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

theorem Th79:
  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 & Ball(p,|.|[x,y]|-p.|-r*a) /\
  y>=0-plane c= +(x,y,r)"].a,1.]
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 > 0;
  let x be Real;
  let a be non negative Real;
  let y,r be positive Real;
  set f = +(x,y,r);
A2: p = |[p`1,p`2]| by EUCLID:53;
  then p in y>=0-plane by 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 A1,A2,Def6;
      then f.p = |.|[x,y]|-p.|/r by A1,A2,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,A1,A2,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 r1 = |.|[x,y]|-p.|-r*a as positive Real by XREAL_1:50;
  let u be object;
  assume
A12: u in Ball(p,|.|[x,y]|-p.|-r*a) /\ y>=0-plane;
  then reconsider z = u as Point of Niemytzki-plane by Lm1,XBOOLE_0:def 4;
  reconsider q = z as Element of TOP-REAL 2 by A12;
  u in Ball(p,|.|[x,y]|-p.|-r*a) by A12,XBOOLE_0:def 4;
  then |.q-p.| < r1 by TOPREAL9:7;
  then
A13: |.|[x,y]|-q.|+|.q-p.| < |.|[x,y]|-q.|+r1 by XREAL_1:6;
A14: q = |[q`1,q`2]| by EUCLID:53;
  then
A15: q`2 >= 0 by Lm1,Th18;
  |.|[x,y]|-p.| <= |.|[x,y]|-q.|+|.q-p.| by TOPRNS_1:34;
  then |.|[x,y]|-p.| < |.|[x,y]|-q.|+r1 by A13,XXREAL_0:2;
  then
A16: |.|[x,y]|-p.|-r1 < |.|[x,y]|-q.|+r1-r1 by XREAL_1:9;
A17: now
    assume q in Ball(|[x,y]|,r);
    then f.q = |.|[x,y]|-q.|/r by A14,A15,Def6;
    hence f.z > a by A16,XREAL_1:81;
  end;
  f.z in [.0,1.] by BORSUK_1:40;
  then
A18: f.z <= 1 by XXREAL_1:1;
  not q in Ball(|[x,y]|,r) implies f.q = 1 by A15,A14,Def6;
  then f.z in ].a,1.] by A18,A5,A17,XXREAL_1:2;
  hence thesis by FUNCT_2:38;
end;
