let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL st f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M & a.e-eq-class f,M = a.e-eq-class f1,M & a.e-eq-class g,M = a.e-eq-class g1,M holds
a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL st f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M & a.e-eq-class f,M = a.e-eq-class f1,M & a.e-eq-class g,M = a.e-eq-class g1,M holds
a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M
let M be sigma_Measure of S; :: thesis: for f, f1, g, g1 being PartFunc of X,REAL st f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M & a.e-eq-class f,M = a.e-eq-class f1,M & a.e-eq-class g,M = a.e-eq-class g1,M holds
a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M
let f, f1, g, g1 be PartFunc of X,REAL ; :: thesis: ( f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M & a.e-eq-class f,M = a.e-eq-class f1,M & a.e-eq-class g,M = a.e-eq-class g1,M implies a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M )
assume that
A1: ( f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M ) and
A2: ( a.e-eq-class f,M = a.e-eq-class f1,M & a.e-eq-class g,M = a.e-eq-class g1,M ) ; :: thesis: a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M
( f a.e.= f1,M & g a.e.= g1,M ) by A1, A2, Th39;
then A3: f + g a.e.= f1 + g1,M by Th31;
( f + g in L1_Functions M & f1 + g1 in L1_Functions M ) by A1, Th23;
hence a.e-eq-class (f + g),M = a.e-eq-class (f1 + g1),M by A3, Th39; :: thesis: verum