reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve R for Ring;
reserve G,H for LeftMod of R;
reserve V for LeftMod_DOMAIN of R;

theorem Th2:
  D is LModMorphism_DOMAIN of G,H iff for x being Element of D
  holds x is strict Morphism of G,H
proof
  thus D is LModMorphism_DOMAIN of G,H implies for x being Element of D holds
  x is strict Morphism of G,H by Def3;
  thus (for x being Element of D holds x is strict Morphism of G,H) implies D
  is LModMorphism_DOMAIN of G,H
  proof
    assume
A1: for x being Element of D holds x is strict Morphism of G,H;
    then for x being Element of D holds x is strict LModMorphism of R;
    then reconsider D9 = D as LModMorphism_DOMAIN of R by Def2;
    for x being Element of D9 holds x is strict Morphism of G,H by A1;
    hence thesis by Def3;
  end;
end;
