let L be non empty LattStr ; :: thesis: ( L is meet-idempotent & L is meet-associative & L is meet-commutative & L is satisfying_QLT1 & L is join-idempotent & L is join-associative & L is join-commutative & L is satisfying_QLT2 & L is satisfying_Bowden_inequality implies L is distributive' )

assume A0: ( L is meet-idempotent & L is meet-associative & L is meet-commutative & L is satisfying_QLT1 & L is join-idempotent & L is join-associative & L is join-commutative & L is satisfying_QLT2 & L is satisfying_Bowden_inequality ) ; :: thesis: L is distributive'

then A2: ( ( for v0 being Element of L holds v0 "/\" v0 = v0 ) & ( 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) ) ) by LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, SHEFFER1:def 9, ROBBINS1:def 7;

for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "\/" ((v0 "\/" v1) "/\" v2) = v0 "\/" (v1 "/\" v2) by BowdenRedef, A0;

then for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) by ThQLT4, A2;

hence L is distributive' by SHEFFER1:def 5; :: thesis: verum

assume A0: ( L is meet-idempotent & L is meet-associative & L is meet-commutative & L is satisfying_QLT1 & L is join-idempotent & L is join-associative & L is join-commutative & L is satisfying_QLT2 & L is satisfying_Bowden_inequality ) ; :: thesis: L is distributive'

then A2: ( ( for v0 being Element of L holds v0 "/\" v0 = v0 ) & ( 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) ) ) by LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, SHEFFER1:def 9, ROBBINS1:def 7;

for v0, v2, v1 being Element of L holds (v0 "\/" (v1 "/\" v2)) "\/" ((v0 "\/" v1) "/\" v2) = v0 "\/" (v1 "/\" v2) by BowdenRedef, A0;

then for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) by ThQLT4, A2;

hence L is distributive' by SHEFFER1:def 5; :: thesis: verum