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;

theorem
  for f being Function of Omega,REAL holds
  set_of_random_variables_on (Sigma,Borel_Sets) =
  Real-Valued-Random-Variables-Set Sigma
  proof
    let f be Function of Omega,REAL;
    for x be object holds x in set_of_random_variables_on (Sigma,Borel_Sets)
    iff x in Real-Valued-Random-Variables-Set Sigma
    proof
      let x be object;
      hereby assume x in set_of_random_variables_on (Sigma,Borel_Sets);
        then consider f being Function of Omega,REAL such that
        A1: x=f & f is (Sigma,Borel_Sets)-random_variable-like;
        f is (Sigma,Borel_Sets)-random_variable-like implies
        f is Real-Valued-Random-Variable of Sigma by Th7;
        hence x in Real-Valued-Random-Variables-Set Sigma by A1;
      end;
      assume x in Real-Valued-Random-Variables-Set Sigma; then
      consider q be Real-Valued-Random-Variable of Sigma such that
      A2: x=q;
      q is (Sigma,Borel_Sets)-random_variable-like by Th7;
      hence x in set_of_random_variables_on (Sigma,Borel_Sets) by A2;
    end;
    hence thesis by TARSKI:2;
  end;
