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 Th27:
 for r being Real holds
  f is_integrable_on P implies expect (r(#)f,P) = r* expect (f,P)
proof let r be Real;
  set h=r(#)f;
  assume
A1: f is_integrable_on P;
  then
A2: Integral(P2M(P),f) =expect (f,P) by Def4;
A3: f is_integrable_on P2M(P) by A1;
  then h is_integrable_on P2M(P) by MESFUNC6:102;
  then h is_integrable_on P;
  hence expect (h,P) =Integral(P2M(P),r(#)f) by Def4
    .=( r)*Integral(P2M(P),f) by A3,MESFUNC6:102
    .= r* expect (f,P) by A2,EXTREAL1:1;
end;
