reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  for P being compact connected Subset of TOP-REAL 2 st P c= C & W-min C
  in P & E-max C in P holds Upper_Arc C c= P or Lower_Arc C c= P
proof
  let P be compact connected Subset of TOP-REAL 2 such that
A1: P c= C and
A2: W-min C in P and
A3: E-max C in P;
A4: now
    given p being Point of TOP-REAL 2 such that
A5: P = {p};
    W-min C = p & E-max C = p by A2,A3,A5,TARSKI:def 1;
    hence contradiction by TOPREAL5:19;
  end;
  per cases by A1,A2,A4,BORSUK_4:56;
  suppose
    ex p1, p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2;
    hence thesis by A1,A2,A3,JORDAN16:22;
  end;
  suppose
    P = C;
    hence thesis by JORDAN6:61;
  end;
end;
