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

theorem Th61:
  for x,y being Real, r being positive Real st x
  <> y holds +(x,r).(|[y,0]|) = 1
proof
  let x,y be Real;
  let r be positive Real;
A1: |.x-y.| = x-y or |.x-y.| = - (x-y) by ABSVALUE:1;
A2: |[x-y,r]|`2 = r by EUCLID:52;
  |[x-y,r]|`1 = x-y by EUCLID:52;
  then
A3: |.|[x-y,r]|.|^2 = r^2+(x-y)^2 by A2,JGRAPH_1:29;
  assume
A4: x <> y;
  then x-y <> 0;
  then
A5: |.x-y.| > 0 by COMPLEX1:47;
  then |.x-y.|^2 <> 0;
  then |.|[x-y,r]|.|^2 > r^2 by A1,A5,A3,XREAL_1:29;
  then |.|[x-y,r-0]|.| > r by SQUARE_1:15;
  then |.|[x,r]|-|[y,0]|.| > r by EUCLID:62;
  then |.|[y,0]|-|[x,r]|.| > r by TOPRNS_1:27;
  then not |[y,0]| in Ball(|[x,r]|,r) by TOPREAL9:7;
  hence thesis by A4,Def5;
end;
