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.(q,hcflat.(q,p))=q & lcmlat.(hcflat.(p,q),q)=q & lcmlat.(q,
  hcflat.(p,q))=q & lcmlat.(hcflat.(q,p),q)=q
proof
  set r=p"/\"q;
  thus
A1: lcmlat.(q,hcflat.(q,p))=lcmlat.(q,q"/\"p) .=(p"/\"q)"\/"q by LATTICES:def 1
    .=q by NEWTON:53;
  thus
A2: lcmlat.(hcflat.(p,q),q)=r"\/"q .=q by NEWTON:53;
  thus lcmlat.(q,hcflat.(p,q))=q by A1,Th6;
  thus thesis by A2,Th6;
end;
