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;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem Th48:
  Ball(|[a,b]|,r) = inside_of_circle(a,b,r)
proof
  reconsider e = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
  thus Ball(|[a,b]|,r) = Ball(e,r) by Th11
    .= inside_of_circle(a,b,r) by Th46;
end;
