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 Th25:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n
    st Ar = At
  holds
    for X be object holds
      X is Linear_Combination of Ar
        iff
      X is Linear_Combination of At
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n;
    assume
    A1: Ar = At;

    let X be object;
    hereby
      assume X is Linear_Combination of Ar; then
      reconsider L=X as Linear_Combination of Ar;
      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;
      hence X is Linear_Combination of At by A1, RLVECT_2:def 6;
    end;

    assume X is Linear_Combination of At; then
    reconsider L = X as Linear_Combination of At;
    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;
    hence X is Linear_Combination of Ar by A1,RLVECT_2:def 6;
  end;
