let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,REAL
for A being Element of S st A c= (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A holds
(max- (f + g)) + (max+ f) is_measurable_on A
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,REAL
for A being Element of S st A c= (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A holds
(max- (f + g)) + (max+ f) is_measurable_on A
let f, g be PartFunc of X,REAL; :: thesis: for A being Element of S st A c= (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A holds
(max- (f + g)) + (max+ f) is_measurable_on A
let A be Element of S; :: thesis: ( A c= (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A implies (max- (f + g)) + (max+ f) is_measurable_on A )
assume that
A1: A c= (dom f) /\ (dom g) and
A2: f is_measurable_on A and
A3: g is_measurable_on A ; :: thesis: (max- (f + g)) + (max+ f) is_measurable_on A
A4: max+ f is_measurable_on A by A2, Th46;
A5: dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def 1;
f + g is_measurable_on A by A2, A3, Th26;
then max- (f + g) is_measurable_on A by A1, A5, Th47;
hence (max- (f + g)) + (max+ f) is_measurable_on A by A4, Th26; :: thesis: verum