reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th30:
  Carrier (T@l) c= T .: (Carrier l)
proof
  T@l is Linear_Combination of T .: (Carrier l) by Th29;
  hence thesis by VECTSP_6:def 4;
end;
