let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
f - g is_integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
f - g is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
f - g is_integrable_on M
let f, g be PartFunc of X,REAL; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies f - g is_integrable_on M )
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M ; :: thesis: f - g is_integrable_on M
R_EAL g is_integrable_on M by A2, MESFUNC6:def 4;
then (- 1) (#) (R_EAL g) is_integrable_on M by MESFUNC5:110;
then - (R_EAL g) is_integrable_on M by MESFUNC2:9;
then A3: R_EAL (- g) is_integrable_on M by MESFUNC6:28;
R_EAL f is_integrable_on M by A1, MESFUNC6:def 4;
then (R_EAL f) + (R_EAL (- g)) is_integrable_on M by A3, MESFUNC5:108;
then R_EAL (f - g) is_integrable_on M by MESFUNC6:23;
hence f - g is_integrable_on M by MESFUNC6:def 4; :: thesis: verum