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

theorem Th3:
  for X, Y being non empty TopSpace, f being Function of X,Y
  holds f is open iff for p being Point of X, V being open Subset of X
  st p in V
  ex W being open Subset of Y st f.p in W & W c= f.:V
  proof
    let X, Y be non empty TopSpace, f be Function of X,Y;
    thus f is open implies for p being Point of X, V being open Subset of X
    st p in V
    ex W being open Subset of Y st f.p in W & W c= f.:V
    proof
      assume
A1:   f is open;
      let p be Point of X, V be open Subset of X such that
A2:   p in V;
      V is a_neighborhood of p by A2,CONNSP_2:3;
      then consider W being open a_neighborhood of f.p such that
A3:   W c= f.:V by A1,TOPGRP_1:22;
      take W;
      thus f.p in W by CONNSP_2:4;
      thus thesis by A3;
    end;
    assume
A4: for p being Point of X, V being open Subset of X st p in V
    ex W being open Subset of Y st f.p in W & W c= f.:V;
    for p being Point of X, P being open a_neighborhood of p
    ex R being a_neighborhood of f.p st R c= f.:P
    proof
      let p be Point of X;
      let P be open a_neighborhood of p;
      consider W being open Subset of Y such that
A5:   f.p in W and
A6:   W c= f.:P by A4,CONNSP_2:4;
      W is a_neighborhood of f.p by A5,CONNSP_2:3;
      hence thesis by A6;
    end;
    hence thesis by TOPGRP_1:23;
  end;
