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 Th19:
  for g,f being Element of Morphs(V) st dom(g) = cod(f) ex G1,G2,
  G3 being Element of V st G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is
  Morphism of G1,G2
proof
  set X = Morphs(V);
  defpred P[Element of X,Element of X] means dom($1) = cod($2);
  let g,f be Element of X such that
A1: P[g,f];
  consider G2,G3 being Element of V such that
A2: G2 <= G3 & g is Morphism of G2,G3 by Def17;
  consider G1,G29 being Element of V such that
A3: G1 <= G29 & f is Morphism of G1,G29 by Def17;
A4: G29 = cod(f) by A3,Def8;
  G2 = dom(g) by A2,Def8;
  hence thesis by A1,A2,A3,A4;
end;
