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 Th6:
  for Af be Subset of n-VectSp_over F_Real,
      Ar be Subset of TOP-REAL n st Af = Ar
  holds [#]Lin Ar = [#]Lin Af
proof
  set V=n-VectSp_over F_Real;
  set T=TOP-REAL n;
  let Af be Subset of V;
  let Ar be Subset of T such that
   A1: Af=Ar;
  hereby let x be object;
   assume x in [#]Lin Ar;
   then x in Lin Ar;
   then consider L be Linear_Combination of Ar such that
    A2: x=Sum L by RLVECT_3:14;
   reconsider L1=L as Linear_Combination of V by Th1;
   Carrier L1=Carrier L & Carrier L c=Ar by Th2,RLVECT_2:def 6;
   then A3: L1 is Linear_Combination of Af by A1,VECTSP_6:def 4;
   Sum L1=Sum L by Th5;
   then x in Lin Af by A2,A3,VECTSP_7:7;
   hence x in [#]Lin Af;
  end;
  let x be object;
  assume x in [#]Lin Af;
  then x in Lin Af;
  then consider L be Linear_Combination of Af such that
   A4: x=Sum L by VECTSP_7:7;
  reconsider L1=L as Linear_Combination of T by Th1;
  Carrier L1=Carrier L & Carrier L c=Af by Th2,VECTSP_6:def 4;
  then A5: L1 is Linear_Combination of Ar by A1,RLVECT_2:def 6;
  Sum L1=Sum L by Th5;
  then x in Lin Ar by A4,A5,RLVECT_3:14;
  hence thesis;
end;
