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 Th26:
  f is_integrable_on P & g is_integrable_on P implies expect (f+g,
  P) = expect (f,P) + expect (g,P)
proof
  set h=f+g;
  assume
A1: f is_integrable_on P & g is_integrable_on P;
  then
A2: Integral(P2M(P),f) =expect (f,P) & Integral(P2M(P),g) =expect (g,P) by Def4
;
A3: f is_integrable_on P2M(P) & g is_integrable_on P2M(P) by A1;
  then consider E be Element of Sigma such that
A4: E = dom f /\ dom g and
A5: Integral(P2M(P),f+g) =Integral(P2M(P),f|E)+Integral(P2M(P),g|E) by
MESFUNC6:101;
A6: dom f = Omega & dom g=Omega by FUNCT_2:def 1;
  h is_integrable_on P2M(P) by A3,MESFUNC6:100;
  then h is_integrable_on P;
  hence expect (h,P) = Integral(P2M(P),f+g) by Def4
    .=Integral(P2M(P),f) +Integral(P2M(P),g) by A4,A5,A6
    .= expect (f,P)+expect (g,P) by A2,SUPINF_2:1;
end;
