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 Th96:
  |[-1,0]|,|[1,0]| realize-max-dist-in C implies
  for Jc, Jd being compact with_the_max_arc Subset of TOP-REAL 2 st
  Jc is_an_arc_of |[-1,0]|,|[1,0]| & Jd is_an_arc_of |[-1,0]|,|[1,0]| &
  C = Jc \/ Jd & Jc /\ Jd = {|[-1,0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd &
  W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
  BDD C = Component_of Down ((1/2) *
  ((UMP (LSeg(LMP Jc,|[0,-3]|) /\ Jd)) + LMP Jc), C`)
proof
  assume
A1: a,b realize-max-dist-in C;
  let Jc, Jd being compact with_the_max_arc Subset of T2 such that
A2: Jc is_an_arc_of a,b and
A3: Jd is_an_arc_of a,b and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {a,b} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc;
  reconsider
  Ux = Component_of Down((1/2) * ((UMP (LSeg(LMP Jc,d) /\ Jd)) + LMP Jc),C`)
  as Subset of T2 by PRE_TOPC:11;
  Ux = BDD C
  proof
    Ux is_inside_component_of C by A1,A2,A3,A4,A5,A6,A7,A8,A9,Th95;
    hence Ux c= BDD C by JORDAN2C:22;
    set F = {B where B is Subset of T2: B is_inside_component_of C};
    let q be object;
    assume q in BDD C;
    then consider Z being set such that
A10: q in Z and
A11: Z in F by TARSKI:def 4;
    ex B being Subset of T2 st Z = B & B is_inside_component_of C by A11;
    hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,Th95;
  end;
  hence thesis;
end;
