reserve L for non empty LattStr;
reserve v3,v101,v100,v102,v103,v2,v1,v0 for Element of L;

theorem Cluster2:
  L is meet-idempotent meet-associative meet-commutative
       join-idempotent join-associative satisfying_QLT2 distributive implies
    L is distributive'
  proof
    assume L is meet-idempotent meet-associative meet-commutative
       join-idempotent join-associative satisfying_QLT2 distributive; then
    (for v0 holds v0"/\"v0 = v0) &
    (for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
    (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
    (for v0 holds v0"\/"v0 = v0) &
    (for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
    (for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
    (for v0,v2,v1 holds v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2))
      by LATTICES:def 5, def 6, def 7, def 11,
      SHEFFER1:def 9, ROBBINS1:def 7; then
    for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3) by ThQLT2;
    hence thesis by SHEFFER1:def 5;
  end;
