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

theorem Th3:
  maxreal.(p,(maxreal.(q,r)))=maxreal.((maxreal.(q,r)),p) &
maxreal.(p,(maxreal.(q,r)))=maxreal.((maxreal.(p,q)),r) & maxreal.(p,(maxreal.(
  q,r)))=maxreal.((maxreal.(q,p)),r) & maxreal.(p,(maxreal.(q,r)))=maxreal.((
  maxreal.(r,p)),q) & maxreal.(p,(maxreal.(q,r)))=maxreal.((maxreal.(r,q)),p) &
  maxreal.(p,(maxreal.(q,r)))=maxreal.((maxreal.(p,r)),q)
proof
  set s=q"\/"r;
  thus
A1: maxreal.(p,(maxreal.(q,r))) = s"\/"p by LATTICES:def 1
    .= maxreal.((maxreal.(q,r)),p);
  thus maxreal.(p,(maxreal.(q,r))) = p"\/"s .= (p"\/"q)"\/"r by XXREAL_0:34
    .= maxreal.(maxreal.(p,q),r);
  thus maxreal.(p,(maxreal.(q,r))) = p"\/"s .= (q"\/"p)"\/"r by XXREAL_0:34
    .= maxreal.(maxreal.(q,p),r);
  thus
A2: maxreal.(p,(maxreal.(q,r))) = p"\/"(q"\/"r) .= (r"\/"p)"\/"q by XXREAL_0:34
    .= maxreal.(maxreal.(r,p),q);
  thus maxreal.(p,(maxreal.(q,r)))=maxreal.((maxreal.(r,q)),p) by A1,Th1;
  thus thesis by A2,Th1;
end;
