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,R^1 holds f is open iff
  for p being Point of R^1, r being positive Real
  ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[
  proof
    let f be Function of R^1,R^1;
    thus f is open implies
    for p being Point of R^1, r being positive Real
    ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[
    proof
      assume
A1:   f is open;
      let p be Point of R^1, r be positive Real;
      reconsider p1 = p, q1 = f.p as Point of RealSpace;
      consider s being positive Real such that
A2:   Ball(q1,s) c= f.:Ball(p1,r) by A1,Th6;
      ].p-r,p+r.[ = Ball(p1,r) & ].f.p-s,f.p+s.[ = Ball(q1,s)
      by FRECHET:7;
      hence thesis by A2;
    end;
    assume
A3: for p being Point of R^1, r being positive Real
    ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[;
    for p, q being Point of RealSpace, r being positive Real
    st q = f.p
    ex s being positive Real st Ball(q,s) c= f.:Ball(p,r)
    proof
      let p, q be Point of RealSpace,
          r be positive Real such that
A4:   q = f.p;
      consider s being positive Real such that
A5:   ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[ by A3;
      ].p-r,p+r.[ = Ball(p,r) & ].f.p-s,f.p+s.[ = Ball(q,s)
      by A4,FRECHET:7;
      hence thesis by A5;
    end;
    hence thesis by Th6;
  end;
