let L be non empty LattStr ; :: thesis: ( L is meet-idempotent & L is meet-associative & L is meet-commutative & L is join-idempotent & L is join-associative & L is satisfying_QLT2 & L is distributive implies L is distributive' )
assume ( L is meet-idempotent & L is meet-associative & L is meet-commutative & L is join-idempotent & L is join-associative & L is satisfying_QLT2 & L is distributive ) ; :: thesis:
then ( ( 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) ) ) by ;
then for v1, v2, v3 being Element of L holds v1 "\/" (v2 "/\" v3) = (v1 "\/" v2) "/\" (v1 "\/" v3) by ThQLT2;
hence L is distributive' by SHEFFER1:def 5; :: thesis: verum