reserve x,y,z for set;

theorem Th4:
  for Q being non empty QuantaleStr st the LattStr of Q = BooleLatt
  {} holds Q is associative commutative unital with_zero complete
  right-distributive left-distributive Lattice-like
proof
  set B = BooleLatt {};
  let Q be non empty QuantaleStr;
  set a = the Element of Q;
  assume
A1: the LattStr of Q = B;
A2: now
    let x, y be Element of Q;
A3: carr(B) = {{}} by LATTICE3:def 1,ZFMISC_1:1;
    then x = {} by A1,TARSKI:def 1;
    hence x = y by A1,A3,TARSKI:def 1;
  end;
  set o = times(Q);
  thus times(Q) is associative
  proof
    thus for a,b,c being Element of Q holds o.(a,o.(b,c)) = o.(o.(a,b ),c)
    by A2;
  end;
A4: ( for p,q,r being Element of Q holds p"/\"(q"/\"r) = (p"/\"q)"/\"r)&
  for p,q being Element of Q holds p"/\"(p"\/" q) = p by A2;
  thus times(Q) is commutative
  proof
    thus for a,b be Element of Q holds o.(a,b) = o.(b,a) by A2;
  end;
  thus times(Q) is having_a_unity
  proof
    take a;
    thus a is_a_left_unity_wrt times(Q)
    proof
      let b be Element of Q;
      thus times(Q).(a,b) = b by A2;
    end;
    let b be Element of Q;
    thus times(Q).(b,a) = b by A2;
  end;
  thus Q is with_zero
  proof
    thus Q is with_left-zero
    proof
      take a;
      thus thesis by A2;
    end;
    take a;
    thus thesis by A2;
  end;
  now
    let X be set;
    consider p being Element of B such that
A5: X is_less_than p and
A6: for r being Element of B st X is_less_than r holds p [= r by
LATTICE3:def 18;
    reconsider p9 = p as Element of Q by A1;
    take p9;
    thus X is_less_than p9 by A1,A5,Th2;
    let r9 be Element of Q;
    reconsider r = r9 as Element of B by A1;
    assume X is_less_than r9;
    then X is_less_than r by A1,Th2;
    then p [= r by A6;
    hence p9 [= r9 by A1;
  end;
  hence for X being set ex p being Element of Q st X is_less_than p & for r
  being Element of Q st X is_less_than r holds p [= r;
  thus Q is right-distributive
  by A2;
  thus Q is left-distributive
  by A2;
A7: ( for p,q being Element of Q holds (p"/\"q)"\/" q = q)& for p,q being
  Element of Q holds p"/\"q = q"/\"p by A2;
  ( for p,q being Element of Q holds p"\/"q = q"\/"p)& for p,q,r being
  Element of Q holds p"\/"(q"\/"r) = (p"\/"q)"\/"r by A2;
  then Q is join-commutative join-associative meet-absorbing meet-commutative
  meet-associative join-absorbing by A7,A4;
  hence thesis;
end;
