
theorem
  for X1,X2 being set, A being Subset-Family of [:X1,X2:]
  for A1 being non empty Subset-Family of X1
  for A2 being non empty Subset-Family of X2
  st A = {[:a,b:] where a is Subset of X1, b is Subset of X2: a in A1 & b in
  A2} holds Intersect A = [:Intersect A1, Intersect A2:]
proof
  let X1,X2 be set, A be Subset-Family of [:X1,X2:];
  let A1 be non empty Subset-Family of X1,
  A2 be non empty Subset-Family of X2 such that
A1: A = {[:a,b:] where a is Subset of X1, b is Subset of X2: a in A1 & b in
  A2};
  hereby
    let x be object;
    assume
A2: x in Intersect A;
    then consider x1,x2 being object such that
A3: x1 in X1 and
A4: x2 in X2 and
A5: [x1,x2] = x by ZFMISC_1:def 2;
    set a1 = the Element of A1,a2 = the Element of A2;
    reconsider a1 as Subset of X1;
    reconsider a2 as Subset of X2;
    now
      let a be set;
      assume a in A1;
      then [:a,a2:] in A by A1;
      then x in [:a,a2:] by A2,SETFAM_1:43;
      hence x1 in a by A5,ZFMISC_1:87;
    end;
    then
A6: x1 in Intersect A1 by A3,SETFAM_1:43;
    now
      let a be set;
      assume a in A2;
      then [:a1,a:] in A by A1;
      then x in [:a1,a:] by A2,SETFAM_1:43;
      hence x2 in a by A5,ZFMISC_1:87;
    end;
    then x2 in Intersect A2 by A4,SETFAM_1:43;
    hence x in [:Intersect A1, Intersect A2:] by A5,A6,ZFMISC_1:87;
  end;
  let x be object;
  assume
A7: x in [:Intersect A1, Intersect A2:];
  then consider x1,x2 being object such that
A8: x1 in Intersect A1 and
A9: x2 in Intersect A2 and
A10: [x1,x2] = x by ZFMISC_1:def 2;
  now
    let c be set;
    assume c in A;
    then consider a being Subset of X1, b being Subset of X2 such that
A11: c = [:a,b:] and
A12: a in A1 and
A13: b in A2 by A1;
A14: x1 in a by A8,A12,SETFAM_1:43;
    x2 in b by A9,A13,SETFAM_1:43;
    hence x in c by A10,A11,A14,ZFMISC_1:87;
  end;
  hence thesis by A7,SETFAM_1:43;
end;
