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

theorem Th9:
  for C,D being Category, E being Subcategory of D, F being Functor of C,E
  for G being Functor of C,D st F = G holds Image F = Image G
proof
  let C,D be Category, E be Subcategory of D;
  let F be Functor of C,E, G be Functor of C,D such that
A1: F = G;
  reconsider S = Image F as strict Subcategory of D by Th4;
A2: now
    thus dom Obj F = the carrier of C & dom Obj G = the carrier of C
    by FUNCT_2:def 1;
    let x be object;
    assume x in the carrier of C;
    then reconsider a = x as Object of C;
    reconsider b = (Obj F).a as Object of D by CAT_2:6;
    G.id a = id ((Obj F).a) by A1,CAT_1:68
      .= id b by CAT_2:def 4;
    hence (Obj G).x = (Obj F).x by CAT_1:67;
  end;
  then
A3: Obj F = Obj G;
  then
A4: the carrier of S = rng Obj G by Def3;
A5: rng G c= the carrier' of S by A1,Def3;
  now
    let T be Subcategory of D;
    assume that
A6: the carrier of T = rng Obj G and
A7: rng G c= the carrier' of T;
    set x = the Object of C;
A8: (Obj G).x in rng Obj G by A2,FUNCT_1:def 3;
A9: (Obj G).x = (Obj F).x by A2;
    then (Obj G).x in (the carrier of T) /\ the carrier of E
    by A6,A8,XBOOLE_0:def 4;
    then
A10: (the carrier of T) meets the carrier of E;
    then reconsider E1 = T /\ E as Subcategory of E by Th6;
    the carrier of E1 = (the carrier of T) /\ the carrier of E by A6,A8,A9,Def2
;
    then
A11: the carrier of E1 = rng Obj F by A3,A6,XBOOLE_1:28;
    the carrier' of E1 = (the carrier' of T) /\ the carrier' of E by A6,A8,A9
,Def2;
    then rng F c= the carrier' of E1 by A1,A7,XBOOLE_1:19;
    then
A12: Image F is Subcategory of E1 by A11,Def3;
    E1 is Subcategory of T by A10,Th6;
    hence S is Subcategory of T by A12,Th4;
  end;
  hence thesis by A4,A5,Def3;
end;
