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 Th50:
(ex x be VECTOR of Pre-Lp-Space(M,k) st f in x & g in x) implies
  f a.e.= g,M & f in Lp_Functions(M,k) & g in Lp_Functions(M,k)
proof
   assume ex x be VECTOR of Pre-Lp-Space(M,k) st f in x & g in x; then
   consider x be VECTOR of Pre-Lp-Space(M,k) such that
A1: f in x & g in x;
   x in the carrier of Pre-Lp-Space(M,k); then
   x in CosetSet(M,k) by Def11; then
   consider h be PartFunc of X,REAL such that
A2: x=a.e-eq-class_Lp(h,M,k) & h in Lp_Functions(M,k);
   (ex i be PartFunc of X,REAL st f=i & i in Lp_Functions(M,k) & h a.e.= i,M) &
   (ex j be PartFunc of X,REAL st g=j & j in Lp_Functions(M,k) & h a.e.= j,M)
     by A1,A2; then
   f a.e.= h,M & h a.e.= g,M;
   hence thesis by A1,A2,LPSPACE1:30;
end;
