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 in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) holds
a.e-eq-class ((a (#) f),M) = a.e-eq-class ((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 in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) holds
a.e-eq-class ((a (#) f),M) = a.e-eq-class ((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 in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) holds
a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M)
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) holds
a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M)
let f, g be PartFunc of X,REAL; :: thesis: ( f in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) implies a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M) )
assume that
A1: ( f in L1_Functions M & g in L1_Functions M ) and
A2: a.e-eq-class (f,M) = a.e-eq-class (g,M) ; :: thesis: a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M)
f a.e.= g,M by A1, A2, Th39;
then A3: a (#) f a.e.= a (#) g,M by Th32;
( a (#) f in L1_Functions M & a (#) g in L1_Functions M ) by A1, Th24;
hence a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M) by A3, Th39; :: thesis: verum