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 v being Element of V

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

Sum L = (L . v) * 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 v being Element of V

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

Sum L = (L . v) * v

let v be Element of V; :: thesis: for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

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

assume Carrier L = {v} ; :: thesis: Sum L = (L . v) * v

then L is Linear_Combination of {v} by Def4;

hence Sum L = (L . v) * v by Th17; :: thesis: verum

for v being Element of V

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

Sum L = (L . v) * 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 v being Element of V

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

Sum L = (L . v) * v

let v be Element of V; :: thesis: for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

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

assume Carrier L = {v} ; :: thesis: Sum L = (L . v) * v

then L is Linear_Combination of {v} by Def4;

hence Sum L = (L . v) * v by Th17; :: thesis: verum