let p be Point of (TOP-REAL 2); :: thesis: for C being compact Subset of (TOP-REAL 2) st p in BDD C holds
inf (proj2 .: (C /\ (north_halfline p))) > sup (proj2 .: (C /\ (south_halfline p)))
let C be compact Subset of (TOP-REAL 2); :: thesis: ( p in BDD C implies inf (proj2 .: (C /\ (north_halfline p))) > sup (proj2 .: (C /\ (south_halfline p))) )
assume p in BDD C ; :: thesis: inf (proj2 .: (C /\ (north_halfline p))) > sup (proj2 .: (C /\ (south_halfline p)))
then A1: ( (South-Bound p,C) `2 < p `2 & p `2 < (North-Bound p,C) `2 ) by Th23;
( (North-Bound p,C) `2 = inf (proj2 .: (C /\ (north_halfline p))) & (South-Bound p,C) `2 = sup (proj2 .: (C /\ (south_halfline p))) ) by EUCLID:56;
hence inf (proj2 .: (C /\ (north_halfline p))) > sup (proj2 .: (C /\ (south_halfline p))) by A1, XXREAL_0:2; :: thesis: verum