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 Th43:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n
    st Ar = At
  holds
    Affin Ar = Affin At
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n;
    assume
    A1: Ar = At;

    set AM = { B where B is Affine Subset of REAL-NS n : Ar c= B };
    set BM = { B where B is Affine Subset of TOP-REAL n : At c= B };

    A2: Affin Ar = meet AM by RLAFFIN1:def 6;
    A3: Affin At = meet BM by RLAFFIN1:def 6;

    At c= Affin At by RLAFFIN1:49; then
    A4: Affin At in BM;

    Ar c= Affin Ar by RLAFFIN1:49; then
    A5: Affin Ar in AM;

    for x be object holds
      x in Affin Ar
        iff
      x in Affin At
    proof
      let x be object;
      hereby
        assume
        A6: x in Affin Ar;

        now
          let Y be set;
          assume Y in BM;
          then
          consider B be Affine Subset of TOP-REAL n such that
          A7: Y = B & At c= B;
          reconsider A = B as Subset of REAL-NS n by Th4;
          reconsider A as Affine Subset of REAL-NS n by Th40;
          Y = A & Ar c= A by A1,A7; then
          Y in AM;
          hence x in Y by A2,A6,SETFAM_1:def 1;
        end;
        hence x in Affin At by A4;
      end;
      assume
      A8: x in Affin At;

      now
        let Y be set;
        assume Y in AM; then
        consider B be Affine Subset of REAL-NS n such that
        A9: Y = B & Ar c= B;
        reconsider A = B as Subset of TOP-REAL n by Th4;
        reconsider A as Affine Subset of TOP-REAL n by Th40;
        Y = B & At c= A by A1,A9;
        then Y in BM;
        hence x in Y by A3,A8,SETFAM_1:def 1;
      end;
      hence x in Affin Ar by A5;
    end;

    hence thesis by TARSKI:2;
  end;
