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 Th25:
f in Lp_Functions(M,k) & g in Lp_Functions(M,k) implies
  f + g in Lp_Functions(M,k)
proof
   set W = Lp_Functions(M,k);
   assume A1: f in W & g in W;
   then consider f1 be PartFunc of X,REAL such that
A2: 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
A3: M.(Ef`) = 0 & dom f1 = Ef &
    f1 is Ef-measurable & (abs f1) to_power k is_integrable_on M by A2;
   consider g1 be PartFunc of X,REAL such that
A4: g1=g & ex Eg1 be Element of S st M.(Eg1`) = 0 & dom g1 = Eg1 &
     g1 is Eg1-measurable & (abs g1) to_power k is_integrable_on M by A1;
   consider Eg be Element of S such that
A5: M.(Eg`) = 0 & dom g1 = Eg &
    g1 is Eg-measurable & (abs g1) to_power k is_integrable_on M by A4;
A6:dom (f1+g1)= Ef /\ Eg by A3,A5,VALUED_1:def 1;
   set Efg = Ef /\ Eg;
   set s = abs(f1 + g1) to_power k;
   set t = (2 to_power k)(#)((abs f1) to_power k + (abs g1) to_power k);
A7:Efg` = (X \ Ef) \/ (X \ Eg) by XBOOLE_1:54;
   Ef` is Element of S & Eg` is Element of S by MEASURE1:34; then
   Ef` is measure_zero of M & Eg` is measure_zero of M
     by A3,A5,MEASURE1:def 7; then
   Ef` \/ Eg` is measure_zero of M by MEASURE1:37; then
A8:M.(Efg`) = 0 by A7,MEASURE1:def 7;
   f1 is Efg-measurable & g1 is Efg-measurable
     by A3,A5,MESFUNC6:16,XBOOLE_1:17; then
A9: f1 + g1 is Efg-measurable by MESFUNC6:26; then
A10: abs(f1 + g1) is Efg-measurable by A6,MESFUNC6:48;
   (abs f1) to_power k + (abs g1) to_power k is_integrable_on M
     by A1,A2,A4,Th22; then
A11: t is_integrable_on M by MESFUNC6:102;
A12:dom abs f1 = dom f1 & dom abs g1 = dom g1 &
   dom abs(f1 + g1) = dom (f1+g1) by VALUED_1:def 11; then
A13: s is Efg-measurable by A6,A10,MESFUN6C:29;
A14: abs s = (abs(f1 + g1) to_power k) by Th14;
A15: dom s = Efg by A6,A12,MESFUN6C:def 4;
A16:dom t = dom ((abs f1) to_power k + (abs g1) to_power k) by VALUED_1:def 5
    .= dom((abs f1) to_power k) /\ dom((abs g1) to_power k) by VALUED_1:def 1
    .= dom (abs f1) /\ dom ((abs g1) to_power k) by MESFUN6C:def 4
    .= dom (abs f1) /\ dom (abs g1) by MESFUN6C:def 4
    .= dom (f1+g1) by A12,VALUED_1:def 1
    .= dom s by A12,MESFUN6C:def 4;
   now let x be Element of X;
    assume x in dom s; then
    (abs s).x <= t.x by A14,Th24,A3,A5,A15;
    hence |.s.x qua Complex.| <= t.x by VALUED_1:18;
   end; then
   s is_integrable_on M by A13,A15,A16,A11,MESFUNC6:96;
   hence thesis by A2,A4,A8,A6,A9;
end;
