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 & 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 & 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 & 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 & 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 and
A2: f is_measurable_on A and
A3: g is_measurable_on A ; :: thesis: (max+ (f + g)) + (max- f) is_measurable_on A
f + g is_measurable_on A by A2, A3, Th26;
then A4: max+ (f + g) is_measurable_on A by Th46;
max- f is_measurable_on A by A1, A2, Th47;
hence (max+ (f + g)) + (max- f) is_measurable_on A by A4, Th26; :: thesis: verum