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

theorem Th76:
  for F being Subset-Family of T holds (for B being Subset of T st
  B in F holds B c= Cl(union F)) & for A being Subset of T st A is
closed_condensed holds (for B being Subset of T st B in F holds B c= A) implies
  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= 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= Cl(union F) by PRE_TOPC:18;
    hence thesis by A1;
  end;
  thus for A being Subset of T st A is closed_condensed holds (for B being
  Subset of T st B in F holds B c= A) implies Cl(union F) c= A
  proof
    let A be Subset of T;
    reconsider A1 = A as Subset of T;
    assume A is closed_condensed;
    then
A2: A1 is closed by TOPS_1:66;
    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 A2,PRE_TOPC:22;
  end;
end;
