reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th25:
  D is GroupMorphism_DOMAIN of G,H iff for x being Element of D
  holds x is strict Morphism of G,H
proof
  thus D is GroupMorphism_DOMAIN of G,H implies for x being Element of D holds
  x is strict Morphism of G,H by Def17;
  thus (for x being Element of D holds x is strict Morphism of G,H) implies D
  is GroupMorphism_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 object st x in D holds x is strict GroupMorphism;
    then reconsider D9 = D as GroupMorphism_DOMAIN by Def16;
    for x being Element of D9 holds x is strict Morphism of G,H by A1;
    hence thesis by Def17;
  end;
end;
