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.(q,lcmlat.(q,p))=q & hcflat.(lcmlat.(p,q),q)=q & hcflat.(q,
  lcmlat.(p,q))=q & hcflat.(lcmlat.(q,p),q)=q
proof
  set s=q"\/"p;
  thus
A1: hcflat.(q,lcmlat.(q,p))=q"/\"s .=q by NEWTON:54;
  thus
A2: hcflat.(lcmlat.(p,q),q)=hcflat.(p"\/"q,q) .=q"/\"(q"\/"p) by LATTICES:def 2
    .=q by NEWTON:54;
  thus hcflat.(q,lcmlat.(p,q))=q by A1,Th5;
  thus thesis by A2,Th5;
end;
