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;
reserve x,y for Point of L-1-Space M;

theorem Th54:
  Integral(M,abs(X-->0)) = 0
proof
  set f=X --> 0;
A1: now
    let x be Element of X;
    f.x = 0 by FUNCOP_1:7;
    then
A2: |.f.x qua Complex.| = 0 by ABSVALUE:2;
    assume x in dom abs f;
    hence (abs f).x = 0 by A2,VALUED_1:def 11;
  end;
  dom f = X by FUNCOP_1:13;
  then dom abs f = X by VALUED_1:def 11;
  hence thesis by A1,Lm2;
end;
