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 Th20:
  for q1,q2 being Point of TOP-REAL 2 st LE q1,q2,C
  holds Segment(q1,q2,C) is compact
proof
  let q1,q2 be Point of TOP-REAL 2 such that
A1: LE q1,q2,C;
A2: q1 in C by A1,JORDAN7:5;
  per cases;
  suppose q2 = W-min C;
    hence thesis by A2,Th19;
  end;
  suppose q1 = q2 & q2 <> W-min C;
    then Segment(q1,q2,C) = {q1} by A2,JORDAN7:8;
    hence thesis;
  end;
  suppose q1 <> q2 & q2 <> W-min C;
    hence thesis by A1,Th17,JORDAN5A:1;
  end;
end;
