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

theorem Th69:
  for p being Point of TOP-REAL 2 st p`2 > 0 for x,a being Real
, r being positive Real st 0 <= a & a < +(x,r).p ex r1 being
  positive Real st r1 <= p`2 & Ball(p,r1) c= +(x,r)"].a,1.]
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 > 0;
  let x,a be Real;
  let r be positive Real;
  set f = +(x,r);
  assume that
A2: 0 <= a and
A3: a < f.p;
A4: p = |[p`1,p`2]| by EUCLID:53;
  then p in the carrier of Niemytzki-plane by A1,Lm1;
  then f.p in the carrier of I[01] by FUNCT_2:5;
  then f.p <= 1 by BORSUK_1:40,XXREAL_1:1;
  then
A5: a < 1 by A3,XXREAL_0:2;
  per cases by A2;
  suppose
A6: a = 0;
    reconsider r1 = p`2 as positive Real by A1;
    reconsider A = Ball(p,r1) as Subset of Niemytzki-plane by A4,Lm1,Th20;
    take r1;
    thus r1 <= p`2;
    let u be object;
    assume
A7: u in Ball(p,r1);
    then reconsider q = u as Point of TOP-REAL 2;
A8: q = |[q`1,q`2]| by EUCLID:53;
    q in A by A7;
    then
A9: q`2 >= 0 by A8,Lm1,Th18;
    q in A by A7;
    then reconsider z = q as Element of Niemytzki-plane;
A10: f.z >= 0 by BORSUK_1:40,XXREAL_1:1;
    y=0-line misses Ball(p,r1) by A4,Th21;
    then not q in y=0-line by A7,XBOOLE_0:3;
    then
A11: q`2 <> 0 by A8;
A12: f.z <= 1 by BORSUK_1:40,XXREAL_1:1;
    |[x,0]|`2 = 0 by EUCLID:52;
    then f.q <> 0 by A11,A9,Th60;
    then f.z in ].a,1.] by A10,A12,A6,XXREAL_1:2;
    hence thesis by FUNCT_2:38;
  end;
  suppose
    a > 0;
    then reconsider b = a as positive Real;
    set r1 = min(|.p-|[x,r*a]|.|-r*a, p`2);
A13: r1 = |.p-|[x,r*a]|.|-r*a or r1 = p`2 by XXREAL_0:def 9;
A14: b <> f.p by A3;
    not |.p-|[x,r*a]|.| < r*a by A3,A5,Th63;
    then |.p-|[x,r*a]|.| > r*a by A14,A1,A5,Th62,XXREAL_0:1;
    then reconsider r1 as positive Real by A13,A1,XREAL_1:50;
    take r1;
    thus r1 <= p`2 by XXREAL_0:17;
    then reconsider A = Ball(p,r1) as Subset of Niemytzki-plane by A4,Lm1,Th20;
    let u be object;
    assume
A15: u in Ball(p,r1);
    then reconsider q = u as Point of TOP-REAL 2;
    u in A by A15;
    then reconsider z = q as Point of Niemytzki-plane;
A16: q = |[q`1,q`2]| by EUCLID:53;
    q in A by A15;
    then
A17: q`2 >= 0 by A16,Lm1,Th18;
A18: now
      |.p-|[x,r*a]|.|-r*a >= r1 by XXREAL_0:17;
      then
A19:  r*a+r1 <= |.p-|[x,r*a]|.|-r*a+r*a by XREAL_1:6;
      assume not f.z > a;
      then |.q-|[x,r*a]|.| <= r*a by A2,A5,A17,Th64;
      then
A20:  |.|[x,r*a]|-q.| <= r*a by TOPRNS_1:27;
A21:  |.|[x,r*a]|-q.|+|.q-p.| >= |.|[x,r*a]|-p.| by TOPRNS_1:34;
      |.q-p.| < r1 by A15,TOPREAL9:7;
      then |.|[x,r*a]|-q.|+|.q-p.| < r*a+r1 by A20,XREAL_1:8;
      then |.|[x,r*a]|-p.| < r*a+r1 by A21,XXREAL_0:2;
      hence contradiction by A19,TOPRNS_1:27;
    end;
    f.z <= 1 by BORSUK_1:40,XXREAL_1:1;
    then f.z in ].a,1.] by A18,XXREAL_1:2;
    hence thesis by FUNCT_2:38;
  end;
end;
