 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;

theorem ThProdPoset01:
  for P,Q being non empty RelStr,
      x being object holds
    x is Element of [:P,Q:]
     iff ex p being Element of P,q being Element of Q st x = [p,q]
  proof
    let P,Q be non empty RelStr;
    let x be object;
    x in the carrier of [:P,Q:]
       iff x in [:the carrier of P, the carrier of Q:] by YELLOW_3:def 2;
    then x in the carrier of [:P,Q:] iff
    ex p,q being object
       st p in the carrier of P & q in the carrier of Q & x = [p,q]
                     by ZFMISC_1:def 2;
    hence thesis;
  end;
