reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem Th3: :: SRINGS_2:Lem 3
  for X1,X2 being set, S1 be Subset-Family of X1, S2 be Subset-Family of X2
  holds {s where s is Subset of [:X1,X2:]: ex s1,s2 be set st s1 in S1 &
  s2 in S2 & s = [:s1,s2:]} is Subset-Family of [:X1,X2:]
  proof
    let X1,X2 be set, S1 be Subset-Family of X1, S2 be Subset-Family of X2;
    set S = {s where s is Subset of [:X1,X2:]:
    ex s1,s2 be set st s1 in S1 & s2 in S2 & s=[:s1,s2:]};
    S c= bool [:X1,X2:]
    proof
      let x be object;
      assume x in S;
      then consider s0 be Subset of [:X1,X2:] such that
A1:   x=s0 & ex s1,s2 be set st s1 in S1 & s2 in S2 & s0=[:s1,s2:];
      thus thesis by A1;
    end;
    hence thesis;
  end;
