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;

theorem Th7:
  for f,g being strict RingMorphism st dom g = cod f holds ex G1,
  G2,G3 st ex f0 being Function of G1,G2, g0 being Function of G2,G3 st f =
  RingMorphismStr(#G1,G2,f0#) & g = RingMorphismStr(#G2,G3,g0#) & g*f =
  RingMorphismStr(#G1,G3,g0*f0#)
proof
  let f,g be strict RingMorphism such that
A1: dom g = cod f;
  set G1 = dom f,G2 = cod f, G3 = cod g;
  reconsider g9 = g as Morphism of G2,G3 by A1,Th3;
  consider g0 being Function of G2,G3 such that
A2: g9 = RingMorphismStr(#G2,G3,g0#) by A1;
  reconsider f9 = f as Morphism of G1,G2 by Th3;
  consider f0 being Function of G1,G2 such that
A3: f9 = RingMorphismStr(#G1,G2,f0#);
  take G1,G2,G3,f0,g0;
  thus thesis by A1,A3,A2,Def9;
end;
