reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;

theorem Th21:
  v in W implies a * v 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 a * v in the carrier of W by A1;
  hence thesis;
end;
