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 being Element of L holds v0 "\/" v0 = v0 ) & ( for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2) ) & ( for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) ) & ( for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) ) implies for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" 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 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 v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" 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 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 v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A4:
for v1, v0 being Element of L holds v0 "/\" v1 = v1 "/\" v0
; ( 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 v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A5:
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 v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A6:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "\/" v2 = v0 "\/" (v1 "\/" v2)
; ( ex v0, v2, v1 being Element of L st not (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2) or ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
assume A7:
for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "/\" (v0 "\/" v1) = v0 "\/" (v1 "/\" v2)
; ( ex v0, v2, v1 being Element of L st not v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2) or for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) )
A9:
for v0, v2, v1 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
assume A10:
for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = (v0 "/\" v1) "\/" (v0 "/\" v2)
; for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3)
A15:
for v102, v101 being Element of L holds v101 "/\" v102 = v101 "/\" (v101 "/\" v102)
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) "\/" (v0 "/\" v2) = v0 "/\" (v1 "\/" v2)
A28:
for v102, v0, v2, v1 being Element of L holds (v0 "/\" (v1 "\/" v2)) "\/" v102 = (v0 "/\" v1) "\/" ((v0 "/\" v2) "\/" v102)
A32:
for v102, v1, v100 being Element of L holds (v100 "/\" v1) "\/" (v100 "/\" v102) = v100 "/\" ((v100 "/\" v1) "\/" v102)
A35:
for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = v0 "/\" ((v0 "/\" v1) "\/" v2)
A39:
for v1, v101, v100 being Element of L holds (v100 "/\" v101) "\/" (v100 "/\" v1) = v100 "/\" (v101 "\/" (v100 "/\" v1))
A42:
for v0, v2, v1 being Element of L holds v0 "/\" (v1 "\/" v2) = v0 "/\" (v1 "\/" (v0 "/\" v2))
A46:
for v102, v1, v100 being Element of L holds (v100 "\/" v1) "/\" (v100 "\/" ((v100 "\/" v1) "/\" v102)) = v100 "\/" ((v100 "\/" v1) "/\" v102)
A49:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" ((v0 "\/" v1) "/\" v2)
A53:
for v2, v1, v101 being Element of L holds (v101 "\/" v1) "/\" (((v101 "\/" v1) "/\" v101) "\/" (v1 "/\" v2)) = v101 "\/" (v1 "/\" v2)
A56:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" ((v0 "/\" (v0 "\/" v1)) "\/" (v1 "/\" v2)) = v0 "\/" (v1 "/\" v2)
A58:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" ((v0 "/\" v0) "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
A60:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" ((v0 "/\" v1) "\/" (v1 "/\" v2))) = v0 "\/" (v1 "/\" v2)
A62:
for v1, v2, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" (v1 "/\" (v0 "\/" v2))) = v0 "\/" (v1 "/\" v2)
A64:
for v1, v2, v0 being Element of L holds v0 "\/" (v1 "/\" (v0 "\/" v2)) = v0 "\/" (v1 "/\" v2)
A66:
for v1, v2, v0 being Element of L holds v0 "\/" ((v0 "\/" v2) "/\" v1) = v0 "\/" (v1 "/\" v2)
A69:
for v2, v1, v0 being Element of L holds (v0 "\/" v1) "/\" (v0 "\/" v2) = v0 "\/" (v2 "/\" v1)
let v1, v2, v3 be Element of L; v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3)
v1 "\/" (v2 "/\" v3) = (v1 "\/" v3) "/\" (v1 "\/" v2)
by A69;
hence
v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3)
by A4; verum