reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve f for RingMorphismStr;
reserve G,H,G1,G2,G3,G4 for Ring;
reserve F for RingMorphism;
reserve V for Ring_DOMAIN;

theorem Th22:
  for f being (Morphism of RingCat(UN)), f9 being Element of
  Morphs(RingObjects(UN)), b being Object of RingCat(UN), b9 being Element of
  RingObjects(UN) holds f is strict Element of Morphs(RingObjects(UN)) & f9 is
Morphism of RingCat(UN) & b is strict Element of RingObjects(UN) & b9 is Object
  of RingCat(UN)
proof
  set C = RingCat(UN), V = RingObjects(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 such that
  x in UN and
A1: GO x,b by Def16;
  ex G,H being Element of V st G <= H & f is Morphism of G,H by Def17;
  hence f is strict Element of X;
  thus f9 is Morphism of C;
  ex x1,x2,x3,x4,x5,x6 being set st x = [[x1,x2,x3,x4],x5,x6] & ex G being
strict Ring st b = G & x1 = the carrier of G & x2 = the addF of G & x3 = comp G
  & x4 = 0.G & x5 = the multF of G & x6 = 1.G by A1;
  hence b is strict Element of V;
  thus thesis;
end;
