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

theorem Th66:
  for x being Real, a,r being positive Real holds
  Ball(|[x,r*a]|,r*a) c= +(x,r)"].0,a.[
proof
  let x be Real;
  let a,r be positive Real, u be object;
  assume
A1: u in Ball(|[x,r*a]|,r*a);
  then reconsider p = u as Point of TOP-REAL 2;
  Ball(|[x,r*a]|,r*a) c= y>=0-plane by Th20;
  then reconsider q = p as Point of Niemytzki-plane by A1,Def3;
  q = |[p`1,p`2]| by EUCLID:53;
  then
A2: p`2 >= 0 by Lm1,Th18;
A3: now
    assume +(x,r).p = 0;
    then p = |[x,0]| by A2,Th60;
    then
A4: p in y=0-line;
    Ball(|[x,r*a]|,r*a) misses y=0-line by Th21;
    hence contradiction by A4,A1,XBOOLE_0:3;
  end;
A5: +(x,r).q <= 1 by BORSUK_1:40,XXREAL_1:1;
  per cases;
  suppose
A6: a > 1;
A7: +(x,r).q > 0 by A3,BORSUK_1:40,XXREAL_1:1;
    +(x,r).q < a by A6,A5,XXREAL_0:2;
    then +(x,r).q in ].0,a.[ by A7,XXREAL_1:4;
    hence thesis by FUNCT_2:38;
  end;
  suppose
A8: a <= 1;
    |.p-|[x,r*a]|.| < r*a by A1,TOPREAL9:7;
    then
A9: +(x,r).p < a by A8,Th63;
    +(x,r).q > 0 by A3,BORSUK_1:40,XXREAL_1:1;
    then +(x,r).q in ].0,a.[ by A9,XXREAL_1:4;
    hence thesis by FUNCT_2:38;
  end;
end;
