reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th26:
f in Lp_Functions(M,k) implies a(#)f in Lp_Functions(M,k)
proof
   assume f in Lp_Functions(M,k); then
   consider f1 be PartFunc of X,REAL such that
A1: f1=f & ex Ef1 be Element of S st M.(Ef1`) = 0 & dom f1 = Ef1 &
    f1 is Ef1-measurable & (abs f1) to_power k is_integrable_on M;
   consider Ef be Element of S such that
A2: M.(Ef`) = 0 & dom f1 = Ef &
    f1 is Ef-measurable & (abs f1) to_power k is_integrable_on M by A1;
A3:dom(a(#)f1) = Ef & a(#)f1 is Ef-measurable
     by A2,MESFUNC6:21,VALUED_1:def 5;
   (|.a qua Complex.| to_power k)(#)((abs f1) to_power k) is_integrable_on M
     by A1,MESFUNC6:102; then
   (abs(a(#)f1)) to_power k is_integrable_on M by Th18;
   hence thesis by A1,A2,A3;
end;
