
theorem Th05:
  for X1,X2 be set, S1 be non empty Subset-Family of X1,
      S2 be non empty Subset-Family of X2
  holds the set of all [:a,b:] where a is Element of S1, b is Element of S2
    is non empty Subset-Family of [:X1,X2:]
proof
   let X1,X2 be set;
   let S1 be non empty Subset-Family of X1;
   let S2 be non empty Subset-Family of X2;
   set L = the set of all [:a,b:] where a is Element of S1, b is Element of S2;
A1:L = {s where s is Subset of [:X1,X2:]: ex x1,x2 be set
         st x1 in S1 & x2 in S2 & s=[:x1,x2:]} by SRINGS_2:2;
   consider a be object such that
A2: a in S1 by XBOOLE_0:def 1;
   consider b be object such that
A3: b in S2 by XBOOLE_0:def 1;
   reconsider a as Element of S1 by A2;
   reconsider b as Element of S2 by A3;
A4:[:a,b:] in L;
   now let z be object;
    assume z in L; then
    ex s be Subset of [:X1,X2:] st
     z = s & ex x1,x2 be set st x1 in S1 & x2 in S2 & s = [:x1,x2:] by A1;
    hence z in bool [:X1,X2:];
   end; then
   L c= bool [:X1,X2:];
   hence thesis by A4;
end;
