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 ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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, f1, g, 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_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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, f1, g, 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_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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, f1, g, 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_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom f1 & f1 is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & E = dom g1 & g1 is_measurable_on E ) & 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 EQC02a;
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