let V be non empty CLSStruct ; :: thesis: for L1, L2 being C_Linear_Combination of V st L1 + L2 = ZeroCLC V holds
L2 = - L1
let L1, L2 be C_Linear_Combination of V; :: thesis: ( L1 + L2 = ZeroCLC V implies L2 = - L1 )
assume A1: L1 + L2 = ZeroCLC V ; :: thesis: L2 = - L1
let v be VECTOR of V; :: according to CONVEX4:def 7 :: thesis: L2 . v = (- L1) . v
(L1 . v) + (L2 . v) = (ZeroCLC V) . v by A1, Def8
.= 0c by Th2 ;
hence L2 . v = - (L1 . v)
.= (- L1) . v by Th31 ;
:: thesis: verum