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
(ex E be Element of S st M.(E`)=0 & E = dom f & f is E-measurable) &
(ex E be Element of S st M.(E`)=0 & E = dom f1 & f1 is E-measurable) &
(ex E be Element of S st M.(E`)=0 & E = dom g & g is E-measurable) &
(ex E be Element of S st M.(E`)=0 & E = dom g1 & g1 is E-measurable) &
a.e-eq-class_Lp(f,M,k) is non empty &
a.e-eq-class_Lp(g,M,k) is non empty &
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
 (ex E be Element of S st M.(E`)=0 & E = dom f & f is E-measurable) &
    (ex E be Element of S st M.(E`)=0 & E = dom f1 & f1 is E-measurable) &
    (ex E be Element of S st M.(E`)=0 & E = dom g & g is E-measurable) &
    (ex E be Element of S st M.(E`)=0 & E = dom g1 & g1 is E-measurable) &
    a.e-eq-class_Lp(f,M,k) is non empty &
    a.e-eq-class_Lp(g,M,k) is non empty &
    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);
   then f a.e.= f1,M & g a.e.= g1,M by Th39;
   hence thesis by Th41,LPSPACE1:31;
end;
