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

theorem Cluster1:
  L is meet-commutative satisfying_QLT1 join-idempotent join-associative
       join-commutative satisfying_QLT2 QLT-distributive implies
    L is distributive
proof
  assume that
A1: L is meet-commutative and
A2: L is satisfying_QLT1 and
A3: L is join-idempotent and
A4: L is join-associative and
A5: L is join-commutative and
A6: L is satisfying_QLT2 and
A7: L is QLT-distributive;
S:  for v1,v0 holds v0"/\"v1 = v1"/\"v0 by LATTICES:def 6,A1;
S2: for v0 holds v0"\/"v0 = v0 by A3,ROBBINS1:def 7;
S3: for v2,v1,v0 holds
       (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A4,LATTICES:def 5;
S4: for v1,v0 holds v0"\/"v1 = v1"\/"v0 by A5,LATTICES:def 4;
  let v1,v2,v3 be Element of L;
  thus thesis by Lemma1,S,A2,A6,S2,S3,S4,A7;
end;
