let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds

L2 = - L1

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: 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 Element of V; :: according to VECTSP_6:def 7 :: thesis: L2 . v = (- L1) . v

(L1 . v) + (L2 . v) = (ZeroLC V) . v by A1, Th22

.= 0. GF by Th3 ;

hence L2 . v = - (L1 . v) by RLVECT_1:6

.= (- L1) . v by Th36 ;

:: thesis: verum

for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds

L2 = - L1

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: 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 Element of V; :: according to VECTSP_6:def 7 :: thesis: L2 . v = (- L1) . v

(L1 . v) + (L2 . v) = (ZeroLC V) . v by A1, Th22

.= 0. GF by Th3 ;

hence L2 . v = - (L1 . v) by RLVECT_1:6

.= (- L1) . v by Th36 ;

:: thesis: verum