
theorem ImportTop:
  for X being set,
      f being Function of bool X, bool X st
    f is preinterior holds
    GenTop f is topology-like
  proof
    let X be set,
        f be Function of bool X, bool X;
    assume
AA: f is preinterior;
    set F = GenTop f;
a1: ex S being Subset of X st S = X & S is f-closed
    proof
      take S = [#]X;
      f is universe-preserving by AA;
      hence thesis;
    end;
a0: ex S being Subset of X st S = {} & S is f-closed
    proof
      take S = {}X;
      f is intensive by AA; then
      f.S c= S;
      hence thesis;
    end;
A2: F is cap-closed
    proof
      let a,b be Subset of X;
      assume
Y0:   a in F & b in F; then
      consider A being Subset of X such that
Y1:   A = a & A is f-closed by GTDef;
      consider B being Subset of X such that
Y2:   B = b & B is f-closed by GTDef,Y0;
      f is intensive /\-preserving universe-preserving by AA; then
      A /\ B is f-closed by Y1,Y2;
      hence thesis by GTDef,Y1,Y2;
    end;
    for a being Subset-Family of X st a c= F holds union a in F
    proof
      let a be Subset-Family of X;
      assume
J0:   a c= F;
      reconsider ua = union a as Subset of X;
      set b = COMPLEMENT a;
f3:     f is intensive by AA;
        union a c= f.union a
        proof
          for bb being set st bb in a holds bb c= f.union a
          proof
            let bb be set;
            assume
FG:         bb in a;
            reconsider b1 = bb as Subset of X by FG;
fh:         f is c=-monotone by AA;
            consider F being Subset of X such that
J1:         F = b1 & F is f-closed by GTDef,FG,J0;
            thus thesis by fh,J1,FG,ZFMISC_1:74;
          end;
          hence thesis by ZFMISC_1:76;
        end; then
        union a = f.union a by f3; then
        ua is f-closed;
        hence thesis by GTDef;
    end; then
    F is union-closed;
    hence thesis by a0,a1,A2,GTDef;
  end;
