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 Th5:
  G1 <= G2 & G2 <= G3 implies G1 <= G3
proof
  assume that
A1: G1 <= G2 and
A2: G2 <= G3;
  consider F0 being RingMorphism such that
A3: dom(F0) = G1 and
A4: cod(F0) = G2 by A1;
  reconsider F = the RingMorphismStr of F0 as RingMorphism by Lm2;
  dom(F) = G1 & cod(F) = G2 by A3,A4;
  then reconsider F9 = F as Morphism of G1,G2 by A1,Def8;
  consider f being Function of G1,G2 such that
A5: F9 = RingMorphismStr(#G1,G2,f#) by A1,Lm8;
  consider G0 being RingMorphism such that
A6: dom(G0) = G2 and
A7: cod(G0) = G3 by A2;
  reconsider G = the RingMorphismStr of G0 as RingMorphism by Lm2;
  dom(G) = G2 & cod(G) = G3 by A6,A7;
  then reconsider G9 = G as Morphism of G2,G3 by A2,Def8;
  consider g being Function of G2,G3 such that
A8: G9 = RingMorphismStr(#G2,G3,g#) by A2,Lm8;
  dom(G) = cod(F) by A4,A6;
  then G*F = RingMorphismStr(#G1,G3,g*f#) by A8,A5,Def9;
  then dom(G*F) = G1 & cod(G*F) = G3;
  hence thesis;
end;
