reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  for C being compact Subset of TOP-REAL 2 st p in C /\ Vertical_Line ((
  W-bound C + E-bound C) / 2) holds (LMP C)`2 <= p`2
proof
  let C be compact Subset of TOP-REAL 2 such that
A1: p in C /\ Vertical_Line ((W-bound C + E-bound C) / 2);
  p`2 = proj2.p by PSCOMP_1:def 6;
  then
A2: p`2 in proj2.:(C /\ Vertical_Line ((E-bound C + W-bound C) / 2)) by A1,
FUNCT_2:35;
  (LMP C)`2 =
   lower_bound (proj2.:(C /\ Vertical_Line ((E-bound C + W-bound C) / 2
  ))) & proj2.:(C /\ Vertical_Line ((E-bound C + W-bound C) / 2)) is non empty
  bounded_below by A1,Lm1,Th13,EUCLID:52,RELAT_1:119;
  hence thesis by A2,SEQ_4:def 2;
end;
