theorem Th28:
f in Lp_Functions(M,k) implies abs f in Lp_Functions(M,k)
proof
   set W = Lp_Functions(M,k);
   assume f in W; 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;
   dom (abs f1) = Ef by A2,VALUED_1:def 11; then
Z1: M.(Ef`) = 0 & dom (abs f1) = Ef &
   (abs f1) is Ef-measurable & (abs(abs f1)) to_power k is_integrable_on M
       by A2,MESFUNC6:48;
   thus thesis by A1,Z1;
end;
