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 Def17; :: 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 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 D is GroupMorphism_DOMAIN of G,H by Def17; :: thesis: verum
end;