reserve X for non empty set;

theorem Th3:
  for B being Subset-Family of X st (for B1,B2 being Element of B holds
  ex BB being Subset of B st B1 /\ B2 = union BB) & X = union B
  holds UniCl B = UniCl FinMeetCl B & TopStruct(#X,UniCl B#) is TopSpace-like
  proof
    let B be Subset-Family of X;
    assume that
A1: (for B1,B2 be Element of B
    ex BB being Subset of B st B1/\B2=union BB) and
A2: X = union B;
    thus UniCl B=UniCl FinMeetCl B
    proof
      hereby let x be object;
        assume
A3:     x in UniCl B;
        then reconsider x0=x as Subset of X;
        consider Y being Subset-Family of X such that
A4:     Y c= B and
A5:     x=union Y by A3,CANTOR_1:def 1;
        B c= FinMeetCl B by CANTOR_1:4;
        then Y c= FinMeetCl B by A4;
        hence x in UniCl FinMeetCl B by A5,CANTOR_1:def 1;
      end;
      let x be object;assume
A6:   x in UniCl FinMeetCl B;
      then reconsider x0=x as Subset of X;
      consider Y1 being Subset-Family of X such that
A7:   Y1 c= FinMeetCl B and
A8:   x=union Y1 by A6,CANTOR_1:def 1;
      FinMeetCl B c= UniCl B
      proof
        per cases;
        suppose B is empty;
          hence thesis by A1,A2,Th1;
        end;
        suppose B is non empty;
          hence thesis by A1,A2,Th2;
        end;
      end;
      then Y1 c= UniCl B by A7;
      then union Y1 in UniCl UniCl B by CANTOR_1:def 1;
      hence x in UniCl B by A8,YELLOW_9:15;
    end;
    hence TopStruct(#X,UniCl B#) is TopSpace-like by CANTOR_1:15;
  end;
