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

theorem Th60:
  for p being Point of TOP-REAL 2 st p`2 >= 0 for x being Real
, r being positive Real st +(x,r).p = 0 holds p = |[x,0]|
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 >= 0;
  set p1 = p`1, p2 = p`2;
  let x be Real;
  let r be positive Real;
  assume
A2: +(x,r).p = 0;
A3: p = |[p1,p2]| by EUCLID:53;
  assume
A4: p <> |[x,0]|;
  then p1 <> x or p2 <> 0 by EUCLID:53;
  then
A5: |[p1,p2]| in Ball(|[x,r]|,r) by A1,A2,A3,Def5;
  Ball(|[x,r]|,r) misses y=0-line by Th21;
  then not |[p1,p2]| in y=0-line by A5,XBOOLE_0:3;
  then p2 <> 0;
  then reconsider p2 as positive Real by A1;
  0/(2*r*p2) = |.|[x,0]|-|[p1,p2]|.|^2/(2*r*p2) by A2,A3,A5,Def5;
  then 0 = |.|[x,0]|-p.| by A3;
  hence contradiction by A4,TOPRNS_1:28;
end;
