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

theorem Th16:
  for Omega1, Omega2 be non empty finite set,
  P1 be Probability of Trivial-SigmaField Omega1,
  P2 be Probability of Trivial-SigmaField Omega2,
  Y1 be non empty finite Subset of Omega1,
  Y2 be non empty finite Subset of 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 be non empty finite Subset of Omega1,
    Y2 be non empty finite Subset of Omega2;
    set P=Product-Probability (Omega1,Omega2,P1,P2);
    consider Q be Function of [:Omega1,Omega2:],REAL such that
    A1: (for x,y be set st x in Omega1 & y in Omega2
    holds Q.(x,y) = (P1.{x})*(P2.{y}) ) &
    (for z be finite Subset of [:Omega1,Omega2:]
    holds P.z = setopfunc(z,[:Omega1,Omega2:],REAL,Q,addreal)) by Def2;
    thus P.([:Y1,Y2:]) = (P1.Y1)*(P2.Y2) by Lm7,A1;
  end;
