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

theorem Th67:
  for x being Real, a,r being positive Real holds
  Ball(|[x,r*a]|,r*a) \/ {|[x,0]|} 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) \/ {|[x,0]|};
  then
A2: u in Ball(|[x,r*a]|,r*a) or u in {|[x,0]|} by XBOOLE_0:def 3;
  reconsider p = u as Point of TOP-REAL 2 by A1;
A3: |[x,0]| in y>=0-plane;
  Ball(|[x,r*a]|,r*a) c= y>=0-plane by Th20;
  then reconsider q = p as Point of Niemytzki-plane by A3,A2,Lm1,TARSKI:def 1;
A4: +(x,r).q <= 1 by BORSUK_1:40,XXREAL_1:1;
A5: +(x,r).q >= 0 by BORSUK_1:40,XXREAL_1:1;
  per cases by A2,TARSKI:def 1;
  suppose
    a > 1;
    then +(x,r).q < a by A4,XXREAL_0:2;
    then +(x,r).q in [.0,a.[ by A5,XXREAL_1:3;
    hence thesis by FUNCT_2:38;
  end;
  suppose
A6: a <= 1 & u in Ball(|[x,r*a]|,r*a);
    then |.p-|[x,r*a]|.| < r*a by TOPREAL9:7;
    then +(x,r).p < a by A6,Th63;
    then +(x,r).q in [.0,a.[ by A5,XXREAL_1:3;
    hence thesis by FUNCT_2:38;
  end;
  suppose
    u = |[x,0]|;
    then +(x,r).u = 0 by Def5;
    then +(x,r).q in [.0,a.[ by XXREAL_1:3;
    hence thesis by FUNCT_2:38;
  end;
end;
