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
for k being positive Real st f in Lp_Functions M,k & f1 in Lp_Functions M,k & g in Lp_Functions M,k & g1 in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k holds
a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions M,k & f1 in Lp_Functions M,k & g in Lp_Functions M,k & g1 in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k holds
a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k
let M be sigma_Measure of S; :: thesis: for f, f1, g, g1 being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions M,k & f1 in Lp_Functions M,k & g in Lp_Functions M,k & g1 in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k holds
a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k
let f, f1, g, g1 be PartFunc of X,REAL ; :: thesis: for k being positive Real st f in Lp_Functions M,k & f1 in Lp_Functions M,k & g in Lp_Functions M,k & g1 in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k holds
a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k
let k be positive Real; :: thesis: ( f in Lp_Functions M,k & f1 in Lp_Functions M,k & g in Lp_Functions M,k & g1 in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k implies a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k )
assume that
A1: f in Lp_Functions M,k and
A2: f1 in Lp_Functions M,k and
A3: g in Lp_Functions M,k and
A4: g1 in Lp_Functions M,k and
A5: ( a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp f1,M,k & a.e-eq-class_Lp g,M,k = a.e-eq-class_Lp g1,M,k ) ; :: thesis: a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k
A8: ( ex E being Element of S st
( M . (E ` ) = 0 & dom f1 = E & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & dom g1 = E & g1 is_measurable_on E ) ) by A2, A4, EQC00a;
( f in a.e-eq-class_Lp f,M,k & g in a.e-eq-class_Lp g,M,k ) by A1, A3, EQC01;
then ( f a.e.= f1,M & g a.e.= g1,M ) by A5, A8, EQC00c;
hence a.e-eq-class_Lp (f + g),M,k = a.e-eq-class_Lp (f1 + g1),M,k by EQC02b, LPSPACE1:31; :: thesis: verum