let V be Z_Module; :: thesis: for L1, L2 being Z_Linear_Combination of V st L1 + L2 = Z_ZeroLC V holds
L2 = - L1
let L1, L2 be Z_Linear_Combination of V; :: thesis: ( L1 + L2 = Z_ZeroLC V implies L2 = - L1 )
assume A1: L1 + L2 = Z_ZeroLC V ; :: thesis: L2 = - L1
let v be VECTOR of V; :: according to ZMODUL02:def 24 :: thesis: L2 . v = (- L1) . v
(L1 . v) + (L2 . v) = (Z_ZeroLC V) . v by A1, Def25
.= 0 by Th9 ;
hence L2 . v = - (L1 . v)
.= (- L1) . v by Th35 ;
:: thesis: verum