theorem ThMak3:
  for S being TarskiGeometryStruct st
    S is (RE) & S is (TE) holds
      (S is (FS) iff S is (FS'))
  proof
    let S be TarskiGeometryStruct;
    assume that
A1: S is (RE) and
A2: S is (TE);
    hereby
      assume
A3:   S is (FS);
      thus S is (FS')
      proof
        let a,b,c,d,a9,b9,c9,d9 be POINT of S;
        assume a <> b & between a,b,c & between a9,b9,c9 &
          a,b equiv a9,b9 & b,c equiv b9,c9 & a,d equiv a9,d9 &
          b,d equiv b9,d9;
        then c,d equiv c9,d9 & c,d equiv d,c by A1,A3;
        hence d,c equiv c9,d9 by A2;
      end;
    end;
    assume
A4: S is (FS');
    let a,b,c,d,a9,b9,c9,d9 be POINT of S;
    assume a <> b & between a,b,c & between a9,b9,c9 &
      a,b equiv a9,b9 & b,c equiv b9,c9 & a,d equiv a9,d9 & b,d equiv b9,d9;
    then d,c equiv c9,d9 & d,c equiv c,d by A1,A4;
    hence c,d equiv c9,d9 by A2;
  end;
