reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;

theorem Th12:
  x = e implies cl_Ball(e,r) = cl_Ball(x,r)
proof
  assume
A1: x = e;
  hereby
    let q be object;
    assume
A2: q in cl_Ball(e,r);
    then reconsider f = q as Point of Euclid n;
    reconsider p = f as Point of TOP-REAL n by TOPREAL3:8;
    dist(f,e) <= r by A2,METRIC_1:12;
    then |. p-x .| <= r by A1,JGRAPH_1:28;
    hence q in cl_Ball(x,r);
  end;
  let q be object;
  assume
A3: q in cl_Ball(x,r);
  then reconsider q as Point of TOP-REAL n;
  reconsider f = q as Point of Euclid n by TOPREAL3:8;
  |. q-x .| <= r by A3,Th6;
  then dist(f,e) <= r by A1,JGRAPH_1:28;
  hence thesis by METRIC_1:12;
end;
