let D be non empty set ; :: thesis: for G, H being AddGroup holds
( D is GroupMorphism_DOMAIN of G,H iff for x being Element of D holds x is strict Morphism of G,H )
let G, H be AddGroup; :: thesis: ( D is GroupMorphism_DOMAIN of G,H iff for x being Element of D holds x is strict Morphism of G,H )
thus ( D is GroupMorphism_DOMAIN of G,H implies for x being Element of D holds x is strict Morphism of G,H ) by Def24; :: thesis: ( ( for x being Element of D holds x is strict Morphism of G,H ) implies D is GroupMorphism_DOMAIN of G,H )
thus ( ( for x being Element of D holds x is strict Morphism of G,H ) implies D is GroupMorphism_DOMAIN of G,H ) :: thesis: verum

proof

assume A1: for x being Element of D holds x is strict Morphism of G,H ; :: thesis: D is GroupMorphism_DOMAIN of G,H
then for x being set st x in D holds
x is strict GroupMorphism ;
then reconsider D' = D as GroupMorphism_DOMAIN by Def23;
for x being Element of D' holds x is strict Morphism of G,H by A1;
hence D is GroupMorphism_DOMAIN of G,H by Def24; :: thesis: verum

end;