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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() and
A2: g is V82() 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 V82() by A2, Th10;
A6: (- 1) (#) g is_measurable_on A by A4, MESFUNC1:37;
A7: - g is V82() by A5, Th9;
- g is_measurable_on A by A6, Th9;
then f + (- g) is_measurable_on A by A1, A3, A7, Th7;
hence f - g is_measurable_on A by Th8; :: thesis: verum