reserve i for Integer,
  a, b, r, s for Real;

theorem Th20:
  for x being Point of R^1, M being a_neighborhood of x ex N being
  Neighbourhood of x st N c= M
proof
  let x be Point of R^1, M be a_neighborhood of x;
  consider V being Subset of R^1 such that
A1: V is open and
A2: V c= M and
A3: x in V by CONNSP_2:6;
  consider r being Real such that
A4: r > 0 and
A5: ].x-r,x+r.[ c= V by A1,A3,FRECHET:8;
  reconsider N = ].x-r,x+r.[ as Neighbourhood of x by A4,RCOMP_1:def 6;
  take N;
  thus thesis by A2,A5;
end;
