reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th2:
  for Lv be Linear_Combination of n-VectSp_over F_Real,
      Lr be Linear_Combination of TOP-REAL n st Lr = Lv
  holds Carrier Lr = Carrier Lv
proof
  set V=n-VectSp_over F_Real;
  set T=TOP-REAL n;
  let Lv be Linear_Combination of V,
      Lr be Linear_Combination of T such that
   A1: Lr=Lv;
  thus Carrier Lr c=Carrier Lv
  proof
   let x be object;
   assume A2: x in Carrier Lr;
   then reconsider v=x as Element of T;
   reconsider u=v as Element of V by Lm1;
   Lv.u<>0.F_Real by A1,A2,RLVECT_2:19;
   hence thesis by VECTSP_6:1;
  end;
  let x be object;
  assume x in Carrier Lv;
  then consider u be Element of V such that
   A3: x=u and
   A4: Lv.u<>0.F_Real by VECTSP_6:1;
  reconsider v=u as Element of T by Lm1;
  v in Carrier Lr by A1,A4,RLVECT_2:19;
  hence thesis by A3;
end;
