let L be non empty LattStr ; ( ( for v0 being Element of L holds v0 "/\" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2) ) & ( for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0 ) & ( for v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) ) & ( for v0 being Element of L holds v0 "\/" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ) & ( for v1, v0 being Element of L holds v0 "\/" v1 = v1 "\/" v0 ) & ( for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ) & ( for v0, v1, v2 being Element of L st v0 "\/" v1 = v1 holds
v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) implies for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A2:
for v0 being Element of L holds v0 "/\" v0 = v0
; ( ex v2, v1, v0 being Element of L st not (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2) or ex v1, v0 being Element of L st not v0 "/\" v1 = v1 "/\" v0 or ex v0, v2, v1 being Element of L st not (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A3:
for v2, v1, v0 being Element of L holds (v0 "/\" v1) "/\" v2 = v0 "/\" (v1 "/\" v2)
; ( ex v1, v0 being Element of L st not v0 "/\" v1 = v1 "/\" v0 or ex v0, v2, v1 being Element of L st not (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A4:
for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0
; ( ex v0, v2, v1 being Element of L st not (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2) or ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A5:
for v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" (v0 "/\" v1) = v0 "/\" (v1 "\/" v2)
; ( ex v0 being Element of L st not v0 "\/" v0 = v0 or ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A6:
for v0 being Element of L holds v0 "\/" v0 = v0
; ( ex v2, v1, v0 being Element of L st not (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) or ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A7:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2)
; ( ex v1, v0 being Element of L st not v0 "\/" v1 = v1 "\/" v0 or ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A8:
for v1, v0 being Element of L holds v0 "\/" v1 = v1 "\/" v0
; ( ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
assume A9:
for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2)
; ( ex v0, v1, v2 being Element of L st
( v0 "\/" v1 = v1 & not v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1 ) or for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3)) )
A11:
for v0, v2, v1 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
assume A12:
for v0, v1, v2 being Element of L st v0 "\/" v1 = v1 holds
v0 "\/" (v2 "/\" v1) = (v0 "\/" v2) "/\" v1
; for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3))
assume
not for v1, v2, v3 being Element of L holds (v1 "/\" v2) "\/" (v1 "/\" v3) = v1 "/\" (v2 "\/" (v1 "/\" v3))
; contradiction
then consider c1, c2, c3 being Element of L such that
A14:
(c1 "/\" c2) "\/" (c1 "/\" c3) <> c1 "/\" (c2 "\/" (c1 "/\" c3))
;
A17:
for v0, v2, v1 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" (v1 "\/" v2)) = v0 "/\" (v1 "\/" v2)
A20:
for v102, v101 being Element of L holds v101 "/\" v102 = v101 "/\" (v101 "/\" v102)
A24:
for v2, v0, v1 being Element of L holds (v1 "/\" v0) "/\" v2 = v0 "/\" (v1 "/\" v2)
A27:
for v0, v2, v1 being Element of L holds v0 "/\" (v1 "/\" v2) = v1 "/\" (v0 "/\" v2)
A30:
for v102, v101 being Element of L holds v101 "\/" v102 = v101 "\/" (v101 "\/" v102)
A35:
for v102, v100 being Element of L holds (v100 "\/" v102) "\/" v102 = v100 "\/" v102
A38:
for v1, v0 being Element of L holds v1 "\/" (v0 "\/" v1) = v0 "\/" v1
A41:
for v0, v2, v1 being Element of L holds (v1 "\/" v0) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
A45:
for v2, v100 being Element of L holds (v100 "\/" v2) "/\" v100 = v100 "\/" (v2 "/\" v100)
A48:
for v1, v0 being Element of L holds v0 "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v0)
A52:
for v102, v1, v100 being Element of L holds (v100 "/\" v1) "\/" (v100 "/\" ((v100 "/\" v1) "\/" v102)) = v100 "/\" ((v100 "/\" v1) "\/" v102)
A56:
for v101, v2, v100 being Element of L holds (v100 "\/" v2) "/\" (v100 "\/" v101) = v100 "\/" (v2 "/\" (v100 "\/" v101))
A60:
for v101, v2, v100 being Element of L holds (v100 "\/" v2) "/\" (v101 "\/" v100) = v100 "\/" (v2 "/\" (v101 "\/" v100))
A63:
for v1, v0 being Element of L holds v0 "\/" (v0 "/\" v1) = v0 "/\" (v0 "\/" v1)
A67:
for v100, v1, v101 being Element of L holds v100 "/\" (v101 "\/" (v101 "/\" v1)) = v101 "/\" (v100 "/\" (v101 "\/" v1))
A71:
for v1, v2, v0 being Element of L holds v1 "\/" (v0 "/\" v2) = (v1 "\/" (v0 "/\" v2)) "/\" (v0 "\/" v1)
A74:
for v0, v2, v1 being Element of L holds v0 "\/" (v1 "/\" v2) = v0 "\/" ((v1 "/\" v2) "/\" (v1 "\/" v0))
A76:
for v0, v2, v1 being Element of L holds v0 "\/" (v1 "/\" v2) = v0 "\/" (v1 "/\" (v2 "/\" (v1 "\/" v0)))
A79:
for v2, v0, v1 being Element of L holds (v1 "\/" v0) "/\" (v0 "\/" v2) = v0 "\/" (v1 "/\" (v0 "\/" v2))
A83:
for v101, v2, v102 being Element of L holds ((v102 "/\" v2) "\/" v101) "/\" (v102 "\/" (v102 "/\" v2)) = (v102 "/\" v2) "\/" (v102 "/\" (v101 "/\" (v102 "\/" v2)))
A86:
for v2, v1, v0 being Element of L holds (v0 "\/" (v0 "/\" v1)) "/\" ((v0 "/\" v1) "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" (v2 "/\" (v0 "\/" v1)))
A88:
for v2, v1, v0 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" ((v0 "/\" v1) "\/" v2)) = (v0 "/\" v1) "\/" (v0 "/\" (v2 "/\" (v0 "\/" v1)))
A90:
for v2, v1, v0 being Element of L holds v0 "/\" ((v0 "/\" v1) "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" (v2 "/\" (v0 "\/" v1)))
A94:
for v102, v2, v101 being Element of L holds (v101 "/\" v2) "\/" (v101 "/\" (v101 "/\" (v102 "/\" (v101 "\/" v2)))) = (v101 "/\" v2) "\/" (v101 "/\" v102)
A97:
for v2, v1, v0 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" (v2 "/\" (v0 "\/" v1))) = (v0 "/\" v1) "\/" (v0 "/\" v2)
A99:
for v2, v1, v0 being Element of L holds (v0 "/\" v1) "\/" (v0 "/\" v2) = v0 "/\" ((v0 "/\" v1) "\/" v2)
A102:
for v2, v1, v0 being Element of L holds v0 "/\" (v2 "\/" (v0 "/\" v1)) = (v0 "/\" v1) "\/" (v0 "/\" v2)
(c1 "/\" c3) "\/" (c1 "/\" c2) <> (c1 "/\" c2) "\/" (c1 "/\" c3)
by A102, A14;
hence
contradiction
by A8; verum