
theorem
  for X,Y be set, M be with_empty_element Subset-Family of X,
      N be with_empty_element Subset-Family of Y
  holds the set of all [:A,B:] where A is Element of M, B is Element of N
     is with_empty_element Subset-Family of [:X,Y:]
proof
   let X,Y be set;
   let M be with_empty_element Subset-Family of X,
       N be with_empty_element Subset-Family of Y;
   set L = the set of all [:A,B:] where A is Element of M, B is Element of N;
   reconsider E1 = {} as Element of M by SETFAM_1:def 8;
   reconsider E2 = {} as Element of N by SETFAM_1:def 8;
   {} = [:E1,E2:] by ZFMISC_1:90; then
   {} in L;
   hence thesis by Th05;
end;
