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);

theorem Th31:
f=u implies u + (-1)*u = (X --> 0)|dom f & ex v,g be PartFunc of X,REAL st
    v in Lp_Functions(M,k) & g in Lp_Functions(M,k) &
    v = u + (-1)*u & g = X --> 0 & v a.e.= g,M
proof
   reconsider u2=u as VECTOR of RLSp_PFunct X by TARSKI:def 3;
   assume A1: f=u;
   (-1)*u =(-1)*u2 by LPSPACE1:5; then
A2:u+(-1)*u =u2+(-1)*u2 by LPSPACE1:4;
   hence u+(-1)*u = (X --> 0)|dom f by A1,LPSPACE1:16;
   u+(-1)*u in Lp_Functions(M,k); then
   consider v be PartFunc of X,REAL such that
A3: v=u+(-1)*u &
    ex ND be Element of S st M.ND`= 0 & dom v = ND & v is ND-measurable &
    (abs v)to_power k is_integrable_on M;
   u in Lp_Functions(M,k); then
   ex uu1 be PartFunc of X,REAL st
    uu1=u & ex ND be Element of S st M.ND`=0 & dom uu1 = ND &
    uu1 is ND-measurable & (abs uu1) to_power k is_integrable_on M; then
   consider ND be Element of S such that
A4: M.ND`=0 & dom f = ND & f is ND-measurable &
    (abs f) to_power k is_integrable_on M by A1;
   set g = X-->0;
A5:ND` is Element of S & (ND`)`= ND by MEASURE1:34;
A6:g in Lp_Functions(M,k) by Th23;
   v|ND = g|ND|ND by A2,A3,A4,A1,LPSPACE1:16; then
   v|ND = g|ND by FUNCT_1:51; then
   v a.e.= g,M by A4,A5;
   hence thesis by A3,A6;
end;
