
theorem 1A5:
  for Omega, Omega2 being non empty set,
      Sigma being SigmaField of Omega,
      Sigma2 being SigmaField of Omega2,
      s being Element of Omega2 holds
   Omega --> s is (Sigma,Sigma2)-random_variable-like
 proof
  let Omega, Omega2 be non empty set;
  let Sigma be SigmaField of Omega;
  let Sigma2 be SigmaField of Omega2;
  let s be Element of Omega2;
  set X = Omega --> s;
   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;
    hence thesis;
    end; then
    {w where w is Element of Omega: X.w is Element of x}=Omega by TARSKI:2;
    then X"x = Omega by A1,Lm1B; then
    X"x is Element of Sigma by PROB_1:23;
    hence thesis;
   end;
   suppose
    A3: not s in x;
    X"x c= {}
    proof
     let q be object;
     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;
    end; then
    X"x = {}; then
    X"x is Element of Sigma by PROB_1:22;
    hence thesis;
   end;
 end;
