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;
reserve V for Ring_DOMAIN;

theorem Th13:
  D is RingMorphism_DOMAIN of G,H iff for x being Element of D
  holds x is Morphism of G,H
proof
  thus D is RingMorphism_DOMAIN of G,H implies for x being Element of D holds
  x is Morphism of G,H by Def13;
  thus (for x being Element of D holds x is Morphism of G,H) implies D is
  RingMorphism_DOMAIN of G,H
  proof
    assume
A1: for x being Element of D holds x is Morphism of G,H;
    then for x being object st x in D holds x is strict RingMorphism;
    then reconsider D9 = D as RingMorphism_DOMAIN by Def12;
    for x being Element of D9 holds x is Morphism of G,H by A1;
    hence thesis by Def13;
  end;
end;
