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

theorem
  for C being Category, D1,D2 being Categorial Category
  for F1 being Functor of C,D1 for F2 being Functor of C,D2 st
  F1 = F2 holds Image F1 = Image F2
proof
  let C be Category, D1,D2 be Categorial Category;
  let F1 be Functor of C,D1;
  let F2 be Functor of C,D2;
  assume
A1: F1 = F2;
  reconsider DD = (the carrier of D1) \/ the carrier of D2 as non empty set;
  DD is categorial
  proof
    let d be Element of DD;
    d is Object of D1 or d is Object of D2 by XBOOLE_0:def 3;
    hence thesis by Th12;
  end;
  then reconsider DD = (the carrier of D1) \/ the carrier of D2 as
  non empty categorial set;
  consider D being full Categorial strict Category such that
A2: the carrier of D = DD by Th20;
  reconsider D1, D2 as Subcategory of D by A2,Th21,XBOOLE_1:7;
  reconsider F1 as Functor of C,D1;
  reconsider F2 as Functor of C,D2;
  rng F1 c= the carrier' of D1;
  then F1 = (incl D1)*F1 by RELAT_1:53;
  then reconsider G1 = F1 as Functor of C,D;
  Image F1 = Image G1 by Th9
    .= Image F2 by A1,Th9;
  hence thesis;
end;
