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 Th72:
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies
  P misses LSeg(|[-1,3]|,|[1,3]|)
proof
  assume
A1: a,b realize-max-dist-in P;
  assume P meets LSeg(lg,pg);
  then consider x being object such that
A2: x in P and
A3: x in LSeg(lg,pg) by XBOOLE_0:3;
  reconsider x as Point of T2 by A2;
  lg in LSeg(lg,pg) by RLTOPSP1:68;
  then
A4: x`2 = rg by A3,Lm25,Lm55;
A5: dist(a,x) = sqrt ((a`1-x`1)^2 + (a`2-x`2)^2) by TOPREAL6:92
    .= sqrt ((rl-x`1)^2 + rg^2) by A4,Lm18,EUCLID:52;
  0+4 < (rl-x`1)^2+9 by XREAL_1:8;
  then 2 < dist(a,x) by A5,SQUARE_1:20,27;
  hence thesis by A1,A2,Lm66;
end;
