reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th19:
  cl_Ball(x,r) \ Ball(x,r) = Sphere(x,r)
proof
  thus cl_Ball(x,r) \ Ball(x,r) c= Sphere(x,r)
  proof
    let a be object;
    assume
A1: a in cl_Ball(x,r) \ Ball(x,r);
    then reconsider a as Point of TOP-REAL n;
A2: a in cl_Ball(x,r) by A1,XBOOLE_0:def 5;
A3: not a in Ball(x,r) by A1,XBOOLE_0:def 5;
A4: |. a-x .| <= r by A2,TOPREAL9:8;
    |. a-x .| >= r by A3,TOPREAL9:7;
    then |. a-x .| = r by A4,XXREAL_0:1;
    hence thesis by TOPREAL9:9;
  end;
  let a be object;
  assume
A5: a in Sphere(x,r);
  then reconsider a as Point of TOP-REAL n;
A6: |. a-x .| = r by A5,TOPREAL9:9;
  then
A7: a in cl_Ball(x,r) by TOPREAL9:8;
  not a in Ball(x,r) by A6,TOPREAL9:7;
  hence thesis by A7,XBOOLE_0:def 5;
end;
