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

theorem Th80:
  for p being Point of TOP-REAL 2 st p`2 = 0 for x being Real
, a being non negative Real for y,r being positive Real
st +(x,y,r).p > a holds |.|[x,y]|-p.| > r*a & ex r1 being positive Real
 st r1 = (|.|[x,y]|-p.|-r*a)/2 & Ball(|[p`1,r1]|,r1) \/ {p} c= +(x,y,r)"].a,1
  .]
proof
  let p be Point of TOP-REAL 2;
A1: p = |[p`1,p`2]| by EUCLID:53;
  assume
A2: p`2 = 0;
  then reconsider p9 = p as Point of Niemytzki-plane by A1,Lm1,Th18;
  let x be Real;
  let a be non negative Real;
  let y,r be positive Real;
  set f = +(x,y,r);
  p in y>=0-plane by A2,A1;
  then f.p in [.0,1.] by Lm1,BORSUK_1:40,FUNCT_2:5;
  then
A3: f.p <= 1 by XXREAL_1:1;
  assume
A4: f.p > a;
  then
A5: a < 1 by A3,XXREAL_0:2;
A6: |.|[x,y]|-p.| = |.p-|[x,y]|.| by TOPRNS_1:27;
  thus |.|[x,y]|-p.| > r*a
  proof
    per cases by A3,XXREAL_0:1;
    suppose
      f.p < 1;
      then p in Ball(|[x,y]|,r) by A2,A1,Def6;
      then f.p = |.|[x,y]|-p.|/r by A2,A1,Def6;
      hence thesis by A4,XREAL_1:79;
    end;
    suppose
A7:   f.p = 1;
      now
A8:     r/r = 1 by XCMPLX_1:60;
        assume
A9:     p in Ball(|[x,y]|,r);
        then
A10:    |.p-|[x,y]|.| < r by TOPREAL9:7;
        1 = |.|[x,y]|-p.|/r by A9,A2,A1,A7,Def6;
        hence contradiction by A10,A8,A6,XREAL_1:74;
      end;
      then
A11:  |.p-|[x,y]|.| >= r by TOPREAL9:7;
      r*1 > r*a by A5,XREAL_1:68;
      hence thesis by A11,A6,XXREAL_0:2;
    end;
  end;
  then reconsider r9 = |.|[x,y]|-p.|-r*a as positive Real by XREAL_1:50;
  take r1 = r9/2;
  thus r1 = (|.|[x,y]|-p.|-r*a)/2;
  let u be object;
A12: Ball(|[p`1,r1]|,r1) c= y>=0-plane by Th20;
  assume
A13: u in Ball(|[p`1,r1]|,r1) \/ {p};
  then u in Ball(|[p`1,r1]|,r1) or u = p9 by ZFMISC_1:136;
  then reconsider z = u as Point of Niemytzki-plane by A12,Def3;
  reconsider q = z as Element of TOP-REAL 2 by A13;
A14: q = |[q`1,q`2]| by EUCLID:53;
  then
A15: q`2 >= 0 by Lm1,Th18;
  then
A16: not q in Ball(|[x,y]|,r) implies f.q = 1 by A14,Def6;
  per cases by A13,ZFMISC_1:136;
  suppose
    u = p9;
    then f.z in ].a,1.] by A4,A3,XXREAL_1:2;
    hence thesis by FUNCT_2:38;
  end;
  suppose
    u in Ball(|[p`1,r1]|,r1);
    then |.q-|[p`1,r1]|.| < r1 by TOPREAL9:7;
    then
A17: |.q-|[p`1,r1]|.|+|.|[p`1,r1]|-p.| < r1+|.|[p`1,r1]|-p.| by XREAL_1:6;
    |.q-p.| <= |.q-|[p`1,r1]|.|+|.|[p`1,r1]|-p.| by TOPRNS_1:34;
    then
A18: |.q-p.| < r1+|.|[p`1,r1]|-p.| by A17,XXREAL_0:2;
     reconsider r1 as Real;
A19: |.|[0,r1]|.| = |.r1.| by TOPREAL6:23;
A20: |.|[x,y]|-p.| <= |.|[x,y]|-q.|+|.q-p.| by TOPRNS_1:34;
A21: |.r1.| = r1 by ABSVALUE:def 1;
    |.|[p`1,r1]|-p.| = |.|[p`1-p`1,r1-0 ]|.| by A2,A1,EUCLID:62;
    then |.|[x,y]|-q.|+|.q-p.| < |.|[x,y]|-q.|+(r1+r1)
     by A18,A19,A21,XREAL_1:6;
    then |.|[x,y]|-p.| < |.|[x,y]|-q.|+(r1+r1) by A20,XXREAL_0:2;
    then
A22: |.|[x,y]|-p.|-(r1+r1) < |.|[x,y]|-q.|+(r1+r1)-(r1+r1) by XREAL_1:9;
A23: now
      assume q in Ball(|[x,y]|,r);
      then f.q = |.|[x,y]|-q.|/r by A14,A15,Def6;
      hence f.z > a by A22,XREAL_1:81;
    end;
    f.z in [.0,1.] by BORSUK_1:40;
    then f.z <= 1 by XXREAL_1:1;
    then f.z in ].a,1.] by A5,A16,A23,XXREAL_1:2;
    hence thesis by FUNCT_2:38;
  end;
end;
