reserve T,T1,T2,S for non empty TopSpace;

theorem Th2:
  for n being Element of NAT,q2 being Point of Euclid n, q being
  Point of TOP-REAL n, r being Real st q=q2 holds Ball(q2,r) = {q3 where
  q3 is Point of TOP-REAL n: |.q-q3.|<r}
proof
  let n be Element of NAT,q2 be Point of (Euclid n), q be Point of TOP-REAL n,
  r be Real;
  assume
A1: q=q2;
A2: {q4 where q4 is Element of Euclid n: dist(q2,q4) < r} c= {q3 where q3 is
  Point of TOP-REAL n: |.q-q3.|<r}
  proof
    let x be object;
    assume x in {q4 where q4 is Element of Euclid n: dist(q2,q4) < r};
    then consider q4 being Element of Euclid n such that
A3: q4=x & dist(q2,q4) < r;
    reconsider q44=q4 as Point of TOP-REAL n by TOPREAL3:8;
    dist(q2,q4)=|.q-q44.| by A1,JGRAPH_1:28;
    hence thesis by A3;
  end;
A4: {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r} c={q4 where q4 is
  Element of Euclid n: dist(q2,q4) < r}
  proof
    let x be object;
    assume x in {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r};
    then consider q3 being Point of TOP-REAL n such that
A5: x=q3 & |.q-q3.|<r;
    reconsider q34=q3 as Point of Euclid n by TOPREAL3:8;
    dist(q2,q34)=|.q-q3.| by A1,JGRAPH_1:28;
    hence thesis by A5;
  end;
  Ball(q2,r)= {q4 where q4 is Element of Euclid n: dist(q2,q4) < r} by
METRIC_1:17;
  hence thesis by A2,A4;
end;
