theorem Th34:
  (-v)*x = -v*x
proof
  (-v)*x = (v*(-1_R))*x by Th32
    .= v*((-1_R) *x) by Def8
    .= v*(-(1_R *x)) by VECTSP_1:9
    .= v*(-x);
  hence thesis by Th33;
end;
