reserve n for Nat,
  i,j for Nat,
  r,s,r1,s1,r2,s2,r9,s9 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board,
  x,y for set,
  v for Point of Euclid 2;

theorem Th4:
  for M being non empty MetrSpace, u being Point of M, P being
Subset of TopSpaceMetr(M) holds u in Int P iff ex r being Real st r > 0
  & Ball(u,r) c= P
proof
  let M be non empty MetrSpace, u be Point of M, P be Subset of TopSpaceMetr(M
  );
  hereby
    assume u in Int P;
    then consider r be Real such that
A1: r > 0 and
A2: Ball(u,r) c= Int P by TOPMETR:15;
    take r;
    thus r > 0 by A1;
    Int P c= P by TOPS_1:16;
    hence Ball(u,r) c= P by A2;
  end;
  given r being Real such that
A3: r > 0 and
A4: Ball(u,r) c= P;
  TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by
PCOMPS_1:def 5;
  then reconsider B = Ball(u,r) as Subset of TopSpaceMetr(M);
A5: B is open by Lm4;
  u in Ball(u,r) by A3,Th1;
  hence thesis by A4,A5,TOPS_1:22;
end;
