reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;

theorem Th83:
  for F being Subset-Family of T holds (F is open implies for B
being Subset of T st B in F holds B c= Int Cl(union F)) & for A being Subset of
T st A is open_condensed holds (for B being Subset of T st B in F holds B c= A)
  implies Int Cl(union F) c= A
proof
  let F be Subset-Family of T;
  thus F is open implies for B being Subset of T st B in F holds B c= Int Cl(
  union F)
  proof
    assume F is open;
    then union F is open by TOPS_2:19;
    then
A1: Int(union F) = union F by TOPS_1:23;
    let B be Subset of T;
A2: Int(union F) c= Int Cl(union F) by PRE_TOPC:18,TOPS_1:19;
    assume B in F;
    then B c= union F by ZFMISC_1:74;
    hence thesis by A2,A1;
  end;
  thus for A being Subset of T st A is open_condensed holds (for B being
  Subset of T st B in F holds B c= A) implies Int Cl(union F) c= A
  proof
    let A be Subset of T;
    assume A is open_condensed;
    then
A3: A = Int Cl A by TOPS_1:def 8;
    assume for B being Subset of T st B in F holds B c= A;
    then for P be set st P in F holds P c= A;
    then union F c= A by ZFMISC_1:76;
    then Cl(union F) c= Cl A by PRE_TOPC:19;
    hence thesis by A3,TOPS_1:19;
  end;
end;
