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

theorem Th68:
  for F being Subset-Family of T holds (for B being Subset of T st
B in F holds B c= (union F) \/ (Int Cl(union F))) & for A being Subset of T st
A is condensed holds (for B being Subset of T st B in F holds B c= A) implies (
  union F) \/ (Int Cl(union F)) c= A
proof
  let F be Subset-Family of T;
  thus for B being Subset of T st B in F holds B c= (union F) \/ (Int Cl(union
  F))
  proof
    let B be Subset of T;
    assume B in F;
    then
A1: B c= union F by ZFMISC_1:74;
    union F c= (union F) \/ (Int Cl(union F)) by XBOOLE_1:7;
    hence thesis by A1;
  end;
  thus for A being Subset of T st A is condensed holds (for B being Subset of
  T st B in F holds B c= A) implies (union F) \/ (Int Cl(union F)) c= A
  proof
    let A be Subset of T;
    assume A is condensed;
    then
A2: Int Cl A c= A by TOPS_1:def 6;
    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
A3: union F c= A by ZFMISC_1:76;
    then Cl(union F) c= Cl A by PRE_TOPC:19;
    then Int Cl(union F) c= Int Cl A by TOPS_1:19;
    then Int Cl(union F) c= A by A2;
    hence thesis by A3,XBOOLE_1:8;
  end;
end;
