reserve n,m for Nat;
reserve p,q,r for Element of Nat_Lattice;
reserve p,q,r for Element of Nat_Lattice;

theorem
  hcflat.(p,hcflat.(q,r)) = hcflat.(hcflat.(q,p),r) & hcflat.(p,hcflat.(
q,r)) = hcflat.(hcflat.(p,r),q) & hcflat.(p,hcflat.(q,r)) = hcflat.(hcflat.(r,q
  ),p) & hcflat.(p,hcflat.(q,r)) = hcflat.(hcflat.(r,p),q)
proof
  set s=r"/\"q;
  thus hcflat.(p,hcflat.(q,r)) =hcflat.(hcflat.(p,q),r) by Th9
    .= hcflat.(hcflat.(q,p),r) by Th6;
  thus
A1: hcflat.(p,hcflat.(q,r)) = hcflat.(p,hcflat.(r,q)) by Th6
    .=hcflat.(hcflat.(p,r),q) by Th9;
  thus hcflat.(p,hcflat.(q,r)) = hcflat.(p,s) by LATTICES:def 2
    .=hcflat.(hcflat.(r,q),p) by Th6;
  thus thesis by A1,Th6;
end;
