let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )

let M be sigma_Measure of S; :: thesis: for k being positive Real holds
( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )

let k be positive Real; :: thesis: ( Lp_Functions (M,k) is add-closed & Lp_Functions (M,k) is multi-closed )
set W = Lp_Functions (M,k);
now :: thesis: for v, u being Element of the carrier of () st v in Lp_Functions (M,k) & u in Lp_Functions (M,k) holds
v + u in Lp_Functions (M,k)
let v, u be Element of the carrier of (); :: thesis: ( v in Lp_Functions (M,k) & u in Lp_Functions (M,k) implies v + u in Lp_Functions (M,k) )
assume A1: ( v in Lp_Functions (M,k) & u in Lp_Functions (M,k) ) ; :: thesis: v + u in Lp_Functions (M,k)
then consider v1 being PartFunc of X,REAL such that
A2: ( v1 = v & ex ND being Element of S st
( M . (ND `) = 0 & dom v1 = ND & v1 is ND -measurable & (abs v1) to_power k is_integrable_on M ) ) ;
consider u1 being PartFunc of X,REAL such that
A3: ( u1 = u & ex ND being Element of S st
( M . (ND `) = 0 & dom u1 = ND & u1 is ND -measurable & (abs u1) to_power k is_integrable_on M ) ) by A1;
reconsider h = v + u as Element of PFuncs (X,REAL) ;
( dom h = (dom v1) /\ (dom u1) & ( for x being object st x in dom h holds
h . x = (v1 . x) + (u1 . x) ) ) by ;
then v + u = v1 + u1 by VALUED_1:def 1;
hence v + u in Lp_Functions (M,k) by A1, A2, A3, Th25; :: thesis: verum
end;
hence Lp_Functions (M,k) is add-closed by IDEAL_1:def 1; :: thesis: Lp_Functions (M,k) is multi-closed
now :: thesis: for a being Real
for u being VECTOR of () st u in Lp_Functions (M,k) holds
a * u in Lp_Functions (M,k)
let a be Real; :: thesis: for u being VECTOR of () st u in Lp_Functions (M,k) holds
a * u in Lp_Functions (M,k)

let u be VECTOR of (); :: thesis: ( u in Lp_Functions (M,k) implies a * u in Lp_Functions (M,k) )
assume A4: u in Lp_Functions (M,k) ; :: thesis: a * u in Lp_Functions (M,k)
then consider u1 being PartFunc of X,REAL such that
A5: ( u1 = u & ex ND being Element of S st
( M . (ND `) = 0 & dom u1 = ND & u1 is ND -measurable & (abs u1) to_power k is_integrable_on M ) ) ;
reconsider h = a * u as Element of PFuncs (X,REAL) ;
A6: ( dom h = dom u1 & ( for x being Element of X st x in dom u1 holds
h . x = a * (u1 . x) ) ) by ;
then for x being object st x in dom h holds
h . x = a * (u1 . x) ;
then a * u = a (#) u1 by ;
hence a * u in Lp_Functions (M,k) by Th26, A4, A5; :: thesis: verum
end;
hence Lp_Functions (M,k) is multi-closed ; :: thesis: verum