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
  for y being Element of S2 st y <> {} holds
  {z where z is Element of Omega1: F.z is Element of y} = F"y
  proof
    let y be Element of S2;
    assume A1: y <> {};
    set D = {z where z is Element of Omega1: F.z is Element of y};
    for x be object holds x in D iff x in F"y
    proof
      let x be object;
      hereby assume x in D;
        then consider z be Element of Omega1 such that
        A2: x=z & F.z is Element of y;
        z in Omega1; then
        z in dom F by FUNCT_2:def 1;
        hence x in F"y by A2,FUNCT_1:def 7,A1;
      end;
      assume x in F"y;
      then x in dom F & F.x in y by FUNCT_1:def 7;
      hence x in D;
    end;
    hence thesis by TARSKI:2;
  end;
