
theorem
  for X,Y being non empty set
  for P being a_partition of X, Q being a_partition of Y holds
  the set of all [:p,q:] where p is Element of P, q is Element of Q
  is a_partition of [:X,Y:]
proof
  let X,Y be non empty set;
  let P be a_partition of X, Q be a_partition of Y;
  set PQ = the set of all [:p,q:] where p is Element of P, q is Element of Q;
  PQ c= bool [:X,Y:]
  proof
    let x be object;
    assume x in PQ;
    then ex p being Element of P, q being Element of Q st ( x = [:p,q :]);
    hence thesis;
  end;
  then reconsider PQ as Subset-Family of [:X,Y:];
  PQ is a_partition of [:X,Y:]
  proof
    thus union PQ = [:X,Y:]
    proof
      let x,y be object;
      thus [x,y] in union PQ implies [x,y] in [:X,Y:];
      assume
A1:   [x,y] in [:X,Y:];
      then
A2:   x in X by ZFMISC_1:87;
A3:   y in Y by A1,ZFMISC_1:87;
      X = union P by EQREL_1:def 4;
      then consider p being set such that
A4:   x in p and
A5:   p in P by A2,TARSKI:def 4;
      Y = union Q by EQREL_1:def 4;
      then consider q being set such that
A6:   y in q and
A7:   q in Q by A3,TARSKI:def 4;
A8:   [x,y] in [:p,q:] by A4,A6,ZFMISC_1:87;
      [:p,q:] in PQ by A5,A7;
      hence thesis by A8,TARSKI:def 4;
    end;
    let A be Subset of [:X,Y:];
    assume A in PQ;
    then consider p being Element of P, q being Element of Q such that
A9: A = [:p,q:];
    thus A <> {} by A9;
    let B be Subset of [:X,Y:];
    assume B in PQ;
    then consider p1 being Element of P, q1 being Element of Q such that
A10: B = [:p1,q1:];
    assume A <> B;
    then p <> p1 or q <> q1 by A9,A10;
    then p misses p1 or q misses q1 by EQREL_1:def 4;
    then p /\ p1 = {} or q /\ q1 = {};
    then A /\ B = [:{}, q /\ q1:] or A /\ B = [:p /\
    p1,{}:] by A9,A10,ZFMISC_1:100;
    then A /\ B = {};
    hence thesis;
  end;
  hence thesis;
end;
