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 Th2:
  for g,f being RingMorphism st dom(g) = cod(f) ex G1,G2,G3 st G1
  <= G2 & G2 <= G3 & the RingMorphismStr of g is Morphism of G2,G3 & the
  RingMorphismStr of f is Morphism of G1,G2
proof
  defpred P[RingMorphism,RingMorphism] means dom($1) = cod($2);
  let g,f be RingMorphism such that
A1: P[g,f];
  ( ex G2,G3 st G2 <= G3 & dom(g) = G2 & cod(g) = G3 & the RingMorphismStr
  of g is Morphism of G2,G3)& ex G1,G29 being Ring st G1 <= G29 & dom(f) = G1 &
  cod(f ) = G29 & the RingMorphismStr of f is Morphism of G1,G29 by Lm6;
  hence thesis by A1;
end;
