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 Th75:
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies W-bound P = -1
proof
  assume
A1: a,b realize-max-dist-in P;
  then
A2: P c= R by Th71;
A3: P = the carrier of (T2|P) by PRE_TOPC:8;
A4: a in P by A1;
  reconsider P as non empty Subset of T2 by A1;
  reconsider Z = (proj1|P).:the carrier of (T2|P) as Subset of REAL;
A5: for p be Real st p in Z holds p >= rl
  proof
    let p be Real;
    assume p in Z;
    then consider p0 being object such that
A6: p0 in the carrier of T2|P and p0 in the carrier of T2|P and
A7: p = (proj1|P).p0 by FUNCT_2:64;
    p0 in R by A2,A3,A6;
    then ex p1 st p0 = p1 & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
    hence thesis by A3,A6,A7,PSCOMP_1:22;
  end;
  for q being Real st
  for p being Real st p in Z holds p >= q holds rl >= q
  proof
    let q be Real such that
A8: for p being Real st p in Z holds p >= q;
    (proj1|P).a = a`1 by A4,PSCOMP_1:22;
    hence thesis by A3,A4,A8,Lm16,FUNCT_2:35;
  end;
  hence thesis by A5,SEQ_4:44;
end;
