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 being TopSpace, T being non empty TopSpace, f being Function of
S, T holds f is continuous iff for P being Subset of T holds f"(Int P) c= Int(f
  "P)
proof
  let S be TopSpace, T be non empty TopSpace, f be Function of S, T;
A1: [#]T <> {};
  hereby
    assume
A2: f is continuous;
    let P be Subset of T;
    f"Int P c= f"P by RELAT_1:143,TOPS_1:16;
    then
A3: Int(f"Int P) c= Int(f"P) by TOPS_1:19;
    f"(Int P) is open by A1,A2,TOPS_2:43;
    hence f"(Int P) c= Int(f"P) by A3,TOPS_1:23;
  end;
  assume
A4: for P being Subset of T holds f"(Int P) c= Int(f"P);
  now
    let P be Subset of T;
    assume P is open;
    then Int P = P by TOPS_1:23;
    then
A5: f"P c= Int (f"P) by A4;
    Int (f"P) c= f"P by TOPS_1:16;
    hence f"P is open by A5,XBOOLE_0:def 10;
  end;
  hence thesis by A1,TOPS_2:43;
end;
