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 Th77:
  for T be RealLinearSpace,
     Af be Subset of RLSp2RVSp(T),
     Ar be Subset of T
    st Af = Ar
  holds [#] (Lin Ar) = [#] (Lin Af)
  proof
    let T be RealLinearSpace;
    let Af be Subset of RLSp2RVSp(T);
    let Ar be Subset of T;
    assume
    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 RLSp2RVSp(T) by Th72;
      Carrier L1 = Carrier L & Carrier L c= Ar
        by Th73,RLVECT_2:def 6; then
      A3: L1 is Linear_Combination of Af by A1,VECTSP_6:def 4;
      Sum L1 = Sum L by Th76;
      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 Th72;
    Carrier L1 = Carrier L & Carrier L c= Af by Th73,VECTSP_6:def 4; then
    A5: L1 is Linear_Combination of Ar by A1,RLVECT_2:def 6;
    Sum L1 = Sum L by Th76;
    then x in Lin Ar by A4,A5,RLVECT_3:14;
    hence x in [#] (Lin Ar);
  end;
