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 Th83:
  |[-1,0]|,|[1,0]| realize-max-dist-in P & p in P implies p`2 < 3
proof
  assume that
A1: a,b realize-max-dist-in P and
A2: p in P;
A3: P /\ dR = {a,b} by A1,Th74;
  P c= R by A1,Th71;
  then p in R by A2;
  then
A4: ex p1 st p1 = p & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
  now
    assume
A5: p`2 = c`2;
    then p in LSeg(lg,pg) by A4,Lm21,Lm24,Lm25,Lm28,Lm29,GOBOARD7:8;
    then p in P /\ dR by A2,Lm40,XBOOLE_0:def 4;
    hence contradiction by A3,A5,Lm18,Lm19,Lm21,TARSKI:def 2;
  end;
  hence thesis by A4,Lm21,XXREAL_0:1;
end;
