
theorem
  for T being with_closure with_properly_defined_topology 1TopStruct,
      A being Subset of T holds
    (the FirstOp of T).A = Cl A
  proof
    let T be with_closure with_properly_defined_topology 1TopStruct,
        A be Subset of T;
    set f = the FirstOp of T;
    consider F being Subset-Family of T such that
A2: (for C being Subset of T holds C in F iff C is closed & A c= C) &
       Cl A = meet F by PRE_TOPC:16;
B1: f is closure by CDef;
Z1: Cl A c= f.A
    proof
      f is idempotent by B1; then
      f.A is op-closed; then
A3:   f.A is closed by Lem1;
      f is extensive by B1; then
      f.A in F by A3,A2;
      hence thesis by A2,SETFAM_1:3;
    end;
    f is closure by CDef; then
N2: f is c=-monotone;
    defpred P[Subset of T] means $1 in F;
    set G = { f.B where B is Subset of T : B in F };
    deffunc T() = bool the carrier of T;
    deffunc F(Element of T()) = f.$1;
    deffunc G(Element of T()) = $1;
TT: for B being Element of T() st P[B] holds F(B) = G(B)
    proof
      let B be Subset of T;
      assume B in F; then
      B is op-closed by Lem1,A2;
      hence thesis;
    end;
    { F(B) where B is Element of T() : P[B] } =
      { G(B) where B is Element of T() : P[B] } from FRAENKEL:sch 6(TT); then
f3: F = G by Lemma;
    [#]T is closed & A c= [#]T; then
j1: G <> {} by f3,A2;
    for Z1 being set st Z1 in G holds f.A c= Z1
    proof
      let Z1 be set;
      assume Z1 in G; then
      ex B being Subset of T st Z1 = f.B & B in F;
      hence thesis by N2,A2;
    end;
    hence thesis by Z1,A2,f3,j1,SETFAM_1:5;
  end;
