reserve x for set;
reserve a,b,d,ra,rb,r0,s1,s2 for Real;
reserve r,s,r1,r2,r3,rc for Real;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve X,Y,Z for non empty TopSpace;

theorem
  for P being compact non empty Subset of TOP-REAL 2 st P is
  being_simple_closed_curve holds S-min(P)<>N-max(P)
proof
  let P be compact non empty Subset of TOP-REAL 2;
  assume
A1: P is being_simple_closed_curve;
  now
A2: |[lower_bound (proj1|S-most P), S-bound P]|=S-min(P) &
|[upper_bound (proj1|N-most P),
    N-bound P]|=N-max(P) by PSCOMP_1:def 22,def 26;
    assume S-min(P)=N-max(P);
    then S-bound P= N-bound P by A2,SPPOL_2:1;
    hence contradiction by A1,Th16;
  end;
  hence thesis;
end;
