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 Th7:
  for f being Function of Omega,REAL holds
  f is (Sigma,Borel_Sets)-random_variable-like iff
  f is Real-Valued-Random-Variable of Sigma
  proof
    let f be Function of Omega,REAL;
    thus f is (Sigma,Borel_Sets)-random_variable-like implies
    f is Real-Valued-Random-Variable of Sigma by FINANCE1:15;
    assume
    A1: f is Real-Valued-Random-Variable of Sigma;
    set B = { x where x is Element of Borel_Sets : f"x is Element of Sigma };
    A2:B= Borel_Sets by Th6,A1;
    for x being set st x in Borel_Sets holds f"x in Sigma
    proof
      let x be set;
      assume x in Borel_Sets; then
      ex z be Element of Borel_Sets st z=x & f"z is Element of Sigma by A2;
      hence thesis;
    end;
    hence thesis;
  end;
