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 Th10:
  for f,g,h being strict RingMorphism st dom h = cod g & dom g =
  cod f holds h*(g*f) = (h*g)*f
proof
  let f,g,h be strict RingMorphism such that
A1: dom h = cod g and
A2: dom g = cod f;
  set G1 = dom f,G2 = cod f, G3 = cod g, G4 = cod h;
  reconsider h9 = h as Morphism of G3,G4 by A1,Th3;
  reconsider f9 = f as Morphism of G1,G2 by Th3;
  reconsider g9 = g as Morphism of G2,G3 by A2,Th3;
A3: G1 <= G2;
  G2 <= G3 & G3 <= G4 by A1,A2;
  then h9*(g9*f9) = (h9*g9)*f9 by A3,Th9;
  hence thesis;
end;
