reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th34:
  for f being (Morphism of GroupCat(UN)), f9 being Element of
  Morphs(GroupObjects(UN)), b being Object of GroupCat(UN), b9 being Element of
GroupObjects(UN) holds f is strict Element of Morphs(GroupObjects(UN)) & f9 is
  Morphism of GroupCat(UN) & b is strict Element of GroupObjects(UN) & b9 is
  Object of GroupCat(UN)
proof
  set C = GroupCat(UN), V = GroupObjects(UN);
  set X = Morphs(V);
  let f be (Morphism of C), f9 be Element of X, b be Object of C, b9 be
  Element of V;
  consider x being object such that
  x in UN and
A1: GO x,b by Def22;
  ex G,H being strict Element of V st f is strict Morphism of G,H by Def23;
  hence f is strict Element of X;
  thus f9 is Morphism of C;
  ex x1,x2,x3,x4 being set st x = [x1,x2,x3,x4] & ex G being strict
AddGroup st b = G & x1 = the carrier of G & x2 = the addF of G & x3 = comp G &
  x4 = 0.G by A1;
  hence b is strict Element of V;
  thus thesis;
end;
