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 Th18:
  r <= s implies Ball(x,r) c= Ball(x,s)
proof
  reconsider xe = x as Point of Euclid n by TOPREAL3:8;
A1: Ball(x,r) = Ball(xe,r) by TOPREAL9:13;
  Ball(x,s) = Ball(xe,s) by TOPREAL9:13;
  hence thesis by A1,PCOMPS_1:1;
end;
