let L be non empty LattStr ; ( L is join-commutative & L is join-associative & L is meet-commutative & L is meet-associative & ( for v1, v0 being Element of L holds v0 "/\" (v0 "\/" v1) = v0 ) & ( for v1, v0 being Element of L holds v0 "\/" (v0 "/\" v1) = v0 ) implies ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A2:
L is join-commutative
; ( not L is join-associative or not L is meet-commutative or not L is meet-associative or ex v1, v0 being Element of L st not v0 "/\" (v0 "\/" v1) = v0 or ex v1, v0 being Element of L st not v0 "\/" (v0 "/\" v1) = v0 or ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A3:
L is join-associative
; ( not L is meet-commutative or not L is meet-associative or ex v1, v0 being Element of L st not v0 "/\" (v0 "\/" v1) = v0 or ex v1, v0 being Element of L st not v0 "\/" (v0 "/\" v1) = v0 or ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A4:
L is meet-commutative
; ( not L is meet-associative or ex v1, v0 being Element of L st not v0 "/\" (v0 "\/" v1) = v0 or ex v1, v0 being Element of L st not v0 "\/" (v0 "/\" v1) = v0 or ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A5:
L is meet-associative
; ( ex v1, v0 being Element of L st not v0 "/\" (v0 "\/" v1) = v0 or ex v1, v0 being Element of L st not v0 "\/" (v0 "/\" v1) = v0 or ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A6:
for v1, v0 being Element of L holds v0 "/\" (v0 "\/" v1) = v0
; ( ex v1, v0 being Element of L st not v0 "\/" (v0 "/\" v1) = v0 or ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) ) )
assume A7:
for v1, v0 being Element of L holds v0 "\/" (v0 "/\" v1) = v0
; ( ( for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0 ) & ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) )
thus
for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "/\" v2)) = v0
( ( for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0 ) & ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) )
thus
for v1, v2, v0 being Element of L holds v0 "/\" (v1 "\/" (v0 "\/" v2)) = v0
( ( for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1 ) & ( for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1 ) )
thus
for v2, v1, v0 being Element of L holds ((v0 "/\" v1) "\/" (v1 "/\" v2)) "\/" v1 = v1
for v2, v1, v0 being Element of L holds ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1
let v2, v1, v0 be Element of L; ((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1
((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 =
(v0 "\/" v1) "/\" ((v1 "\/" v2) "/\" v1)
by A5
.=
(v0 "\/" v1) "/\" (v1 "/\" (v1 "\/" v2))
by A4
.=
(v0 "\/" v1) "/\" v1
by A6
.=
v1 "/\" (v0 "\/" v1)
by A4
.=
v1 "/\" (v1 "\/" v0)
by A2
.=
v1
by A6
;
hence
((v0 "\/" v1) "/\" (v1 "\/" v2)) "/\" v1 = v1
; verum