom g9 = cod g;
A3: [f9,f] in dom(the Comp of C) & [g9,g] in dom(the Comp of D) by A1,A2,
CAT_1:15;
  dom [f9,g9] = [dom f9,dom g9] & cod [f,g] = [cod f,cod g] by Th22;
  hence
  [f9,g9](*)[f,g] = |:the Comp of C, the Comp of D:|.([f9,g9],[f,g])
     by A1,A2,CAT_1:16
    .= [(the Comp of C).(f9,f),(the Comp of D).(g9,g)] by A3,FUNCT_4:def 3
    .= [f9(*)f,(the Comp of D).(g9,g)] by A1,CAT_1:16
    .= [f9(*)f,g9(*)g] by A2,CAT_1:16;
end;
