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

theorem Th19:
  for x being Point of R^1, N being Subset of REAL, M being Subset
of R^1 st M = N holds N is Neighbourhood of x implies M is a_neighborhood of x
proof
  let x be Point of R^1, N be Subset of REAL, M be Subset of R^1 such that
A1: M = N;
  given r such that
A2: 0 < r and
A3: N = ].x-r,x+r.[;
  M is open by A1,A3,JORDAN6:35;
  hence thesis by A1,A2,A3,CONNSP_2:3,TOPREAL6:15;
end;
