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;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);

theorem Th45:
f in Lp_Functions(M,k) & f1 in Lp_Functions(M,k) &
g in Lp_Functions(M,k) & g1 in Lp_Functions(M,k) &
a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(f1,M,k) &
a.e-eq-class_Lp(g,M,k) = a.e-eq-class_Lp(g1,M,k)
  implies a.e-eq-class_Lp(f+g,M,k) = a.e-eq-class_Lp(f1+g1,M,k)
proof
   assume that
A1:f in Lp_Functions(M,k) and
A2:f1 in Lp_Functions(M,k) and
A3:g in Lp_Functions(M,k) and
A4:g1 in Lp_Functions(M,k) and
A5:a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(f1,M,k) &
   a.e-eq-class_Lp(g,M,k) = a.e-eq-class_Lp(g1,M,k);
A6:(ex E be Element of S st M.(E`) = 0 & dom f1 = E & f1 is E-measurable) &
   (ex E be Element of S st M.(E`) = 0 & dom g1 = E & g1 is E-measurable)
     by A2,A4,Th35;
   f in a.e-eq-class_Lp(f,M,k) &
   g in a.e-eq-class_Lp(g,M,k) by A1,A3,Th38; then
   f a.e.= f1,M & g a.e.= g1,M by A5,A6,Th37;
   hence thesis by Th41,LPSPACE1:31;
end;
