reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;
reserve f,g for Real-Valued-Random-Variable of Sigma;

theorem Th22:
  f(#)g is Real-Valued-Random-Variable of Sigma
proof
  set X = [#]Sigma;
A2: f is X-measurable;
A3: R_EAL f is X-measurable by A2;
  dom(R_EAL f)=X & dom(R_EAL g)=X by FUNCT_2:def 1;
  then
A4: dom(R_EAL f) /\ dom(R_EAL g)=X;
  g is ([#]Sigma)-measurable;
  then R_EAL g is X-measurable;
  then (R_EAL f)(#)(R_EAL g) is X-measurable by A3,A4,MESFUNC7:15;
  then R_EAL(f(#)g) is X-measurable by Th21;
  then f(#)g is X-measurable;
  hence thesis;
end;
