reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem
  F <> {} implies [#] D \ union F = meet (COMPLEMENT(F))
proof
A1: for x st x in D holds (for X st X in F holds not x in X) iff for Y st Y
  in COMPLEMENT(F) holds x in Y
  proof
    let x such that
A2: x in D;
    thus (for X st X in F holds not x in X) implies for Y st Y in COMPLEMENT(F
    ) holds x in Y
    proof
      assume
A3:   for X st X in F holds not x in X;
      let Y;
      assume
A4:   Y in COMPLEMENT(F);
      then reconsider Y as Subset of D;
      Y` in F by A4,Def7;
      then not x in Y` by A3;
      hence thesis by A2,XBOOLE_0:def 5;
    end;
    assume
A5: for Y st Y in COMPLEMENT(F) holds x in Y;
    let X;
    assume
A6: X in F;
    then reconsider X as Subset of D;
    X`` = X;
    then X` in COMPLEMENT(F) by A6,Def7;
    then x in X` by A5;
    hence thesis by XBOOLE_0:def 5;
  end;
  assume F <> {};
  then
A7: COMPLEMENT(F) <> {} by Th32;
A8: [#] D \ union F c= meet COMPLEMENT(F)
  proof
    let x be object;
    assume
A9: x in [#] D \ union F;
    then not x in union F by XBOOLE_0:def 5;
    then for X st X in F holds not x in X by TARSKI:def 4;
    then for Y st Y in COMPLEMENT(F) holds x in Y by A1,A9;
    hence thesis by A7,Def1;
  end;
  meet COMPLEMENT(F) c= [#] D \ union F
  proof
    let x be object;
    assume
A10: x in meet COMPLEMENT(F);
    then for X holds X in COMPLEMENT(F) implies x in X by Def1;
    then for Y st x in Y holds not Y in F by A1;
    then not x in union F by TARSKI:def 4;
    hence thesis by A10,XBOOLE_0:def 5;
  end;
  hence thesis by A8;
end;
