reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem
  for S, T being non empty TopSpace, f being Function of S, T st for p
being Point of S, P being open a_neighborhood of p ex R being a_neighborhood of
  f.p st R c= f.:P holds f is open
proof
  let S, T be non empty TopSpace, f be Function of S, T such that
A1: for p being Point of S, P being open a_neighborhood of p ex R being
  a_neighborhood of f.p st R c= f.:P;
  let A be Subset of S such that
A2: A is open;
  for x being set holds x in f.:A iff ex Q being Subset of T st Q is open
  & Q c= f.:A & x in Q
  proof
    let x be set;
    hereby
      assume x in f.:A;
      then consider a being object such that
A3:   a in dom f and
A4:   a in A and
A5:   x = f.a by FUNCT_1:def 6;
      reconsider p = a as Point of S by A3;
      consider V being Subset of S such that
A6:   V is open and
A7:   V c= A and
A8:   a in V by A2,A4;
      V is a_neighborhood of p by A6,A8,CONNSP_2:3;
      then consider R being a_neighborhood of f.p such that
A9:   R c= f.:V by A1,A6;
      take K = Int R;
      Int R c= R by TOPS_1:16;
      then
A10:  K c= f.:V by A9;
      thus K is open;
      f.:V c= f.:A by A7,RELAT_1:123;
      hence K c= f.:A by A10;
      thus x in K by A5,CONNSP_2:def 1;
    end;
    thus thesis;
  end;
  hence thesis by TOPS_1:25;
end;
