reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th87:
  |[-1,0]|,|[1,0]| realize-max-dist-in D implies
  LSeg(|[0,3]|,UMP D) c= north_halfline UMP D
proof
  set p = UMP D;
  assume
A1: a,b realize-max-dist-in D;
  let x be object;
  assume
A2: x in LSeg(c,p);
  then reconsider x as Point of T2;
A3: p in LSeg(c,p) by RLTOPSP1:68;
  LSeg(c,p) is vertical by A1,Th81;
  then
A4: x`1 = p`1 by A2,A3;
  p`2 <= x`2 by A1,A2,Th85;
  hence thesis by A4,TOPREAL1:def 10;
end;
