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 Th13:
  x = e implies Sphere(e,r) = Sphere(x,r)
proof
  assume
A1: x = e;
  hereby
    let q be object;
    assume
A2: q in Sphere(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_1:13;
    then |. p-x .| = r by A1,JGRAPH_1:28;
    hence q in Sphere(x,r);
  end;
  let q be object;
  assume
A3: q in Sphere(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,Th7;
  then dist(f,e) = r by A1,JGRAPH_1:28;
  hence thesis by METRIC_1:13;
end;
