reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;

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;
