let V be RealLinearSpace; :: thesis: for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds

L2 = - L1

let L1, L2 be Linear_Combination of V; :: thesis: ( L1 + L2 = ZeroLC V implies L2 = - L1 )

assume A1: L1 + L2 = ZeroLC V ; :: thesis: L2 = - L1

let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: L2 . v = (- L1) . v

(L1 . v) + (L2 . v) = (ZeroLC V) . v by A1, Def10

.= 0 by Th20 ;

hence L2 . v = - (L1 . v)

.= (- L1) . v by Th49 ;

:: thesis: verum

L2 = - L1

let L1, L2 be Linear_Combination of V; :: thesis: ( L1 + L2 = ZeroLC V implies L2 = - L1 )

assume A1: L1 + L2 = ZeroLC V ; :: thesis: L2 = - L1

let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: L2 . v = (- L1) . v

(L1 . v) + (L2 . v) = (ZeroLC V) . v by A1, Def10

.= 0 by Th20 ;

hence L2 . v = - (L1 . v)

.= (- L1) . v by Th49 ;

:: thesis: verum