reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;
reserve E1,E2 for Element of S;
reserve x,A for set;
reserve a,b for Real;

theorem Th23:
  max+(chi(A,X)) = chi(A,X)
proof
A1: dom max+(chi(A,X)) = dom chi(A,X) by MESFUNC2:def 2;
  now
    let x be Element of X;
A2: rng chi(A,X) c= {0,1} by FUNCT_3:39;
    assume
A3: x in dom max+(chi(A,X));
    then
A4: (max+(chi(A,X))).x = max((chi(A,X)).x,0.) by MESFUNC2:def 2;
    (chi(A,X)).x in rng chi(A,X) by A1,A3,FUNCT_1:3;
    hence (max+(chi(A,X))).x = (chi(A,X)).x by A4,A2,XXREAL_0:def 10;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
