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

theorem Th77:
  for p being Point of TOP-REAL 2 st p`2 >= 0 for x being Real
, y being non negative Real for r being positive Real
  holds +(x,y,r).p = 0 iff p = |[x,y]|
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p`2 >= 0;
  let x be Real;
  let y be non negative Real;
  let r be positive Real;
A2: p = |[p`1,p`2]| by EUCLID:53;
  hereby
    assume
A3: +(x,y,r).p = 0;
    then p in Ball(|[x,y]|,r) by A1,A2,Def6;
    then 0 = |.|[x,y]|-p.|/r by A1,A2,A3,Def6;
    then 0 * r = |.|[x,y]|-p.|;
    hence p = |[x,y]| by TOPRNS_1:28;
  end;
  assume
A4: p = |[x,y]|;
  then p in Ball(|[x,y]|,r) by Th13;
  hence +(x,y,r).p = |.|[x,y]|-p.|/r by A4,Def6
    .= 0/r by A4,TOPRNS_1:28
    .= 0;
end;
