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 Th20:
  for g,f being Element of Morphs(V) st dom(g) = cod(f) holds g*f in Morphs(V)
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;
  assume P[g,f];
  then consider G1,G2,G3 being Element of V such that
A1: G1 <= G2 & G2 <= G3 and
A2: g is Morphism of G2,G3 and
A3: f is Morphism of G1,G2 by Th19;
  reconsider f9 = f as Morphism of G1,G2 by A3;
  reconsider g9 = g as Morphism of G2,G3 by A2;
  G1 <= G3 & g9*'f9 = g9*f9 by A1,Def10,Th5;
  hence thesis by Def17;
end;
