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
  for X be Real-Valued-Random-Variable of Sigma st 0 < r &
  X is nonnegative & X is_integrable_on P holds
  P.({t where t is Element of Omega : r <= X.t }) <= expect (X,P)/r
proof
  let X be Real-Valued-Random-Variable of Sigma;
  assume that
A1: 0 < r and
A2: X is nonnegative and
A3: X is_integrable_on P;
  set PM=P2M(P);
  set K={t where t is Element of Omega : r <= X.t };
  set S = [#]Sigma;
  now
    let t be object;
    assume t in S /\ great_eq_dom(X,r);
    then t in great_eq_dom(X,r) by XBOOLE_0:def 4;
    then t in dom X & r <= X.t by MESFUNC1:def 14;
    hence t in K;
  end;
  then
A6: S /\ great_eq_dom(X,r) c= K;
A7: dom X = S by FUNCT_2:def 1;
  now
    let x be object;
    assume x in K;
    then
A8: ex t be Element of Omega st x=t & r <= X.t;
    then x in great_eq_dom(X,r) by A7,MESFUNC1:def 14;
    hence x in S /\ great_eq_dom(X,r) by A8,XBOOLE_0:def 4;
  end;
  then K c= S /\ great_eq_dom(X,r);
  then S /\ great_eq_dom(X,r) = K by A6;
  then reconsider K as Element of Sigma by A7,MESFUNC6:13;
  Integral(PM,X|K) <= Integral(PM,X|S) by A2,A7,MESFUNC6:87;
  then
A9: Integral(PM,X|K) <= Integral(PM,X);
  expect (X,P)=Integral(PM,X) by A3,Def4;
  then reconsider IX=Integral(PM,X) as Element of REAL by XREAL_0:def 1;
  reconsider PMK=PM.K as Element of REAL by XXREAL_0:14;
A10: for t be Element of Omega st t in K holds r <= X.t
  proof
    let t be Element of Omega;
    assume t in K;
    then ex s be Element of Omega st s=t & r <= X.s;
    hence thesis;
  end;
A11: jj in REAL by XREAL_0:def 1;
  PM.K <= jj by PROB_1:35;
  then
A12: PM.K < +infty by XXREAL_0:2,9,A11;
  X is_integrable_on P2M(P) by A3;
  then (r)*(PM.K) <= Integral(PM,X|K) by A7,A12,A10,Th2;
  then r*PMK <= Integral(PM,X|K) by EXTREAL1:1;
  then r*PMK <= Integral(PM,X) by A9,XXREAL_0:2;
  then (r*PMK)/r <= IX /r by A1,XREAL_1:72;
  then PMK <= IX /r by A1,XCMPLX_1:89;
  hence thesis by A3,Def4;
end;
