let V be RealLinearSpace; :: thesis: for v being VECTOR of V holds - v = (- 1) * v
let v be VECTOR of V; :: thesis: - v = (- 1) * v
v + ((- 1) * v) = (1 * v) + ((- 1) * v) by Def11
.= (1 + (- 1)) * v by Def9
.= 0. V by Th23 ;
hence - v = (- v) + (v + ((- 1) * v)) by Th10
.= ((- v) + v) + ((- 1) * v) by Def6
.= (0. V) + ((- 1) * v) by Def13
.= (- 1) * v by Th10 ;
:: thesis: verum