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
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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, g, f1, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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, g, f1, g1 being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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; :: thesis: for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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: ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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 ( ex E being Element of S st
( M . (E `) = 0 & E = dom f & f is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom f1 & f1 is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & E = dom g1 & g1 is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & not a.e-eq-class_Lp (g,M,k) is empty & 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)
then ( f a.e.= f1,M & g a.e.= g1,M ) by Th39;
hence a.e-eq-class_Lp ((f + g),M,k) = a.e-eq-class_Lp ((f1 + g1),M,k) by ; :: thesis: verum