let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,(abs f))
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,(abs f))
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,(abs f))
let f be PartFunc of X,REAL; :: thesis: ( f is_integrable_on M implies |.(Integral (M,f)).| <= Integral (M,(abs f)) )
assume f is_integrable_on M ; :: thesis: |.(Integral (M,f)).| <= Integral (M,(abs f))
then R_EAL f is_integrable_on M ;
then |.(Integral (M,f)).| <= Integral (M,|.(R_EAL f).|) by MESFUNC5:101;
hence |.(Integral (M,f)).| <= Integral (M,(abs f)) by Th1; :: thesis: verum