 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem Th4:
  Omega --> r is Real-Valued-Random-Variable of Sigma
  proof
    set E0 = (Omega --> 1);
    set E = (Omega --> r);
    reconsider S= Omega as Element of Sigma by MEASURE1:7;
    A1: dom E0 = Omega & rng E0 c= {1} by FUNCOP_1:13;
    reconsider E0 as Function of Omega, REAL by FUNCT_2:7,NUMBERS:19;
    A2: R_EAL E0 =chi(S,Omega) by Th3;
    chi(S,Omega) is S-measurable by MESFUNC2:29;
    then E0 is ([#]Sigma)-measurable by A2,MESFUNC6:def 1; then
    A3: E0 is Real-Valued-Random-Variable of Sigma;
    A4: dom E = dom E0 by A1,FUNCT_2:def 1;
    now let x be object;
      assume
      A5:x in dom E;
      hence E.x = r*1 by FUNCOP_1:7
      .= r*(E0.x) by A5,FUNCOP_1:7;
    end; then
    E = r(#) E0 by A4,VALUED_1:def 5;
    hence thesis by A3,RANDOM_1:20;
  end;
