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 & A c= dom g ) ; :: thesis: f - g is_measurable_on A
A5: (- 1) (#) g is real-valued by A2, Th12;
A6: (- 1) (#) g is_measurable_on A by A4, MESFUNC1:41;
A7: - g is real-valued by A5, Th11;
A8: - g is_measurable_on A by A6, Th11;
A9: f + (- g) is_measurable_on A by A1, A3, A7, A8, Th7;
thus f - g is_measurable_on A by A9, Th9; :: thesis: verum