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

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