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 Th26:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n st Ar = At
  holds [#]Lin Ar = [#]Lin At
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n;
    assume
    A1: Ar = At;
    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 TOP-REAL n by Th11;
      Carrier L1 = Carrier L & Carrier L c= Ar by RLVECT_2:def 6;
      then A3: L1 is Linear_Combination of At by A1,RLVECT_2:def 6;
      Sum L1 = Sum L by Th23;
      then x in Lin At by A2,A3,RLVECT_3:14;
      hence x in [#] (Lin At);
    end;
    let x be object;
    assume x in [#] (Lin At);
    then x in Lin At;
    then consider L be Linear_Combination of At such that
    A4: x = Sum L by RLVECT_3:14;

    reconsider L1 = L as Linear_Combination of REAL-NS n by Th11;
    Carrier L1 = Carrier L & Carrier L c= At by RLVECT_2:def 6; then
    A5: L1 is Linear_Combination of Ar by A1,RLVECT_2:def 6;
    Sum L1 = Sum L by Th23;
    then x in Lin Ar by A4,A5,RLVECT_3:14;
    hence x in [#] (Lin Ar);
  end;
