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 L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . 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 L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let v be Element of V; :: thesis: for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let L1, L2 be Linear_Combination of V; :: thesis: (L1 - L2) . v = (L1 . v) - (L2 . v)

thus (L1 - L2) . v = (L1 . v) + ((- L2) . v) by Th22

.= (L1 . v) + (- (L2 . v)) by Th36

.= (L1 . v) - (L2 . v) by RLVECT_1:def 11 ; :: thesis: verum

for v being Element of V

for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . 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 L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let v be Element of V; :: thesis: for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let L1, L2 be Linear_Combination of V; :: thesis: (L1 - L2) . v = (L1 . v) - (L2 . v)

thus (L1 - L2) . v = (L1 . v) + ((- L2) . v) by Th22

.= (L1 . v) + (- (L2 . v)) by Th36

.= (L1 . v) - (L2 . v) by RLVECT_1:def 11 ; :: thesis: verum