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 Sum l = 0. 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 Sum l = 0. V

let l be Linear_Combination of {} the carrier of V; :: thesis: Sum l = 0. V

l = ZeroLC V by Th6;

hence Sum l = 0. V by Lm1; :: thesis: verum

for l being Linear_Combination of {} the carrier of V holds Sum l = 0. 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 Sum l = 0. V

let l be Linear_Combination of {} the carrier of V; :: thesis: Sum l = 0. V

l = ZeroLC V by Th6;

hence Sum l = 0. V by Lm1; :: thesis: verum