let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

let l be Linear_Combination of {} the carrier of V; :: thesis: l = ZeroLC V

Carrier l c= {} by Def4;

then Carrier l = {} ;

hence l = ZeroLC V by Def3; :: thesis: verum

for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

let l be Linear_Combination of {} the carrier of V; :: thesis: l = ZeroLC V

Carrier l c= {} by Def4;

then Carrier l = {} ;

hence l = ZeroLC V by Def3; :: thesis: verum