reserve x,y for Real;
reserve a,b,c for Element of Real_Lattice;
reserve p,q,r for Element of Real_Lattice;

theorem Th5:
  maxreal.(minreal.(p,q),q)=q & maxreal.(q,minreal.(p,q))=q &
  maxreal.(q,minreal.(q,p))=q & maxreal.(minreal.(q,p),q)=q
proof
  set s=p"/\"q;
  thus
A1: maxreal.(minreal.(p,q),q) =s"\/"q .=q by XXREAL_0:36;
  thus maxreal.(q,minreal.(p,q)) =(p"/\"q)"\/"q by LATTICES:def 1
    .=q by XXREAL_0:36;
  thus maxreal.(q,minreal.(q,p))=maxreal.(q,q"/\"p)
    .=(p"/\"q)"\/"q by LATTICES:def 1
    .=q by XXREAL_0:36;
  thus thesis by A1,Th2;
end;
