let X be non empty set ; for S being SigmaField of X
for M being sigma_Measure of S
for f, g, f1, 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; for M being sigma_Measure of S
for f, g, f1, 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; for f, g, f1, 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, g, f1, g1 be 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 k be positive Real; ( 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) )
; a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
A6:
( ex E being Element of S st
( M . (E `) = 0 & dom f1 = E & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & dom g1 = E & g1 is E -measurable ) )
by A2, A4, Th35;
( f in a.e-eq-class_Lp (f,M,k) & g in a.e-eq-class_Lp (g,M,k) )
by A1, A3, Th38;
then
( f a.e.= f1,M & g a.e.= g1,M )
by A5, A6, Th37;
hence
a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k)
by Th41, LPSPACE1:31; verum