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

theorem Th57:
  for F being Subset-Family of T holds (for A being Subset of T st
A in F holds Int Cl A c= A) implies Int Cl(meet F) c= meet F & Int Cl Int(meet
  F) = Int(meet F)
proof
  let F be Subset-Family of T;
A1: Int Int (meet F) c= Int Cl Int(meet F) by PRE_TOPC:18,TOPS_1:19;
  assume
A2: for A being Subset of T st A in F holds Int Cl A c= A;
  thus Int Cl(meet F) c= meet F
  proof
    now
      per cases;
      suppose
A3:     F = {};
        Cl {}T = {}T by PRE_TOPC:22;
        hence thesis by A3,SETFAM_1:1;
      end;
      suppose
A4:     F <> {};
        now
          let A0 be set;
          assume
A5:       A0 in F;
          then reconsider A = A0 as Subset of T;
          Cl(meet F) c= Cl A by A5,PRE_TOPC:19,SETFAM_1:3;
          then
A6:       Int Cl(meet F) c= Int Cl A by TOPS_1:19;
          Int Cl A c= A by A2,A5;
          hence Int Cl(meet F) c= A0 by A6;
        end;
        hence thesis by A4,SETFAM_1:5;
      end;
    end;
    hence thesis;
  end;
  then
A7: Int Int Cl(meet F) c= Int(meet F) by TOPS_1:19;
  Cl Int(meet F) c= Cl(meet F) by PRE_TOPC:19,TOPS_1:16;
  then Int Cl Int(meet F) c= Int Cl(meet F) by TOPS_1:19;
  then Int Cl Int(meet F) c= Int(meet F) by A7;
  hence Int Cl Int(meet F) = Int(meet F) by A1,XBOOLE_0:def 10;
end;
