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 Th6:
  for G being Morphism of G2,G3, F being Morphism of G1,G2 st G1 <=
  G2 & G2 <= G3 holds G*F is Morphism of G1,G3
proof
  let G be Morphism of G2,G3, F be Morphism of G1,G2;
  assume that
A1: G1 <= G2 and
A2: G2 <= G3;
  consider g being Function of G2,G3 such that
A3: G = RingMorphismStr(#G2,G3,g#) by A2,Lm8;
  consider f being Function of G1,G2 such that
A4: F = RingMorphismStr(#G1,G2,f#) by A1,Lm8;
  dom(G) = G2 by A2,Def8
    .= cod(F) by A1,Def8;
  then G*F = RingMorphismStr(#G1,G3,g*f#) by A3,A4,Def9;
  then
A5: dom(G*F) = G1 & cod(G*F) = G3;
  G1 <= G3 by A1,A2,Th5;
  hence thesis by A5,Def8;
end;
