theorem Th4:
  v = - v implies v = 0.V
proof
  assume v = - v;
  then 0.V = v - (- v) by RLVECT_1:15
    .= v + v by RLVECT_1:17
    .= 1r * v + v by Def5
    .= 1r * v + 1r * v by Def5
    .= (1r + 1r) * v by Def3;
  hence thesis by Th2;
end;
