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 vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3
let L1, L2, L3 be Linear_Combination of V; :: thesis: L1 + (L2 + L3) = (L1 + L2) + L3
let v be Element of V; :: according to VECTSP_6:def 10 :: thesis: (L1 + (L2 + L3)) . v = ((L1 + L2) + L3) . v
thus (L1 + (L2 + L3)) . v = (L1 . v) + ((L2 + L3) . v) by Def11
.= (L1 . v) + ((L2 . v) + (L3 . v)) by Def11
.= ((L1 . v) + (L2 . v)) + (L3 . v) by RLVECT_1:def 6
.= ((L1 + L2) . v) + (L3 . v) by Def11
.= ((L1 + L2) + L3) . v by Def11 ; :: thesis: verum