reserve Omega, Omega1, Omega2 for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S1 for SigmaField of Omega1;
reserve S2 for SigmaField of Omega2;
reserve F for random_variable of S1,S2;

theorem Th12:
  for P being Probability of S1,
  F being random_variable of S1,S2 holds
  (image_measure(F,P2M(P))).Omega2 = 1
  proof
    let P be Probability of S1,
    F be random_variable of S1,S2;
A1:  for y be set st y in S2 holds
      (image_measure(F,P2M(P))).y = (P2M P).(F"y) by Def6;
    thus (image_measure(F,P2M(P))).Omega2
     = (P2M(P)).(F"Omega2) by PROB_1:5,A1
    .= (P2M(P)).(Omega1) by FUNCT_2:40
    .= 1 by PROB_1:def 8;
  end;
