let x be Real; :: thesis: for y, r being positive Real holds (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane
let y, r be positive Real; :: thesis: (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane
the carrier of Niemytzki-plane = y>=0-plane by Def3;
then reconsider a = |[x,y]| as Point of Niemytzki-plane by Th18;
consider BB being Neighborhood_System of Niemytzki-plane such that
for x being Real holds BB . |[x,0]| = { ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Real : q > 0 } and
A1: for x, y being Real st y > 0 holds
BB . |[x,y]| = { ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Real : q > 0 } by Def3;
BB . |[x,y]| = { ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Real : q > 0 } by A1;
then (Ball (|[x,y]|,r)) /\ y>=0-plane in BB . a ;
hence (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane by YELLOW_8:12; :: thesis: verum