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 Th40:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n
    st Ar = At
  holds
    Ar is Affine
      iff
    At is Affine
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n;
    assume
    A1: Ar = At;

    hereby
      assume
      A2: Ar is Affine;

      for x, y be VECTOR of TOP-REAL n
      for a be Real st x in At & y in At holds
      ((1 - a) * x) + (a * y) in At
      proof
        let x, y be VECTOR of TOP-REAL n;
        let a be Real;
        assume
        A3: x in At & y in At;
        reconsider x0 = x, y0 = y as VECTOR of REAL-NS n by Th4;
        (1 - a) * x = (1 - a) * x0 &
        a * y = a * y0 by Th8; then
        ((1 - a) * x) + (a * y) = ((1 - a) * x0) + (a * y0) by Th7;
        hence ((1 - a) * x) + (a * y) in At by A1,A2,A3,RUSUB_4:def 4;
      end;

      hence At is Affine by RUSUB_4:def 4;
    end;
    assume
    A4: At is Affine;

    for x, y be VECTOR of REAL-NS n
    for a be Real
      st x in Ar & y in Ar
    holds
      ((1 - a) * x) + (a * y) in Ar
    proof
      let x, y be VECTOR of REAL-NS n;
      let a be Real;
      assume
      A5: x in Ar & y in Ar;
      reconsider x0 = x, y0 = y as VECTOR of TOP-REAL n by Th4;

      (1 - a) * x = (1 - a) * x0 &
      a * y = a * y0 by Th8;

      then ((1 - a) * x) + (a * y)
         = ((1 - a) * x0) + (a * y0) by Th7;

      hence ((1 - a) * x) + (a * y) in Ar by A1,A4,A5,RUSUB_4:def 4;
    end;
    hence Ar is Affine by RUSUB_4:def 4;
  end;
