reserve n for Nat;
reserve p for Point of TOP-REAL n, r for Real;

theorem Th3:
  for S being open Subset of TOP-REAL n st p in S holds
  ex B being ball Subset of TOP-REAL n st B c= S & p in B
proof
  let S be open Subset of TOP-REAL n;
  assume A1: p in S;
  A2: TopStruct(# the carrier of TOP-REAL n,the topology of TOP-REAL n #)
  = TopSpaceMetr (Euclid n) by EUCLID:def 8;
  A3: S in Family_open_set Euclid n by A2,PRE_TOPC:def 2;
  reconsider x=p as Element of Euclid n by A2;
  consider r be Real such that
  A4: r > 0 & Ball(x,r) c= S by A1,A3,PCOMPS_1:def 4;
  reconsider r as positive Real by A4;
  reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
  reconsider B = Ball(p,r) as ball Subset of TOP-REAL n by Def1;
  take B;
  reconsider p1=p as Point of TOP-REAL n1;
  Ball(x,r) =  Ball(p1,r) by TOPREAL9:13;
  hence thesis by A4,GOBOARD6:1;
end;
