let a be Real; :: thesis: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f a.e.= g,M holds
a (#) f a.e.= a (#) g,M
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 a.e.= g,M holds
a (#) f a.e.= a (#) g,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 a.e.= g,M holds
a (#) f a.e.= a (#) g,M
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f a.e.= g,M holds
a (#) f a.e.= a (#) g,M
let f, g be PartFunc of X,REAL; :: thesis: ( f a.e.= g,M implies a (#) f a.e.= a (#) g,M )
assume f a.e.= g,M ; :: thesis: a (#) f a.e.= a (#) g,M
then consider E being Element of S such that
A1: M . E = 0 and
A2: f | (E `) = g | (E `) ;
(a (#) f) | (E `) = a (#) (g | (E `)) by A2, RFUNCT_1:49
.= (a (#) g) | (E `) by RFUNCT_1:49 ;
hence a (#) f a.e.= a (#) g,M by A1; :: thesis: verum