let X be non empty set ; :: thesis: for S being SigmaField of X
for A being Element of S
for f being V174() PartFunc of X,ExtREAL
for g being V175() PartFunc of X,ExtREAL st f is_measurable_on A & g is_measurable_on A & A c= dom (f - g) holds
f - g is_measurable_on A
let S be SigmaField of X; :: thesis: for A being Element of S
for f being V174() PartFunc of X,ExtREAL
for g being V175() PartFunc of X,ExtREAL st f is_measurable_on A & g is_measurable_on A & A c= dom (f - g) holds
f - g is_measurable_on A
let A be Element of S; :: thesis: for f being V174() PartFunc of X,ExtREAL
for g being V175() PartFunc of X,ExtREAL st f is_measurable_on A & g is_measurable_on A & A c= dom (f - g) holds
f - g is_measurable_on A
let f be V174() PartFunc of X,ExtREAL; :: thesis: for g being V175() PartFunc of X,ExtREAL st f is_measurable_on A & g is_measurable_on A & A c= dom (f - g) holds
f - g is_measurable_on A
let g be V175() PartFunc of X,ExtREAL; :: thesis: ( f is_measurable_on A & g is_measurable_on A & A c= dom (f - g) implies f - g is_measurable_on A )
assume that
A1: f is_measurable_on A and
A2: g is_measurable_on A and
A3: A c= dom (f - g) ; :: thesis: f - g is_measurable_on A
A4: dom (f - g) = (dom f) /\ (dom g) by MESFUNC5:17;
dom ((- f) + g) = dom (- (f - g)) by Th60;
then A5: dom ((- f) + g) = dom (f - g) by MESFUNC1:def 7;
( (dom f) /\ (dom g) c= dom f & (dom f) /\ (dom g) c= dom g ) by XBOOLE_1:17;
then ( A c= dom f & A c= dom g ) by A3, A4;
then - f is_measurable_on A by A1, Th59;
then A6: (- f) + g is_measurable_on A by A2, A3, A5, Th61;
( dom f = dom (- f) & dom g = dom (- g) ) by MESFUNC1:def 7;
then dom ((- f) + g) = (dom f) /\ (dom g) by MESFUNC9:1;
then dom ((- f) + g) = dom (f - g) by MESFUNC5:17;
then - ((- f) + g) is_measurable_on A by A3, A6, Th59;
hence f - g is_measurable_on A by Th60; :: thesis: verum