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

theorem Th20:
  for a,b,r being Real st r > 0 holds Ball(|[a,b]|,r) c=
  y>=0-plane iff r <= b
proof
  let a,b,r be Real such that
A1: r > 0;
  hereby
A2: |[a,b]| in Ball(|[a,b]|,r) by A1,Th13;
    assume that
A3: Ball(|[a,b]|,r) c= y>=0-plane and
A4: r > b;
    reconsider b as non negative Real by A2,A3,Th18;
    reconsider br=b-r as negative Real by A4,XREAL_1:49;
    set y = br/2;
    reconsider r as positive Real by A1;
    |[a,y]|-|[a,b]| = |[a-a,y-b]| by EUCLID:62;
    then
A5: |.|[a,y]|-|[a,b]|.| = |.y-b.| by TOPREAL6:23
      .= |.b-y.| by COMPLEX1:60
      .= (b+r)/2 by ABSVALUE:def 1;
    b+r < r+r by A4,XREAL_1:6;
    then (b+r)/2 < (r+r)/2 by XREAL_1:74;
    then |[a,y]| in Ball(|[a,b]|,r) by A5,TOPREAL9:7;
    hence contradiction by A3,Th18;
  end;
  assume
A6: r <= b;
  let x be object;
  assume
A7: x in Ball(|[a,b]|,r);
  then reconsider z = x as Element of TOP-REAL 2;
A8: |.z-|[a,b]|.| < r by A7,TOPREAL9:7;
A9: |[z`1-a,z`2-b]|`1 = z`1-a by EUCLID:52;
A10: |[z`1-a,z`2-b]|`2 = z`2-b by EUCLID:52;
A11: z = |[z`1,z`2]| by EUCLID:53;
  then z-|[a,b]| = |[z`1-a,z`2-b]| by EUCLID:62;
  then |.z-|[a,b]|.| = sqrt((z`1-a)^2+(z`2-b)^2) by A9,A10,JGRAPH_1:30;
  then |.z`2-b.| <= |.z-|[a,b]|.| by COMPLEX1:79;
  then |.z`2-b.| < r by A8,XXREAL_0:2;
  then
A12: |.b-z`2.| < r by COMPLEX1:60;
  now
    assume z`2 < 0;
    then b-z`2 > b by XREAL_1:46;
    then b-z`2 > r by A6,XXREAL_0:2;
    hence contradiction by A1,A12,ABSVALUE:def 1;
  end;
  hence thesis by A11;
end;
