
theorem Th7:
  for x being Point of RealSpace, r being Real holds
  Ball(x,r) = ].x-r, x+r.[
proof
  let x be Point of RealSpace, r be Real;
  reconsider x2=x as Element of REAL by METRIC_1:def 13;
  thus Ball(x,r) c= ].x-r, x+r.[
  proof
    let y be object;
    assume
A1: y in Ball(x,r);
    then reconsider y1=y as Element of RealSpace;
    reconsider y2=y1 as Element of REAL by METRIC_1:def 13;
A2: dist(x,y1)=real_dist.(x2,y2) by METRIC_1:def 1,def 13
      .=|.x2 - y2 .| by METRIC_1:def 12
      .=|.-(y2 - x2) .|
      .=|.y2 - x2 .| by COMPLEX1:52;
    dist(x,y1)<r by A1,METRIC_1:11;
    hence thesis by A2,RCOMP_1:1;
  end;
  let y be object;
  assume
A3: y in ].x-r, x+r.[;
  then reconsider y2=y as Element of REAL;
  reconsider x1=x,y1=y2 as Element of RealSpace by METRIC_1:def 13;
  |.y2 - x .|=|.-(y2 - x).| by COMPLEX1:52
    .=|.x - y2 .|
    .=real_dist.(x2,y2) by METRIC_1:def 12;
  then
A4: real_dist.(x2,y2) < r by A3,RCOMP_1:1;
  dist(x1,y1)=real_dist.(x2,y2) by METRIC_1:def 1,def 13;
  hence thesis by A4,METRIC_1:11;
end;
