theorem Th21:
  v in W implies v * a in W
proof
  reconsider VW = the carrier of W as Subset of V by Def2;
  assume v in W;
  then
A1: v in the carrier of W;
  VW is linearly-closed by Lm1;
  then v * a in the carrier of W by A1;
  hence thesis;
end;
