theorem Th39:
  u in W & v in W implies u + v in W
proof
  reconsider VW = the carrier of W as Subset of V by Def8;
  assume u in W & v in W;
  then
A1: u in the carrier of W & v in the carrier of W;
  VW is linearly-closed by Lm3;
  then u + v in the carrier of W by A1;
  hence thesis;
end;
