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 ` ) by Def10;
(a (#) f) | (E ` ) = a (#) (g | (E ` )) by A2, RFUNCT_1:65
.= (a (#) g) | (E ` ) by RFUNCT_1:65 ;
hence a (#) f a.e.= a (#) g,M by A1, Def10; :: thesis: verum