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 Th22:
for k be positive Real, f,g be PartFunc of X,REAL st
  f in Lp_Functions(M,k) & g in Lp_Functions (M,k) holds
    (abs f) to_power k is_integrable_on M &
    (abs g) to_power k is_integrable_on M &
    (abs f) to_power k + (abs g) to_power k is_integrable_on M
proof
   let k be positive Real;
   let f,g be PartFunc of X,REAL;
   assume A1: f in Lp_Functions (M,k) & g in Lp_Functions (M,k); then
A2:ex f1 be PartFunc of X,REAL st
    f=f1 & ex Ev be Element of S st M.(Ev`) = 0 & dom f1 = Ev &
     f1 is Ev-measurable & (abs f1) to_power k is_integrable_on M;
   ex g1 be PartFunc of X, REAL st
    g = g1 & ex Eu be Element of S st M.(Eu`) = 0 & dom g1 = Eu &
     g1 is Eu-measurable & (abs g1) to_power k is_integrable_on M by A1;
   hence thesis by A2,MESFUNC6:100;
end;
