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,COMPLEX
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,COMPLEX
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,COMPLEX
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,COMPLEX; :: 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 A1: f is_integrable_on M ; :: thesis: f | A is_integrable_on M
then Im f is_integrable_on M ;
then (Im f) | A is_integrable_on M by MESFUNC6:91;
then A2: Im (f | A) is_integrable_on M by Th7;
Re f is_integrable_on M by A1;
then (Re f) | A is_integrable_on M by MESFUNC6:91;
then Re (f | A) is_integrable_on M by Th7;
hence f | A is_integrable_on M by A2; :: thesis: verum