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,ExtREAL st f is_simple_func_in S & g is_simple_func_in S holds
f - g is_simple_func_in S
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_simple_func_in S & g is_simple_func_in S holds
f - g is_simple_func_in S
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,ExtREAL st f is_simple_func_in S & g is_simple_func_in S holds
f - g is_simple_func_in S
let f, g be PartFunc of X,ExtREAL; :: thesis: ( f is_simple_func_in S & g is_simple_func_in S implies f - g is_simple_func_in S )
assume that
A1: f is_simple_func_in S and
A4: g is_simple_func_in S ; :: thesis: f - g is_simple_func_in S
(- 1) (#) g is_simple_func_in S by A4, MESFUNC5:39;
then - g is_simple_func_in S by MESFUNC2:9;
then f + (- g) is_simple_func_in S by A1, Th28;
then - (g - f) is_simple_func_in S by MEASUR11:64;
hence f - g is_simple_func_in S by MEASUR11:64; :: thesis: verum