reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;
reserve x,A for set;

theorem Th38:
  dom(|.f.|+|.g.|) = dom f /\ dom g & dom |.f+g.| c= dom |.f.|
proof
  dom(|.f.|+|.g.|) = dom |.f.| /\ dom |.g.| by VALUED_1:def 1;
  then dom(|.f.|+|.g.|) = dom f /\ dom |.g.| by VALUED_1:def 11;
  hence dom(|.f.|+|.g.|) = dom f /\ dom g by VALUED_1:def 11;
  dom |.f+g.| = dom(f+g) by VALUED_1:def 11
    .= dom f /\ dom g by VALUED_1:def 1;
  then dom |.f+g.| c= dom f by XBOOLE_1:17;
  hence thesis by VALUED_1:def 11;
end;
