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

theorem Th21:
  for a,b,r being Real st r > 0 & b >= 0 holds Ball(|[a,b]|
  ,r) misses y=0-line iff r <= b
proof
  let a,b,r be Real;
  assume that
A1: r > 0 and
A2: b >= 0;
  hereby
A3: |[a,0]| in y=0-line;
    assume that
A4: Ball(|[a,b]|,r) misses y=0-line and
A5: r > b;
    |[a,0]|-|[a,b]| = |[a-a,0-b]| by EUCLID:62;
    then |.|[a,0]|-|[a,b]|.| = |.0-b.| by TOPREAL6:23
      .= |.b-0 .| by COMPLEX1:60;
    then |.|[a,0]|-|[a,b]|.| < r by A2,A5,ABSVALUE:def 1;
    then |[a,0]| in Ball(|[a,b]|,r) by TOPREAL9:7;
    hence contradiction by A3,A4,XBOOLE_0:3;
  end;
  assume
A6: r <= b;
  assume Ball(|[a,b]|,r) meets y=0-line;
  then consider x being object such that
A7: x in Ball(|[a,b]|,r) and
A8: x in y=0-line by XBOOLE_0:3;
  reconsider x as Element of TOP-REAL 2 by A7;
A9: x = |[x`1,x`2]| by EUCLID:53;
  then x`2 = 0 by A8,Th15;
  then
A10: x-|[a,b]| = |[x`1-a,0-b]| by A9,EUCLID:62;
  then
A11: (x-|[a,b]|)`2 = 0-b by EUCLID:52;
  (x-|[a,b]|)`1 = x`1-a by A10,EUCLID:52;
  then |.x-|[a,b]|.| = sqrt((x`1-a)^2+(0-b)^2) by A11,JGRAPH_1:30;
  then |.x-|[a,b]|.| >= |.0-b.| by COMPLEX1:79;
  then
A12: |.x-|[a,b]|.| >= |.b-0 .| by COMPLEX1:60;
  |.x-|[a,b]|.| < r by A7,TOPREAL9:7;
  then |.b.| < r by A12,XXREAL_0:2;
  hence contradiction by A1,A6,ABSVALUE:def 1;
end;
