let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1
let L1, L2 be Linear_Combination of V; :: thesis: L1 + L2 = L2 + L1
let v be Element of V; :: according to VECTSP_6:def 10 :: thesis: (L1 + L2) . v = (L2 + L1) . v
thus (L1 + L2) . v = (L2 . v) + (L1 . v) by Def11
.= (L2 + L1) . v by Def11 ; :: thesis: verum