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

theorem Th11:
  for p being Point of TOP-REAL 2, G being Subset of TOP-REAL 2 st
  G is open & p in G ex r being Real st r>0 & {q where q is Point of
  TOP-REAL 2: p`1-r<q`1 & q`1<p`1+r & p`2-r<q`2 & q`2<p`2+r} c= G
proof
  let p be Point of TOP-REAL 2,G being Subset of TOP-REAL 2;
  assume that
A1: G is open and
A2: p in G;
  reconsider GG=G as Subset of the TopStruct of TOP-REAL 2;
  reconsider q2=p as Point of Euclid 2 by TOPREAL3:8;
  TopSpaceMetr Euclid 2 = the TopStruct of TOP-REAL 2 & GG is open by A1,
EUCLID:def 8,PRE_TOPC:30;
  then consider r being Real such that
A3: r>0 and
A4: Ball(q2,r) c= GG by A2,TOPMETR:15;
  set s=r/sqrt(2);
A5: Ball(q2,r)= {q3 where q3 is Point of TOP-REAL 2: |.p-q3.|<r} by Th2;
A6: {q where q is Point of TOP-REAL 2: p`1-s<q`1 & q`1<p`1+s & p`2-s<q`2 & q
  `2<p`2+s} c= Ball(q2,r)
  proof
    let x be object;
    assume x in {q where q is Point of TOP-REAL 2: p`1-s<q`1 & q`1<p`1+s & p
    `2-s<q`2 & q`2<p`2+s};
    then consider q being Point of TOP-REAL 2 such that
A7: q=x and
A8: p`1-s<q`1 and
A9: q`1<p`1+s and
A10: p`2-s<q`2 and
A11: q`2<p`2+s;
    p`1+s-s>q`1-s by A9,XREAL_1:14;
    then
A12: p`1-q`1>q`1+-s-q`1 by XREAL_1:14;
    p`2+s-s>q`2-s by A11,XREAL_1:14;
    then
A13: p`2-q`2>q`2+-s-q`2 by XREAL_1:14;
    p`2-s+s<q`2+s by A10,XREAL_1:8;
    then p`2-q`2<q`2+s-q`2 by XREAL_1:14;
    then
A14: (p`2-q`2)^2<s^2 by A13,SQUARE_1:50;
    s^2=r^2/(sqrt(2))^2 by XCMPLX_1:76
      .=r^2/2 by SQUARE_1:def 2;
    then
A15: s^2+s^2=r^2;
    p`1-s+s<q`1+s by A8,XREAL_1:8;
    then p`1-q`1<q`1+s-q`1 by XREAL_1:14;
    then
A16: (p-q)`2=p`2-q`2 & (p`1-q`1)^2<s^2 by A12,SQUARE_1:50,TOPREAL3:3;
    (|.p-q.|)^2=((p-q)`1)^2+((p-q)`2)^2 & (p-q)`1=p`1-q`1 by JGRAPH_1:29
,TOPREAL3:3;
    then (|.p-q.|)^2<r^2 by A16,A14,A15,XREAL_1:8;
    then |.p-q.|<r by A3,SQUARE_1:48;
    hence thesis by A5,A7;
  end;
  s>0 by A3,XREAL_1:139;
  hence thesis by A4,A6,XBOOLE_1:1;
end;
