let V be Z_Module; :: thesis: for v being VECTOR of V holds - v = (- 1) * v
let v be VECTOR of V; :: thesis: - v = (- 1) * v
reconsider N1 = 1, N0 = 0 , M1 = - 1 as Integer ;
v + ((- 1) * v) = (1 * v) + ((- 1) * v) by Def5
.= (1 + (- 1)) * v by Def3
.= 0. V by Th1 ;
hence - v = (- v) + (v + ((- 1) * v)) by RLVECT_1:4
.= ((- v) + v) + ((- 1) * v) by RLVECT_1:def 3
.= (0. V) + ((- 1) * v) by RLVECT_1:def 10
.= (- 1) * v by RLVECT_1:4 ;
:: thesis: verum