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

theorem Th62:
  for p being Point of TOP-REAL 2 for x being Real, a,r
being positive Real st a <= 1 & |.p-|[x,r*a]|.| = r*a & p`2 <> 0 holds
  +(x,r).p = a
proof
  let p be Point of TOP-REAL 2;
  set p1 = p`1, p2 = p`2;
  let x be Real;
  let a,r be positive Real;
  assume
A1: a <= 1;
A2: |[p1-x,p2-r*a]|`1 = p1-x by EUCLID:52;
A3: |[p1-x,p2-r*a]|`2 = p2-r*a by EUCLID:52;
  assume that
A4: |.p-|[x,r*a]|.| = r*a and
A5: p`2 <> 0;
A6: p = |[p`1,p`2]| by EUCLID:53;
  then |.|[p1-x,p2-r*a]|.|^2 = (r*a)^2 by A4,EUCLID:62;
  then
A7: (p1-x)^2+(p2-r*a)^2 = (r*a)^2 by A2,A3,JGRAPH_1:29;
  then
A8: (p1-x)^2+p2^2 = 2*p2*r*a;
  (p1-x)^2 >= 0 by XREAL_1:63;
  then reconsider p2 as positive Real by A5,A8;
A9: |[p1-x,p2-0]|`1 = p1-x by EUCLID:52;
A10: |[p1-x,p2]|`2 = p2 by EUCLID:52;
  per cases by A1,XXREAL_0:1;
  suppose
    a < 1;
    then r*a < r by XREAL_1:157;
    then reconsider s = r-r*a as positive Real by XREAL_1:50;
    |.p-|[x,r]|.|^2 = |.|[p1-x,p2-r]|.|^2 by A6,EUCLID:62
      .= (p1-x)^2+(p2-r)^2 by Th9
      .= (p1-x)^2+(p2-a*r)^2+((r-a*r)^2+2*(r-a*r)*(a*r-p2))
      .= |.|[p1-x,p2-a*r]|.|^2+((r-a*r)^2+2*(r-a*r)*(a*r-p2)) by Th9
      .= (a*r)^2+(r*r-r*p2+r*a*r-r*p2-(a*r*r-a*r*p2+(a*r)^2-a*r*p2)) by A6,A4,
EUCLID:62
      .= r^2-(1+1)*p2*s;
    then |.p-|[x,r]|.|^2 < r^2 by XREAL_1:44;
    then |.p-|[x,r]|.| < r by SQUARE_1:15;
    then p in Ball(|[x,r]|,r) by TOPREAL9:7;
    then +(x,r).p = |.|[x,0]|-p.|^2/(2*r*p2) by A6,Def5
      .= |.p-|[x,0]|.|^2/(2*r*p2) by TOPRNS_1:27
      .= |.|[p1-x,p2-0]|.|^2/(2*r*p2) by A6,EUCLID:62
      .= ((p1-x)^2+p2^2)/(2*r*p2) by A9,A10,JGRAPH_1:29;
    then
A11: +(x,r).p = (2*p2*r*a)/(2*r*p2) by A7;
    a*((2*p2*r)/(2*r*p2)) = a*1 by XCMPLX_1:60;
    hence thesis by A11,XCMPLX_1:74;
  end;
  suppose
A12: a = 1;
A13: p2 is non negative;
    not p in Ball(|[x,r]|,r) by A12,A4,TOPREAL9:7;
    hence thesis by A13,A6,A12,Def5;
  end;
end;
