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 by Def9;
then |.(Integral M,f).| <= Integral M,|.(R_EAL f).| by MESFUNC5:107;
hence |.(Integral M,f).| <= Integral M,(abs f) by Th1; :: thesis: verum