let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-unital RLSStruct ; :: thesis: for v being Element of V holds - v = (- 1) * v
let v be Element of V; :: thesis: - v = (- 1) * v
v + ((- 1) * v) = (1 * v) + ((- 1) * v) by Def8
.= (1 + (- 1)) * v by Def6
.= 0. V by Th10 ;
hence - v = (- v) + (v + ((- 1) * v))
.= ((- v) + v) + ((- 1) * v) by Def3
.= (0. V) + ((- 1) * v) by Def10
.= (- 1) * v ;
:: thesis: verum