reserve
  n, m for Nat,
  T for non empty TopSpace,
  M, M1, M2 for non empty MetrSpace;

theorem
  for f being Function of R^1,T holds f is open iff
  for p being Point of R^1, r being positive Real
  ex V being open Subset of T st f.p in V & V c= f.:].p-r,p+r.[
  proof
    let f be Function of R^1,T;
    thus f is open implies
    for p being Point of R^1, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:].p-r,p+r.[
    proof
      assume
A1:   f is open;
      let p be Point of R^1, r be positive Real;
      reconsider q = p as Point of RealSpace;
      consider W be open Subset of T such that
A2:   f.p in W & W c= f.:Ball(q,r) by A1,Th5;
      ].q-r,q+r.[ = Ball(q,r) by FRECHET:7;
      hence thesis by A2;
    end;
    assume
A3: for p being Point of R^1, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:].p-r,p+r.[;
    for p being Point of RealSpace, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
    proof
      let p be Point of RealSpace, r be positive Real;
      reconsider q = p as Point of R^1;
      consider W being open Subset of T such that
A4:   f.q in W & W c= f.:].p-r,p+r.[ by A3;
      ].p-r,p+r.[ = Ball(p,r) by FRECHET:7;
      hence thesis by A4;
    end;
    hence thesis by Th5;
  end;
