theorem :: BROUWER2:3

for n being Nat
for p, q being Point of (TOP-REAL n)
for S being Subset of (TOP-REAL n) st p in S & p <> q & S /\ (halfline (p,q)) is bounded holds
ex w being Point of (TOP-REAL n) st
( w in (Fr S) /\ (halfline (p,q)) & ( for u being Point of (TOP-REAL n) st u in S /\ (halfline (p,q)) holds
|.(p - u).| <= |.(p - w).| ) & ( for r being Real st r > 0 holds
ex u being Point of (TOP-REAL n) st
( u in S /\ (halfline (p,q)) & |.(w - u).| < r ) ) )