reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th19:
  for q being Point of TOP-REAL 2 st q in C
  holds Segment(q,W-min C,C) is compact
proof
  let q be Point of TOP-REAL 2 such that
A1: q in C;
  per cases;
  suppose q = W-min C;
    hence thesis by Th18;
  end;
  suppose q <> W-min C;
    hence thesis by A1,Th16,JORDAN5A:1;
  end;
end;
