theorem Th3:
  - v = (- 1r) * v
proof
  v + (- 1r) * v = 1r * v + (- 1r) * v by Def5
    .= (1r + (- 1r)) * v by Def3
    .= 0.V by Th1;
  hence (- v) = (- v) + (v + (- 1r) * v) by RLVECT_1:4
    .= ((- v) + v) + (- 1r) * v by RLVECT_1:def 3
    .= 0.V + (- 1r) * v by RLVECT_1:5
    .= (- 1r) * v by RLVECT_1:4;
end;
