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 Th39:
  T|(Carrier l) is one-to-one implies T .: (Carrier l) = Carrier ( T@l)
proof
  assume
A1: T|(Carrier l) is one-to-one;
A2: T .: (Carrier l) c= Carrier (T@l)
  proof
    let w be object;
    assume w in T .: (Carrier l);
    then consider v being object such that
A3: v in dom T and
A4: v in Carrier l and
A5: T.v = w by FUNCT_1:def 6;
    reconsider v as Element of V by A3;
    (T@l).(T.v) = l.v & l.v <> 0.F by A1,A4,Th37,VECTSP_6:2;
    hence thesis by A5;
  end;
  Carrier (T@l) c= T .: (Carrier l) by Th30;
  hence thesis by A2;
end;
