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

theorem Th63:
  for p being Point of TOP-REAL 2 for x,a being Real, r
being positive Real st a <= 1 & |.p-|[x,r*a]|.| < r*a holds +(x,r).p <
  a
proof
  let p be Point of TOP-REAL 2;
  set p1 = p`1, p2 = p`2;
  let x,a be Real;
  let r be positive Real;
  assume that
A1: a <= 1;
A2: |[p1-x,p2-r*a]|`2 = p2-r*a by EUCLID:52;
A3: p = |[p`1,p`2]| by EUCLID:53;
  then
A4: p2 = 0 implies p in y=0-line;
  set r1 = r*a, r2 = r*1;
A5: |[p1-x,p2-r*a]|`1 = p1-x by EUCLID:52;
  assume
A6: |.p-|[x,r*a]|.| < r*a;
  then reconsider r1 as positive Real;
A7: p in Ball(|[x,r1]|,r1) by A6,TOPREAL9:7;
  |.p-|[x,r*a]|.|^2 < (r*a)^2 by A6,SQUARE_1:16;
  then |.|[p1-x,p2-r*a]|.|^2 < (r*a)^2 by A3,EUCLID:62;
  then (p1-x)^2+(p2-r*a)^2 < (r*a)^2 by A5,A2,JGRAPH_1:29;
  then (p1-x)^2+p2^2-2*p2*(r*a)+(r*a)^2 < (r*a)^2;
  then (p1-x)^2+p2^2-2*p2*r*a < 0 by XREAL_1:31;
  then
A8: (p1-x)^2+p2^2 < 2*p2*r*a by XREAL_1:48;
A9: Ball(|[x,r1]|,r1) misses y=0-line by Th21;
  Ball(|[x,r1]|,r1) c= y>=0-plane by Th20;
  then reconsider p2 as positive Real by A3,A7,Th18,A9,A4,XBOOLE_0:3;
A10: |[p1-x,p2-0]|`1 = p1-x by EUCLID:52;
A11: |[p1-x,p2]|`2 = p2 by EUCLID:52;
  Ball(|[x,r1]|,r1) c= Ball(|[x,r2]|,r2) by A1,Th23,XREAL_1:64;
  then +(x,r).p = |.|[x,0]|-p.|^2/(2*r*p2) by A3,A7,Def5
    .= |.p-|[x,0]|.|^2/(2*r*p2) by TOPRNS_1:27
    .= |.|[p1-x,p2-0]|.|^2/(2*r*p2) by A3,EUCLID:62
    .= ((p1-x)^2+p2^2)/(2*r*p2) by A10,A11,JGRAPH_1:29;
  then
A12: +(x,r).p < (2*p2*r*a)/(2*r*p2) by A8,XREAL_1:74;
  a*((2*p2*r)/(2*r*p2)) = a*1 by XCMPLX_1:60;
  hence thesis by A12,XCMPLX_1:74;
end;
