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 V st Carrier L = {} 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 V st Carrier L = {} holds

Sum L = 0. V

let L be Linear_Combination of V; :: thesis: ( Carrier L = {} implies Sum L = 0. V )

assume Carrier L = {} ; :: thesis: Sum L = 0. V

then L = ZeroLC V by Def3;

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

for L being Linear_Combination of V st Carrier L = {} 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 V st Carrier L = {} holds

Sum L = 0. V

let L be Linear_Combination of V; :: thesis: ( Carrier L = {} implies Sum L = 0. V )

assume Carrier L = {} ; :: thesis: Sum L = 0. V

then L = ZeroLC V by Def3;

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