reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  u=p1 & p1=|[r1,s1]| & p2=|[r2,s2]| & p=|[r2,s1]| & p2 in Ball(u,r)
  implies p in Ball(u,r)
proof
  assume that
A1: u=p1 and
A2: p1=|[r1,s1]| and
A3: p2=|[r2,s2]| and
A4: p=|[r2,s1]| and
A5: p2 in Ball(u,r);
  reconsider p19= p1, p29= p2,p9=p as Element of REAL 2 by EUCLID:22;
  reconsider r1,s1,r2,s2 as Real;
A6: (Pitag_dist 2).(p19,p29)=|.p19-p29.| by EUCLID:def 6;
  p2 in {u6 where u6 is Element of Euclid 2:dist(u,u6)<r} by A5,METRIC_1:17;
  then ex u5 st u5 = p2 & dist(u,u5)<r;
  then
A7: |.p19-p29.| < r by A1,A6,METRIC_1:def 1;
  reconsider up=p as Point of Euclid 2 by EUCLID:22;
  (Pitag_dist 2).(p19,p9)=|.p19-p9.| by EUCLID:def 6;
  then
A8: dist(u,up)=|.p19-p9.| by A1,METRIC_1:def 1;
  (s1-s2)^2 >= 0 by XREAL_1:63;
  then sqrreal.(s1-s2) >= 0 by RVSUM_1:def 2;
  then
A9: sqrreal.(r1-r2)+0 <= sqrreal.(r1-r2) + sqrreal.(s1-s2) by XREAL_1:7;
  p19-p9 = p1-p by EUCLID:69;
  then (p19-p9)=<*r1-r2,s1-s1*> by A2,A4,Th5;
  then sqr (p19-p9) = <* sqrreal.(r1-r2),sqrreal.(s1-s1)*> by FINSEQ_2:36;
  then
A10: Sum sqr (p19-p9) = sqrreal.(r1-r2) + sqrreal.0 by RVSUM_1:77
    .= sqrreal.(r1-r2) + 0^2 by RVSUM_1:def 2
    .= sqrreal.(r1-r2);
  p19-p29 = p1-p2 by EUCLID:69;
  then (p19-p29)=<*r1-r2,s1-s2*> by A2,A3,Th5;
  then sqr (p19-p29) = <* sqrreal.(r1-r2),sqrreal.(s1-s2)*> by FINSEQ_2:36;
  then
A11: |.p19-p29.| = sqrt (sqrreal.(r1-r2) + sqrreal.(s1-s2)) by RVSUM_1:77;
  (r1-r2)^2 >= 0 by XREAL_1:63;
  then sqrreal.(r1-r2) >= 0 by RVSUM_1:def 2;
  then |.p19-p9.| <= sqrt(sqrreal.(r1-r2) + sqrreal.(s1-s2)) by A10,A9,
SQUARE_1:26;
  then |.p19-p9.| < r by A7,A11,XXREAL_0:2;
  then p in {u6 where u6 is Element of Euclid 2:dist(u,u6)<r} by A8;
  hence thesis by METRIC_1:17;
end;
