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 Th16:
  for C being compact non empty Subset of TOP-REAL 2 st C is
  being_simple_closed_curve holds N-bound(C) > S-bound(C)
proof
  let C be compact non empty Subset of TOP-REAL 2;
  assume
A1: C is being_simple_closed_curve;
  now
    assume
A2: N-bound C <= S-bound C;
    for p st p in C holds p`2=S-bound C
    proof
      let p;
      assume p in C;
      then
A3:   S-bound C <= p`2 & p`2 <= N-bound C by PSCOMP_1:24;
      then S-bound C <= N-bound C by XXREAL_0:2;
      then S-bound C = N-bound C by A2,XXREAL_0:1;
      hence thesis by A3,XXREAL_0:1;
    end;
    hence contradiction by A1,Th14;
  end;
  hence thesis;
end;
