let V be non empty CLSStruct ; :: thesis: for L1, L2, L3 being C_Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3
let L1, L2, L3 be C_Linear_Combination of V; :: thesis: L1 + (L2 + L3) = (L1 + L2) + L3
let v be VECTOR of V; :: according to CONVEX4:def 7 :: thesis: (L1 + (L2 + L3)) . v = ((L1 + L2) + L3) . v
thus (L1 + (L2 + L3)) . v = (L1 . v) + ((L2 + L3) . v) by Def8
.= (L1 . v) + ((L2 . v) + (L3 . v)) by Def8
.= ((L1 . v) + (L2 . v)) + (L3 . v)
.= ((L1 + L2) . v) + (L3 . v) by Def8
.= ((L1 + L2) + L3) . v by Def8 ; :: thesis: verum