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
for A being Element of S st f is_integrable_on M holds
f | A is_integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for A being Element of S st f is_integrable_on M holds
f | A is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for A being Element of S st f is_integrable_on M holds
f | A is_integrable_on M
let f be PartFunc of X,REAL; :: thesis: for A being Element of S st f is_integrable_on M holds
f | A is_integrable_on M
let A be Element of S; :: thesis: ( f is_integrable_on M implies f | A is_integrable_on M )
assume f is_integrable_on M ; :: thesis: f | A is_integrable_on M
then R_EAL f is_integrable_on M by Def9;
then R_EAL (f | A) is_integrable_on M by MESFUNC5:103;
hence f | A is_integrable_on M by Def9; :: thesis: verum