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