reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th7:
  for C,D being Category, E be Subcategory of D, F being Functor of C,D st
  rng F c= the carrier' of E holds F is Functor of C, E
proof
  let C,D be Category, E be Subcategory of D, F be Functor of C,D;
  assume
A1: rng F c= the carrier' of E;
A2: dom F = the carrier' of C by FUNCT_2:def 1;
A3: dom Obj F = the carrier of C by FUNCT_2:def 1;
  reconsider G = F as Function of the carrier' of C, the carrier' of E
  by A1,A2,FUNCT_2:def 1,RELSET_1:4;
A4: rng Obj F c= the carrier of E
  proof
    let y be object;
    assume y in rng Obj F;
    then consider x being object such that
A5: x in dom Obj F and
A6: y = (Obj F).x by FUNCT_1:def 3;
    reconsider x as Object of C by A5;
    F.id x = id ((Obj F).x) by CAT_1:68;
    then id ((Obj F).x) in rng F by A2,FUNCT_1:def 3;
    then reconsider f = id ((Obj F).x) as Morphism of E by A1;
A7: dom id ((Obj F).x) = y by A6;
    dom id ((Obj F).x) = dom f by CAT_2:9;
    hence thesis by A7;
  end;
A8: now
    let c be Object of C;
    (Obj F).c in rng Obj F by A3,FUNCT_1:def 3;
    then reconsider d = (Obj F).c as Object of E by A4;
    take d;
    thus G.id c = id ((Obj F).c) by CAT_1:68
      .= id d by CAT_2:def 4;
  end;
A9: now
    let f be Morphism of C;
A10: dom (F.f) = dom (G.f) by CAT_2:9;
A11: cod (F.f) = cod (G.f) by CAT_2:9;
    thus G.id dom f = id (F.dom f) by CAT_1:71
      .= id dom (F.f) by CAT_1:72
      .= id dom (G.f) by A10,CAT_2:def 4;
    thus G.id cod f = id (F.cod f) by CAT_1:71
      .= id cod (F.f) by CAT_1:72
      .= id cod (G.f) by A11,CAT_2:def 4;
  end;
  now
    let f,g be Morphism of C;
    assume
A12: dom g = cod f;
    then
A13: F.(g(*)f) = (F.g)(*)(F.f) by CAT_1:64;
A14: dom (F.g) = cod (F.f) by A12,CAT_1:64;
A15: dom (F.g) = dom (G. g) by CAT_2:9;
    cod (F.f) = cod (G.f) by CAT_2:9;
    hence G.(g(*)f) = (G.g)(*)(G.f) by A13,A14,A15,CAT_2:11;
  end;
  hence thesis by A8,A9,CAT_1:61;
end;
