reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th9:
  for a,b,c being Object of A st Hom(a,b) <> {} & Hom(b,c) <> {}
  for f being Morphism of a,b, g being Morphism of b,c
   holds F/.(g*f) = (F/.g)*(F/.f)
proof
  let a,b,c be Object of A;
  assume that
A1: Hom(a,b) <> {} and
A2: Hom(b,c) <> {};
  let f be Morphism of a,b, g be Morphism of b,c;
A3: dom g = b by A2,CAT_1:5;
A4: cod f = b by A1,CAT_1:5;
A5: F/.g = F.(g qua Morphism of A) by A2,CAT_3:def 10;
A6: Hom(F.a,F.b) <> {} by A1,CAT_1:84;
A7: F/.f = F.(f qua Morphism of A) by A1,CAT_3:def 10;
A8: Hom(F.b,F.c) <> {} by A2,CAT_1:84;
  Hom(a,c) <> {} by A1,A2,CAT_1:24;
  hence F/.(g*f) = F.(g*f qua Morphism of A) by CAT_3:def 10
    .= F.((g qua Morphism of A)(*)(f qua Morphism of A)) by A1,A2,CAT_1:def 13
    .= (F.(g qua Morphism of A))(*)(F.(f qua Morphism of A)) by A3,A4,CAT_1:64
    .= (F/.g)*(F/.f) by A6,A8,A5,A7,CAT_1:def 13;
end;
