let V be ComplexLinearSpace; :: thesis: for v being VECTOR of V holds - v = (- 1r ) * v
let v be VECTOR of V; :: thesis: - v = (- 1r ) * v
v + ((- 1r ) * v) = (1r * v) + ((- 1r ) * v) by Def5
.= (1r + (- 1r )) * v by Def3
.= 0. V by Th2 ;
hence - v = (- v) + (v + ((- 1r ) * v)) by RLVECT_1:10
.= ((- v) + v) + ((- 1r ) * v) by RLVECT_1:def 6
.= (0. V) + ((- 1r ) * v) by RLVECT_1:16
.= (- 1r ) * v by RLVECT_1:10 ;
:: thesis: verum