reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;

theorem
  for f being Function
  for s, t being Element of product f, A be set holds
  s +* t|A is Element of product f
proof
  let f be Function;
  let s, t be Element of product f, A be set;
  per cases;
  suppose f is non-empty;
    then product f <> {};
    then
A1: t in product f;
    product f c= sproduct f by Th51;
    hence thesis by A1,Th53,Th65;
  end;
  suppose f is not non-empty;
    then {} in rng f;
    then
A2: product f = {} by Th26;
    t = {} by A2,SUBSET_1:def 1;
    then t|A = {};
    hence thesis;
  end;
end;
