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