set B = BooleLatt {};
let Q be non empty QuantaleStr ; :: thesis: ( LattStr(# the carrier of Q, the L_join of Q, the L_meet of Q #) = BooleLatt {} implies ( Q is associative & Q is commutative & Q is unital & Q is with_zero & Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like ) )
set a = the Element of Q;
assume A1: LattStr(# the carrier of Q, the L_join of Q, the L_meet of Q #) = BooleLatt {} ; :: thesis: ( Q is associative & Q is commutative & Q is unital & Q is with_zero & Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like )

A2: now

let x, y be Element of Q; :: thesis: x = y
A3: H₁( BooleLatt {}) = {{}} by LATTICE3:def 1, ZFMISC_1:1;
then x = {} by A1, TARSKI:def 1;
hence x = y by A1, A3, TARSKI:def 1; :: thesis: verum

end;

thus H₄(Q) is associative :: according to MONOID_0:def 12 :: thesis: ( Q is commutative & Q is unital & Q is with_zero & Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like )

proof

let a, b, c be Element of Q; :: according to BINOP_1:def 14 :: thesis: H₄(Q) . (a,(H₄(Q) . (b,c))) = H₄(Q) . ((H₄(Q) . (a,b)),c)
thus H₄(Q) . (a,(H₄(Q) . (b,c))) = H₄(Q) . ((H₄(Q) . (a,b)),c) by A2; :: thesis: verum

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 H₄(Q) is commutative :: according to MONOID_0:def 11 :: thesis: ( Q is unital & Q is with_zero & Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like )

proof

let a, b be Element of Q; :: according to BINOP_1:def 13 :: thesis: H₄(Q) . (a,b) = H₄(Q) . (b,a)
thus H₄(Q) . (a,b) = H₄(Q) . (b,a) by A2; :: thesis: verum

end;

thus H₄(Q) is having_a_unity :: according to MONOID_0:def 10 :: thesis: ( Q is with_zero & Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like )

proof

take the Element of Q ; :: according to SETWISEO:def 2 :: thesis: the Element of Q is_a_unity_wrt H₄(Q)
thus the Element of Q is_a_left_unity_wrt H₄(Q) :: according to BINOP_1:def 7 :: thesis: the Element of Q is_a_right_unity_wrt H₄(Q)

proof

let b be Element of Q; :: according to BINOP_1:def 16 :: thesis: H₄(Q) . ( the Element of Q,b) = b
thus H₄(Q) . ( the Element of Q,b) = b by A2; :: thesis: verum

end;

let b be Element of Q; :: according to BINOP_1:def 17 :: thesis: H₄(Q) . (b, the Element of Q) = b
thus H₄(Q) . (b, the Element of Q) = b by A2; :: thesis: verum

end;

thus Q is with_zero :: thesis: ( Q is complete & Q is right-distributive & Q is left-distributive & Q is Lattice-like )

proof

thus Q is with_left-zero :: according to QUANTAL1:def 4 :: thesis: Q is with_right-zero

proof

take the Element of Q ; :: according to QUANTAL1:def 2 :: thesis: for b being Element of Q holds the Element of Q * b = the Element of Q
thus for b being Element of Q holds the Element of Q * b = the Element of Q by A2; :: thesis: verum

end;

take the Element of Q ; :: according to QUANTAL1:def 3 :: thesis: for a being Element of Q holds a * the Element of Q = the Element of Q
thus for a being Element of Q holds a * the Element of Q = the Element of Q by A2; :: thesis: verum

end;

now

let X be set ; :: thesis: ex p9 being Element of Q st
( X is_less_than p9 & ( for r9 being Element of Q st X is_less_than r9 holds
p9 [= r9 ) )
consider p being Element of (BooleLatt {}) such that
A5: X is_less_than p and
A6: for r being Element of (BooleLatt {}) st X is_less_than r holds
p [= r by LATTICE3:def 18;
reconsider p9 = p as Element of Q by A1;
take p9 = p9; :: thesis: ( X is_less_than p9 & ( for r9 being Element of Q st X is_less_than r9 holds
p9 [= r9 ) )
thus X is_less_than p9 by A1, A5, Th2; :: thesis: for r9 being Element of Q st X is_less_than r9 holds
p9 [= r9
let r9 be Element of Q; :: thesis: ( X is_less_than r9 implies p9 [= r9 )
reconsider r = r9 as Element of (BooleLatt {}) by A1;
assume X is_less_than r9 ; :: thesis: p9 [= r9
then X is_less_than r by A1, Th2;
then p [= r by A6;
hence p9 [= r9 by A1, Th1; :: thesis: verum

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 ) ) ; :: according to LATTICE3:def 18 :: thesis: ( Q is right-distributive & Q is left-distributive & Q is Lattice-like )
thus Q is right-distributive :: thesis: ( Q is left-distributive & Q is Lattice-like )

proof

let a be Element of Q; :: according to QUANTAL1:def 5 :: thesis: for X being set holds a [*] ("\/" (X,Q)) = "\/" ( { (a [*] b) where b is Element of Q : b in X } ,Q)
let X be set ; :: thesis: a [*] ("\/" (X,Q)) = "\/" ( { (a [*] b) where b is Element of Q : b in X } ,Q)
thus a [*] ("\/" (X,Q)) = "\/" ( { (a [*] b) where b is Element of Q : b in X } ,Q) by A2; :: thesis: verum

end;

thus Q is left-distributive :: thesis: Q is Lattice-like

proof

let a be Element of Q; :: according to QUANTAL1:def 6 :: thesis: for X being set holds ("\/" (X,Q)) [*] a = "\/" ( { (b [*] a) where b is Element of Q : b in X } ,Q)
let X be set ; :: thesis: ("\/" (X,Q)) [*] a = "\/" ( { (b [*] a) where b is Element of Q : b in X } ,Q)
thus ("\/" (X,Q)) [*] a = "\/" ( { (b [*] a) where b is Element of Q : b in X } ,Q) by A2; :: thesis: verum

end;

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 & Q is join-associative & Q is meet-absorbing & Q is meet-commutative & Q is meet-associative & Q is join-absorbing ) by A7, A4, LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, LATTICES:def 7, LATTICES:def 8, LATTICES:def 9;
hence Q is Lattice-like ; :: thesis: verum