reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th70:
  for p being Point of TOP-REAL 2 st p`2 = 0 for x being Real
, r being positive Real st +(x,r).p = 1 ex r1 being positive
  Real st Ball(|[p`1,r1]|,r1) \/ {p} c= +(x,r)"{1}
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 = 0;
  let x be Real;
  let r be positive Real;
  set r1 = (|.p-|[x,r]|.|-r)/2;
  set f = +(x,r);
A2: p =|[p`1,p`2]| by EUCLID:53;
  assume
A3: f.p = 1;
  then p`1-x <> 0 by A2,A1,Def5;
  then (p`1-x)^2 > 0 by SQUARE_1:12;
  then (p`1-x)^2+(0-r)^2 > 0+(0-r)^2 by XREAL_1:6;
  then |.|[p`1-x,p`2-r]|.|^2 > r^2 by A1,Th9;
  then |.p-|[x,r]|.|^2 > r^2 by A2,EUCLID:62;
  then |.p-|[x,r]|.| > r by SQUARE_1:15;
  then |.p-|[x,r]|.|-r > 0 by XREAL_1:50;
  then reconsider r1 as positive Real;
  take r1;
  let u be object;
  assume
A4: u in Ball(|[p`1,r1]|,r1) \/ {p};
  then reconsider q = u as Point of TOP-REAL 2;
A5: Ball(|[p`1,r1]|,r1) c= y>=0-plane by Th20;
  u in Ball(|[p`1,r1]|,r1) or u = p by A4,ZFMISC_1:136;
  then reconsider z = q as Point of Niemytzki-plane by A5,A1,A2,Lm1,Th18;
A6: q = |[q`1,q`2]| by EUCLID:53;
A7: now
    assume
A8: q in Ball(|[p`1,r1]|,r1);
    then |.q-|[p`1,r1]|.| < r1 by TOPREAL9:7;
    then
A9: |.q-|[p`1,r1]|.|+|.|[p`1,r1]|-p.| < r1+|.|[p`1,r1]|-p.| by XREAL_1:6;
A10: |.r1.| = r1 by ABSVALUE:def 1;
A11: |.q-|[p`1,r1]|.|+|.|[p`1,r1]|-p.| >= |.q-p.| by TOPRNS_1:34;
A12: |.q-p.|+|.|[x,r]|-q.| >= |.|[x,r]|-p.| by TOPRNS_1:34;
A13: |.|[0,r1]|.| = |.r1.| by TOPREAL6:23;
    |[p`1,r1]|-p = |[p`1-p`1,r1-0]| by A1,A2,EUCLID:62;
    then r1+r1 > |.q-p.| by A9,A11,A10,A13,XXREAL_0:2;
    then |.p-|[x,r]|.|-r+|.|[x,r]|-q.| > |.q-p.|+|.|[x,r]|-q.| by XREAL_1:6;
    then |.|[x,r]|-p.| < |.p-|[x,r]|.|-r+|.|[x,r]|-q.| by A12,XXREAL_0:2;
    then
A14: |.|[x,r]|-p.|-(|.p-|[x,r]|.|-r) < |.|[x,r]|-q.| by XREAL_1:19;
    |.p-|[x,r]|.| = |.|[x,r]|-p.| by TOPRNS_1:27;
    then |.q-|[x,r]|.| > r by A14,TOPRNS_1:27;
    hence not q in Ball(|[x,r]|,r) by TOPREAL9:7;
    q`2 = 0 implies q in y=0-line & Ball(|[p`1,r1]|,r1) misses y=0-line
    by A6,Th21;
    hence q`2 <> 0 by A8,XBOOLE_0:3;
  end;
  z in y>=0-plane by Lm1;
  then q`2 >= 0 by A6,Th18;
  then f.z = 1 by A7,A3,A4,A6,Def5,ZFMISC_1:136;
  then f.z in {1} by TARSKI:def 1;
  hence thesis by FUNCT_2:38;
end;
