let X be non empty set ; :: thesis: for S being SigmaField of X

for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let f be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let A be Element of S; :: thesis: ( f is_simple_func_in S implies f | A is_simple_func_in S )

assume A1: f is_simple_func_in S ; :: thesis: f | A is_simple_func_in S

then Im f is_simple_func_in S by MESFUN7C:43;

then R_EAL (Im f) is_simple_func_in S by MESFUNC6:49;

then R_EAL ((Im f) | A) is_simple_func_in S by MESFUNC5:34;

then (Im f) | A is_simple_func_in S by MESFUNC6:49;

then A2: Im (f | A) is_simple_func_in S by MESFUN6C:7;

Re f is_simple_func_in S by A1, MESFUN7C:43;

then R_EAL (Re f) is_simple_func_in S by MESFUNC6:49;

then R_EAL ((Re f) | A) is_simple_func_in S by MESFUNC5:34;

then (Re f) | A is_simple_func_in S by MESFUNC6:49;

then Re (f | A) is_simple_func_in S by MESFUN6C:7;

hence f | A is_simple_func_in S by A2, MESFUN7C:43; :: thesis: verum

for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let f be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_simple_func_in S holds

f | A is_simple_func_in S

let A be Element of S; :: thesis: ( f is_simple_func_in S implies f | A is_simple_func_in S )

assume A1: f is_simple_func_in S ; :: thesis: f | A is_simple_func_in S

then Im f is_simple_func_in S by MESFUN7C:43;

then R_EAL (Im f) is_simple_func_in S by MESFUNC6:49;

then R_EAL ((Im f) | A) is_simple_func_in S by MESFUNC5:34;

then (Im f) | A is_simple_func_in S by MESFUNC6:49;

then A2: Im (f | A) is_simple_func_in S by MESFUN6C:7;

Re f is_simple_func_in S by A1, MESFUN7C:43;

then R_EAL (Re f) is_simple_func_in S by MESFUNC6:49;

then R_EAL ((Re f) | A) is_simple_func_in S by MESFUNC5:34;

then (Re f) | A is_simple_func_in S by MESFUNC6:49;

then Re (f | A) is_simple_func_in S by MESFUN6C:7;

hence f | A is_simple_func_in S by A2, MESFUN7C:43; :: thesis: verum