let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g, f1, 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, g, f1, 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, g, f1, 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, g, f1, 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