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