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

theorem
  for f being Function of T,R^1 holds f is open iff
  for p being Point of T, V being open Subset of T st p in V
  ex r being positive Real st ].f.p-r,f.p+r.[ c= f.:V
  proof
    let f be Function of T,R^1;
    thus f is open implies
    for p being Point of T, V being open Subset of T st p in V
    ex r being positive Real st ].f.p-r,f.p+r.[ c= f.:V
    proof
      assume
A1:   f is open;
      let p be Point of T, V be open Subset of T;
      assume
A2:   p in V;
      reconsider fp = f.p as Point of RealSpace;
      consider r being positive Real such that
A3:   Ball(fp,r) c= f.:V by A1,A2,Th4;
      ].fp-r,fp+r.[ = Ball(fp,r) by FRECHET:7;
      hence thesis by A3;
    end;
    assume
A4: for p being Point of T, V being open Subset of T st p in V
    ex r being positive Real st ].f.p-r,f.p+r.[ c= f.:V;
    for p being Point of T, V being open Subset of T,
    q being Point of RealSpace st q = f.p & p in V
    ex r being positive Real st Ball(q,r) c= f.:V
    proof
      let p be Point of T, V be open Subset of T,
          q be Point of RealSpace such that
A5:   q = f.p;
      assume p in V;
      then consider r being positive Real such that
A6:   ].f.p-r,f.p+r.[ c= f.:V by A4;
      ].q-r,q+r.[ = Ball(q,r) by FRECHET:7;
      hence thesis by A5,A6;
    end;
    hence thesis by Th4;
  end;
