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

theorem Th6:
  minreal.(q,maxreal.(q,p))=q & minreal.(maxreal.(p,q),q)=q &
  minreal.(q,maxreal.(p,q))=q & minreal.(maxreal.(q,p),q)=q
proof
  set s=q"\/"p;
  thus
A1: minreal.(q,maxreal.(q,p)) =q"/\"s .=q by XXREAL_0:35;
  thus
A2: minreal.(maxreal.(p,q),q)=minreal.(p"\/"q,q)
    .=q"/\"(q"\/"p) by LATTICES:def 2
    .=q by XXREAL_0:35;
  thus minreal.(q,maxreal.(p,q))=q by A1,Th1;
  thus thesis by A2,Th1;
end;
