 reserve Omega, Omega2 for non empty set;
 reserve Sigma, F for SigmaField of Omega;
 reserve Sigma2, F2 for SigmaField of Omega2;

theorem
  for s being Real, f being Function of Omega, REAL st f = Omega --> s holds
    f is (Sigma,Borel_Sets)-random_variable-like
proof
 let s be Real;
 let X be Function of Omega, REAL;
 assume A0: X = Omega --> s;
 X is (Sigma,Borel_Sets)-random_variable-like
 proof
   let x be set;
   per cases;
   suppose
A1: s in x;
 for q being object holds
    q in {w where w is Element of Omega: X.w is Element of x} iff q in Omega
 proof
  let q be object;
  hereby assume
    q in {w where w is Element of Omega: X.w is Element of x}; then
    ex w being Element of Omega st w=q & X.w is Element of x;
    hence q in Omega;
  end;
  assume q in Omega;
  then reconsider w = q as Element of Omega;
  X.w is Element of x by A1,FUNCOP_1:7,A0;
  hence thesis;
 end; then
A2: {w where w is Element of Omega: X.w is Element of x}=Omega by TARSKI:2;
  x is non empty by A1; then
  {w where w is Element of Omega: X.w is Element of x}=X"x by Lm1; then
  X"x = Omega by A2; then
  X"x is Element of Sigma by PROB_1:23;
  hence thesis;
   end;
   suppose
A3: not s in x;
  for q being object holds q in X"x iff q in {}
 proof
  let q be object;
  now assume q in X"x; then
    q in dom X & X.q in x by FUNCT_1:def 7;
    hence q in {} by A3,FUNCOP_1:7,A0;
  end;
  hence thesis;
 end; then
 X"x = {} by TARSKI:2; then
 X"x is Element of Sigma by PROB_1:22;
 hence thesis;
 end;
  end;
  hence thesis;
 end;
