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
  |[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(|[0,3]|,|[0,-3]|) meets C
proof
  assume
A1: a,b realize-max-dist-in C;
  set Jc = Upper_Arc C;
  consider Pf being Path of c,d, f being Function
  of I[01], T2|LSeg(c,d) such that
A2: rng f = LSeg(c,d) and
A3: Pf = f by Th43;
A4: a = W-min C by A1,Th79;
  b = E-max C by A1,Th80;
  then Jc is_an_arc_of a,b by A4,JORDAN6:def 8;
  then consider Pg being Path of a,b, g being Function
  of I[01], T2|Jc such that
A5: rng g = Jc and
A6: Pg = g by Th42;
A7: Jc c= C by JORDAN6:61;
A8: C c= R by A1,Th71;
A9: a in C by A1;
A10: b in C by A1;
A11: the carrier of TR = R by PRE_TOPC:8;
  reconsider AR = a, BR = b, CR = c, DR = d
  as Point of TR by A8,A9,A10,Lm62,Lm63,Lm67,PRE_TOPC:8;
  rng Pg c= the carrier of TR by A5,A6,A7,A8,A11;
  then reconsider h = Pg as Path of AR,BR by Th30;
  LSeg(c,d) c= R by Lm62,Lm63,Lm67,JORDAN1:def 1;
  then reconsider v = Pf as Path of CR,DR by A2,A3,A11,Th30;
  consider s, t being Point of I[01] such that
A12: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A13: dom h = the carrier of I[01] by FUNCT_2:def 1;
  dom v = the carrier of I[01] by FUNCT_2:def 1;
  then
A14: v.t in rng Pf by FUNCT_1:def 3;
  h.s in rng Pg by A13,FUNCT_1:def 3;
  hence thesis by A2,A3,A5,A6,A7,A12,A14,XBOOLE_0:3;
end;
