reserve D,D9 for non empty set;
reserve R for Ring;
reserve G,H,S for non empty ModuleStr over R;
reserve UN for Universe;
reserve R for Ring;
reserve G, H for LeftMod of R;
reserve G1, G2, G3 for LeftMod of R;
reserve f for LModMorphismStr over R;

theorem Th12:
  for G being Morphism of G2,G3, F being Morphism of G1,G2 holds G
  *F is strict Morphism of G1,G3
proof
  let G be Morphism of G2,G3, F be Morphism of G1,G2;
  consider g being Function of G2,G3 such that
A1: the LModMorphismStr of G = LModMorphismStr(#G2,G3,g#) and
  g is additive homogeneous by Th7;
  consider f being Function of G1,G2 such that
A2: the LModMorphismStr of F = LModMorphismStr(#G1,G2,f#) and
  f is additive homogeneous by Th7;
  dom(G) = G2 by Def8
    .= cod(F) by Def8;
  then G*F = LModMorphismStr(#G1,G3,g*f#) by A1,A2,Def10;
  then dom(G*F) = G1 & cod(G*F) = G3;
  hence thesis by Def8;
end;
