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

theorem
  lcmlat.(p,lcmlat.(q,r)) = lcmlat.(lcmlat.(q,p),r) & lcmlat.(p,lcmlat.(
q,r)) = lcmlat.(lcmlat.(p,r),q) & lcmlat.(p,lcmlat.(q,r)) = lcmlat.(lcmlat.(r,q
  ),p) & lcmlat.(p,lcmlat.(q,r)) = lcmlat.(lcmlat.(r,p),q)
proof
  set s=r"\/"q;
  thus lcmlat.(p,lcmlat.(q,r)) =lcmlat.(lcmlat.(p,q),r) by Th7
    .= lcmlat.(lcmlat.(q,p),r) by Th5;
  thus
A1: lcmlat.(p,lcmlat.(q,r)) = lcmlat.(p,lcmlat.(r,q)) by Th5
    .=lcmlat.(lcmlat.(p,r),q) by Th7;
  thus lcmlat.(p,lcmlat.(q,r)) = lcmlat.(p,s) by LATTICES:def 1
    .=lcmlat.(lcmlat.(r,q),p) by Th5;
  thus thesis by A1,Th5;
end;
