 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem
  for Omega1, Omega2 be non empty finite set,
  P1 be Probability of Trivial-SigmaField (Omega1),
  P2 be Probability of Trivial-SigmaField (Omega2),
  y1, y2 be set st y1 in Omega1 & y2 in Omega2 holds
  Product-Probability (Omega1,Omega2,P1,P2).({[y1,y2]})
  = (P1.{y1})*(P2.{y2})
  proof
    let Omega1, Omega2 be non empty finite set,
    P1 be Probability of Trivial-SigmaField Omega1,
    P2 be Probability of Trivial-SigmaField Omega2,
    y1, y2 be set;
    assume
A1: y1 in Omega1 & y2 in Omega2; then
A2: {y1} is finite Subset of Omega1 by ZFMISC_1:31;
    for yy be object st yy in {y2} holds yy in Omega2 by A1,TARSKI:def 1;
    then
A3: {y2} is finite Subset of Omega2 by TARSKI:def 3;
    [:{y1},{y2}:] = {[y1,y2]} by ZFMISC_1:29;
    hence Product-Probability (Omega1,Omega2,P1,P2).({[y1,y2]})
    = (P1.{y1})*(P2.{y2}) by Th16,A2,A3;
  end;
