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

theorem Th68:
  for p being Point of TOP-REAL 2 st p`2 >= 0 for x,a being Real
, r being positive Real st 0 < +(x,r).p & +(x,r).p < a & a <= 1
  holds p in Ball(|[x,r*a]|,r*a)
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 >= 0;
  let x,a be Real;
A2: p = |[p`1,p`2]| by EUCLID:53;
  let r be positive Real;
  set r1 = r*a;
  assume that
A3: 0 < +(x,r).p and
A4: +(x,r).p < a and
A5: a <= 1;
A6: x <> p`1 implies p <> |[p`1,0]| by A4,A5,Th61;
A7: p <> |[x,0]| by A3,Def5;
  assume not p in Ball(|[x,r1]|,r1);
  then |.p-|[x,r1]|.| >= r1 by TOPREAL9:7;
  then |.p-|[x,r1]|.| = r1 or |.p-|[x,r1]|.| > r1 & (a < 1 or a = 1) by A5,
XXREAL_0:1;
  then
  +(x,r).p = a or a < 1 & +(x,r).p > a or a = 1 & not p in Ball(|[x,r]|,r
  ) by A1,A2,A3,A5,A7,A6,Th62,Th64,TOPREAL9:7;
  hence thesis by A1,A2,A4,A7,A6,Def5;
end;
