 reserve X for set;
 reserve S for Subset-Family of X;

theorem
  for X1,X2 be set, S1 be Subset-Family of X1, S2 be Subset-Family of X2
  holds {[:a,b:] where a is Element of S1, b is Element of S2:a in S1 &
  b in S2} = {s where s is Subset of [:X1,X2:] : ex a,b be set st a in S1 &
  b in S2 & s=[:a,b:]}
  proof
    let X1,X2 be set;
    let S1 be Subset-Family of X1;
    let S2 be Subset-Family of X2;
    thus {[:a,b:] where a is Element of S1, b is Element of S2:a in S1 &
    b in S2} c= {s where s is Subset of [:X1,X2:] : ex a,b be set st
    a in S1 & b in S2 & s=[:a,b:]}
    proof
      let x be object;
      assume x in {[:a,b:] where a is Element of S1, b is Element of
S2:   a in S1 & b in S2};
      then consider a be Element of S1, b be Element of S2 such that
A1:   x=[:a,b:] & a in S1 & b in S2;
      [:a,b:] c= [:X1,X2:] by A1,ZFMISC_1:96;
      hence thesis by A1;
    end;
    let x be object;
    assume x in {s where s is Subset of [:X1,X2:] : ex a,b be set st
    a in S1 & b in S2 & s=[:a,b:]};
    then ex s0 be Subset of [:X1,X2:] st x=s0 & ex a0,b0 be set st
    a0 in S1 & b0 in S2 & s0=[:a0,b0:];
    hence thesis;
  end;
