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

for f, g being PartFunc of X,COMPLEX

for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

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

for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

let f, g be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

let A be Element of S; :: thesis: ( f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )

assume that

A1: f is A -measurable and

A2: g is A -measurable and

A3: A c= dom g ; :: thesis: f - g is A -measurable

A4: Im g is A -measurable by A2;

A5: A c= dom (Re g) by A3, COMSEQ_3:def 3;

A6: Re g is A -measurable by A2;

A7: A c= dom (Im g) by A3, COMSEQ_3:def 4;

Im f is A -measurable by A1;

then (Im f) - (Im g) is A -measurable by A4, A7, MESFUNC6:29;

then A8: Im (f - g) is A -measurable by Th6;

Re f is A -measurable by A1;

then (Re f) - (Re g) is A -measurable by A6, A5, MESFUNC6:29;

then Re (f - g) is A -measurable by Th6;

hence f - g is A -measurable by A8; :: thesis: verum

for f, g being PartFunc of X,COMPLEX

for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

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

for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

let f, g be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds

f - g is A -measurable

let A be Element of S; :: thesis: ( f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )

assume that

A1: f is A -measurable and

A2: g is A -measurable and

A3: A c= dom g ; :: thesis: f - g is A -measurable

A4: Im g is A -measurable by A2;

A5: A c= dom (Re g) by A3, COMSEQ_3:def 3;

A6: Re g is A -measurable by A2;

A7: A c= dom (Im g) by A3, COMSEQ_3:def 4;

Im f is A -measurable by A1;

then (Im f) - (Im g) is A -measurable by A4, A7, MESFUNC6:29;

then A8: Im (f - g) is A -measurable by Th6;

Re f is A -measurable by A1;

then (Re f) - (Re g) is A -measurable by A6, A5, MESFUNC6:29;

then Re (f - g) is A -measurable by Th6;

hence f - g is A -measurable by A8; :: thesis: verum