reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem
  for X being non empty compact Subset of TOP-REAL 2, p being Point of
  Euclid 2 st p = 0.TOP-REAL 2 & a > 0 holds X
  c= Ball(p, |.E-bound X.|+|.N-bound X.|+|.W-bound X.|+|.S-bound X.|+a)
proof
  let X be non empty compact Subset of TOP-REAL 2, p be Point of Euclid 2 such
  that
A1: p = 0.TOP-REAL 2 and
A2: a > 0;
  set A = X, n = N-bound A, s = S-bound A, e = E-bound A, w = W-bound A, r =
  |.e.|+|.n.|+|.w.|+|.s.|+a;
A3: |.e.| + |.n.| + |.w.| + |.s.| + 0 < |.e.| + |.n.| + |.w.| + |.s.| + a
        by A2,XREAL_1:8;
  let x be object;
  assume
A4: x in X;
  then reconsider b = x as Point of Euclid 2 by TOPREAL3:8;
  reconsider P = p, B = b as Point of TOP-REAL 2 by TOPREAL3:8;
A5: P`1 = 0 by A1,Th22;
A6: B`1 <= e by A4,PSCOMP_1:24;
A7: B`2 <= n by A4,PSCOMP_1:24;
A8: s <= B`2 by A4,PSCOMP_1:24;
A9: P`2 = 0 by A1,Th22;
A10: dist(p,b) = (Pitag_dist 2).(p,b) by METRIC_1:def 1
    .= sqrt ((P`1 - B`1)^2 + (P`2 - B`2)^2) by TOPREAL3:7
    .= sqrt ((B`1)^2 + (B`2)^2) by A5,A9;
A11: 0 <= (B`2)^2 by XREAL_1:63;
  0 <= (B`1)^2 by XREAL_1:63;
  then sqrt ((B`1)^2 + (B`2)^2) <= sqrt(B`1)^2 + sqrt(B`2)^2
      by A11,SQUARE_1:59;
  then sqrt ((B`1)^2 + (B`2)^2) <= |.B`1.| + sqrt(B`2)^2 by COMPLEX1:72;
  then
A12: sqrt ((B`1)^2 + (B`2)^2) <= |.B`1.| + |.B`2.| by COMPLEX1:72;
A13: 0 <= |.n.| by COMPLEX1:46;
A14: 0 <= |.e.| by COMPLEX1:46;
A15: 0 <= |.w.| by COMPLEX1:46;
A16: 0 <= |.s.| by COMPLEX1:46;
A17: w <= B`1 by A4,PSCOMP_1:24;
  now
    per cases;
    case
A18:  B`1 >= 0 & B`2 >= 0;
      |.e.| + |.n.| + 0 <= |.e.| + |.n.| + |.w.| by A15,XREAL_1:7;
      then |.e.| + |.n.| <= |.e.| + |.n.| + |.w.| + |.s.| by A16,XREAL_1:7;
      then
A19:  |.e.| + |.n.| < r by A3,XXREAL_0:2;
A20:  |.B`2.| <= |.n.| by A7,A18,Th1;
      |.B`1.| <= |.e.| by A6,A18,Th1;
      then |.B`1.| + |.B`2.| <= |.e.| + |.n.| by A20,XREAL_1:7;
      then dist(p,b) <= |.e.| + |.n.| by A10,A12,XXREAL_0:2;
      hence dist(p,b) < r by A19,XXREAL_0:2;
    end;
    case
A21:  B`1 < 0 & B`2 >= 0;
      0 + (|.n.| + |.w.|) <= |.e.| + (|.n.| + |.w.|) by A14,XREAL_1:7;
      then |.w.| + |.n.| <= |.e.| + |.n.| + |.w.| + |.s.| by A16,XREAL_1:7;
      then
A22:  |.w.| + |.n.| < r by A3,XXREAL_0:2;
A23:  |.B`2.| <= |.n.| by A7,A21,Th1;
      |.B`1.| <= |.w.| by A17,A21,Th2;
      then |.B`1.| + |.B`2.| <= |.w.| + |.n.| by A23,XREAL_1:7;
      then dist(p,b) <= |.w.| + |.n.| by A10,A12,XXREAL_0:2;
      hence dist(p,b) < r by A22,XXREAL_0:2;
    end;
    case
A24:  B`1 >= 0 & B`2 < 0;
A25:  |.e.| + |.n.| + |.s.| + 0 <= |.e.| + |.n.| + |.s.| + |.w.|
      by A15,XREAL_1:7;
      |.e.| + |.s.| + 0 <= |.e.| + |.s.| + |.n.| by A13,XREAL_1:7;
      then |.e.| + |.s.| <= |.e.| + |.n.| + |.w.| + |.s.| by A25,XXREAL_0:2;
      then
A26:  |.e.| + |.s.| < r by A3,XXREAL_0:2;
A27:  |.B`2.| <= |.s.| by A8,A24,Th2;
      |.B`1.| <= |.e.| by A6,A24,Th1;
      then |.B`1.| + |.B`2.| <= |.e.| + |.s.| by A27,XREAL_1:7;
      then dist(p,b) <= |.e.| + |.s.| by A10,A12,XXREAL_0:2;
      hence dist(p,b) < r by A26,XXREAL_0:2;
    end;
    case
A28:  B`1 < 0 & B`2 < 0;
      then
A29:  |.B`2.| <= |.s.| by A8,Th2;
      |.B`1.| <= |.w.| by A17,A28,Th2;
      then |.B`1.| + |.B`2.| <= |.w.| + |.s.| by A29,XREAL_1:7;
      then
A30:  dist(p,b) <= |.w.| + |.s.| by A10,A12,XXREAL_0:2;
      0 + (|.w.| + |.s.|) <= |.e.| + |.n.| + (|.w.| + |.s.|) by A14,A13,
XREAL_1:7;
      then |.w.| + |.s.| < r by A2,XREAL_1:8;
      hence dist(p,b) < r by A30,XXREAL_0:2;
    end;
  end;
  hence thesis by METRIC_1:11;
end;
