reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-Space M;

theorem Th48:
  f in x & g in x implies f a.e.= g,M & Integral(M,f) = Integral(M
  ,g) & Integral(M,abs f) = Integral(M,abs g)
proof
  assume that
A1: f in x and
A2: g in x;
A3: g in L1_Functions M by A2,Th46;
  f a.e.= g,M & f in L1_Functions M by A1,A2,Th46;
  hence thesis by A3,Th43,Th45;
end;
