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

theorem Th65:
  for p being Point of TOP-REAL 2 st p`2 >= 0 for x,a,b being Real
, r being positive Real st 0 <= a & b <= 1 & +(x,r).p in ].a,b.[
ex r1 being positive Real st r1 <= p`2 & Ball(p,r1) c= +(x,r)"].a,b.[
proof
  let p be Point of TOP-REAL 2 such that
A1: p`2 >= 0;
  let x,a,b be Real, r be positive Real;
A2: p = |[p`1,p`2]| by EUCLID:53;
A3: Ball(|[x,r]|,r) misses y=0-line by Th21;
A4: |[p`1-x,p`2]|`1 = p`1-x by EUCLID:52;
  assume that
A5: 0 <= a and
A6: b <= 1 and
A7: +(x,r).p in ].a,b.[;
A8: +(x,r).p > a by A7,XXREAL_1:4;
A9: +(x,r).p < b by A7,XXREAL_1:4;
  then
A10: p in Ball(|[x,r]|,r) or p`1 = x & p`2 = 0 & p <> |[x,0]| by A8,A1,A2,A5,A6
,Def5;
  then
A11: +(x,r).p = |.|[x,0]|-p.|^2/(2*r*p`2) by A1,A2,Def5;
  p`2 = 0 implies p in y=0-line by A2;
  then reconsider p2 = p`2, b9 = b as positive Real by A3,A1,A5,A8,A7,A10,
EUCLID:53,XBOOLE_0:3,XXREAL_1:4;
A12: |[p`1-x,p2]|`2 = p2 by EUCLID:52;
A13: 2*r*p2 > 0;
  then |.|[x,0]|-p.|^2 > (2*r*p`2)*a by A11,A8,XREAL_1:79;
  then |.p-|[x,0]|.|^2 > (2*r*p`2)*a by TOPRNS_1:27;
  then |.|[p`1-x,p`2-0]|.|^2 > (2*r*p`2)*a by A2,EUCLID:62;
  then (p`1-x)^2+p2^2 > (2*r*p2)*a by A4,A12,JGRAPH_1:29;
  then (p`1-x)^2+p2^2 - (2*r*p2)*a > 0 by XREAL_1:50;
  then (p`1-x)^2+p2^2 - (2*r*p2)*a + (r*a)^2 > (r*a)^2 by XREAL_1:29;
  then
A14: (p`1-x)^2+(p2 - r*a)^2 > (r*a)^2;
  |.|[x,0]|-p.|^2 < (2*r*p`2)*b by A13,A11,A9,XREAL_1:77;
  then |.p-|[x,0]|.|^2 < (2*r*p`2)*b by TOPRNS_1:27;
  then |.|[p`1-x,p`2-0]|.|^2 < (2*r*p`2)*b by A2,EUCLID:62;
  then (p`1-x)^2+p2^2 < (2*r*p2)*b by A4,A12,JGRAPH_1:29;
  then (p`1-x)^2+p2^2 - (2*r*p2)*b < 0 by XREAL_1:49;
  then (p`1-x)^2+p2^2 - (2*r*p2)*b + (r*b)^2 < (r*b)^2 by XREAL_1:30;
  then
A15: (p`1-x)^2+(p2 - r*b)^2 < (r*b)^2;
A16: |[p`1-x, p2-r*b]|`2 = p2-r*b by EUCLID:52;
A17: |[p`1-x, p2-r*a]|`2 = p2-r*a by EUCLID:52;
  |[p`1-x, p2-r*a]|`1 = p`1-x by EUCLID:52;
  then |.|[p`1-x, p2-r*a]|.|^2 > (r*a)^2 by A14,A17,JGRAPH_1:29;
  then |.p-|[x,r*a]|.|^2 > (r*a)^2 by A2,EUCLID:62;
  then
A18: |.p-|[x,r*a]|.| > r*a by SQUARE_1:48;
A19: r*b-|.p-|[x,r*b]|.|+|.p-|[x,r*b]|.| = r*b;
  set r1 = min(r*b-|.p-|[x,r*b]|.|, |.p-|[x,r*a]|.|-r*a);
A20: |.p-|[x,r*b]|.| = |.|[x,r*b]|-p.| by TOPRNS_1:27;
  |[p`1-x, p2-r*b]|`1 = p`1-x by EUCLID:52;
  then |.|[p`1-x, p2-r*b]|.|^2 < (r*b)^2 by A15,A16,JGRAPH_1:29;
  then
A21: |.p-|[x,r*b]|.|^2 < (r*b)^2 by A2,EUCLID:62;
  r*b9 >= 0;
  then |.p-|[x,r*b]|.| < r*b by A21,SQUARE_1:48;
  then
  r*b-|.p-|[x,r*b]|.| > 0 & r1 = r*b-|.p-|[x,r*b]|.| or |.p-|[x,r*a]|.|-r
  *a > 0 & r1 = |.p-|[x,r*a]|.|-r*a by A18,XREAL_1:50,XXREAL_0:15;
  then reconsider r1 as positive Real;
  take r1;
  r1 <= r*b-|.p-|[x,r*b]|.| by XXREAL_0:17;
  then
A22: |.|[x,r*b]|-p.|+r1 <= r*b by A20,A19,XREAL_1:6;
  |.p-|[p`1,0]|.| = |.|[p`1-p`1,p2-0]|.| by A2,EUCLID:62
    .= |.p2.| by TOPREAL6:23
    .= p2 by ABSVALUE:def 1;
  then
A23: |.|[x,r*b]|-|[p`1,0]|.| <= |.|[x,r*b]|-p.|+p2 by TOPRNS_1:34;
  thus now
    assume r1 > p`2;
    then |.|[x,r*b]|-p.|+p2 < |.|[x,r*b]|-p.|+r1 by XREAL_1:8;
    then |.|[x,r*b]|-|[p`1,0]|.| < |.|[x,r*b]|-p.|+r1 by A23,XXREAL_0:2;
    then |.|[x,r*b]|-|[p`1,0]|.| < r*b by A22,XXREAL_0:2;
    then |.|[p`1,0]|-|[x,r*b]|.| < r*b by TOPRNS_1:27;
    then
A24: |[p`1,0]| in Ball(|[x,r*b]|,r*b) by TOPREAL9:7;
    |[p`1,0]| in y=0-line;
    then Ball(|[x,r*b]|,r*b9) meets y=0-line by A24,XBOOLE_0:3;
    hence contradiction by Th21;
  end;
  then
A25: Ball(p,r1) c= y>=0-plane by A2,Th20;
  let u be object;
  assume
A26: u in Ball(p,r1);
  then reconsider q = u as Point of TOP-REAL 2;
A27: |.q-p.| < r1 by A26,TOPREAL9:7;
  then |.q-p.|+|.p-|[x,r*b]|.| < r1+|.p-|[x,r*b]|.| by XREAL_1:8;
  then
A28: |.q-p.|+|.p-|[x,r*b]|.| < r*b by A20,A22,XXREAL_0:2;
  |.q-|[x,r*b]|.| <= |.q-p.|+|.p-|[x,r*b]|.| by TOPRNS_1:34;
  then |.q-|[x,r*b]|.| < r*b by A28,XXREAL_0:2;
  then
A29: +(x,r).q < b by A6,Th63;
A30: |.p-|[x,r*a]|.| <= |.p-q.|+|.q-|[x,r*a]|.| by TOPRNS_1:34;
  a < b by A8,A9,XXREAL_0:2;
  then
A31: a < 1 by A6,XXREAL_0:2;
  |.p-|[x,r*a]|.|-r*a >= r1 by XXREAL_0:17;
  then |.p-|[x,r*a]|.|-r*a > |.q-p.| by A27,XXREAL_0:2;
  then
A32: |.p-|[x,r*a]|.|-|.q-p.| > r*a by XREAL_1:12;
  |.p-q.| = |.q-p.| by TOPRNS_1:27;
  then |.q-|[x,r*a]|.| >= |.p-|[x,r*a]|.|-|.q-p.| by A30,XREAL_1:20;
  then
A33: |.q-|[x,r*a]|.| > r*a by A32,XXREAL_0:2;
  q = |[q`1,q`2]| by EUCLID:53;
  then q`2 >= 0 by A25,A26,Th18;
  then +(x,r).q > a by A33,A5,A31,Th64;
  then +(x,r).q in ].a,b.[ by A29,XXREAL_1:4;
  hence thesis by A25,A26,Lm1,FUNCT_2:38;
end;
