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
  f is_integrable_on P & g is_integrable_on P implies expect (f-g,P) =
  expect (f,P) - expect (g,P)
proof
  assume that
A1: f is_integrable_on P and
A2: g is_integrable_on P;
  g is_integrable_on P2M(P) by A2;
  then (-1)(#)g is_integrable_on P2M(P) by MESFUNC6:102;
  then (-jj )(#)g is_integrable_on P;
  hence expect (f-g,P) = expect (f,P) + expect ((-jj)(#)g,P) by A1,Th26
    .= expect (f,P) + (-1)*expect (g,P) by A2,Th27
    .= expect (f,P) - expect (g,P);
end;
