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 Th91:
  |[-1,0]|,|[1,0]| realize-max-dist-in D implies
  LSeg(|[0,3]|,UMP D) /\ D = {UMP D}
proof
  assume
A1: a,b realize-max-dist-in D;
  set m = UMP D;
  set w = (W-bound D + E-bound D) / 2;
A2: c`1 = w by A1,Lm87;
A3: m`1 = w by EUCLID:52;
A4: m in LSeg(c,m) by RLTOPSP1:68;
A5: m in D by JORDAN21:30;
  thus LSeg(c,m) /\ D c= {m}
  proof
    let x be object;
    assume
A6: x in LSeg(c,m) /\ D;
    then
A7: x in LSeg(c,m) by XBOOLE_0:def 4;
A8: x in D by A6,XBOOLE_0:def 4;
    reconsider x as Point of T2 by A6;
    LSeg(c,m) is vertical by A2,A3,SPPOL_1:16;
    then
A9: x`1 = m`1 by A4,A7;
    then x in Vertical_Line w by A3,JORDAN6:31;
    then x in D /\ Vertical_Line w by A8,XBOOLE_0:def 4;
    then
A10: x`2 <= m`2 by JORDAN21:28;
    m`2 <= x`2 by A1,A7,Th85;
    then x`2 = m`2 by A10,XXREAL_0:1;
    then x = m by A9,TOPREAL3:6;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {m};
  then x = m by TARSKI:def 1;
  hence thesis by A4,A5,XBOOLE_0:def 4;
end;
