let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let f, g be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let A be Element of S; :: thesis: ( f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A & A c= dom g implies f - g is_measurable_on A )
assume that
A1: f is real-valued and
A2: g is real-valued and
A3: f is_measurable_on A and
A4: g is_measurable_on A and
A5: A c= dom g ; :: thesis: f - g is_measurable_on A
( (- 1) (#) g is real-valued & (- 1) (#) g is_measurable_on A ) by A2, A4, A5, Th12, MESFUNC1:41;
then ( - g is real-valued & - g is_measurable_on A ) by Th11;
then f + (- g) is_measurable_on A by A1, A3, Th7;
hence f - g is_measurable_on A by Th9; :: thesis: verum