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 f is
open holds for p being Point of S, P being a_neighborhood of p ex R being open
  a_neighborhood of f.p st R c= f.:P
proof
  let S, T be non empty TopSpace, f be Function of S, T such that
A1: for A being Subset of S st A is open holds f.:A is open;
  let p be Point of S, P be a_neighborhood of p;
A2: p in Int P by CONNSP_2:def 1;
  f.:Int P is open by A1;
  then reconsider R = f.:Int P as open a_neighborhood of f.p by A2,CONNSP_2:3
,FUNCT_2:35;
  take R;
  thus thesis by RELAT_1:123,TOPS_1:16;
end;
