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 Th80:
  for P being compact Subset of TOP-REAL 2 holds
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies
  E-min P = |[1,0]| & E-max P = |[1,0]|
proof
  let P be compact Subset of T2;
  set M = E-most P;
  assume
A1: a,b realize-max-dist-in P;
  then
A2: M = {b} by Th78;
  set f = proj2|M;
A3: dom f = the carrier of (T2|M) by FUNCT_2:def 1;
A4: the carrier of (T2|M) = M by PRE_TOPC:8;
A5: b in {b} by TARSKI:def 1;
A6: f.:the carrier of (T2|M) = Im(f,b) by A1,A4,Th78
    .= {f.b} by A2,A3,A4,A5,FUNCT_1:59
    .= {proj2.b} by A2,A5,FUNCT_1:49
    .= {b`2} by PSCOMP_1:def 6;
  then
A7: lower_bound (proj2|M) = b`2 by SEQ_4:9;
A8: upper_bound (proj2|M) = b`2 by A6,SEQ_4:9;
  b = |[b`1,b`2]| by EUCLID:53;
  hence thesis by A1,A7,A8,Lm17,Th76;
end;
