theorem Th40:
  v in W implies z * v in W
proof
  reconsider VW = the carrier of W as Subset of V by Def8;
  assume v in W;
  then
A1: v in the carrier of W;
  VW is linearly-closed by Lm3;
  then z * v in the carrier of W by A1;
  hence thesis;
end;
