 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_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,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-CSpace M;

theorem Th42:
  f in x & g in x implies f a.e.cpfunc= 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_CFunctions M by A2,Th39;
  f a.e.cpfunc= g,M & f in L1_CFunctions M by A1,A2,Th39;
  hence thesis by A3,Th36,Th38;
end;
