let X be non empty set ; :: thesis: for S being SigmaField of X

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let f, g be PartFunc of X,REAL; :: thesis: for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let a be Real; :: thesis: for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let k be positive Real; :: thesis: ( ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) )

assume ( ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ) ; :: thesis: a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

then a (#) f a.e.= a (#) g,M by Th39, LPSPACE1:32;

hence a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) by Th41; :: thesis: verum

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let f, g be PartFunc of X,REAL; :: thesis: for a being Real

for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let a be Real; :: thesis: for k being positive Real st ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds

a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

let k be positive Real; :: thesis: ( ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) )

assume ( ex E being Element of S st

( M . (E `) = 0 & dom f = E & f is E -measurable ) & ex E being Element of S st

( M . (E `) = 0 & dom g = E & g is E -measurable ) & not a.e-eq-class_Lp (f,M,k) is empty & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) ) ; :: thesis: a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k)

then a (#) f a.e.= a (#) g,M by Th39, LPSPACE1:32;

hence a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) by Th41; :: thesis: verum