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 Th19:
  dom(|.f.|+|.g.|) = dom f /\ dom g & dom |.f+g.| c= dom |.f.|
proof
  set F = |.f.|;
  set G = |.g.|;
  F is without-infty by MESFUNC5:12;
  then not -infty in rng F;
  then
A1: F"{-infty} = {} by FUNCT_1:72;
  G is without-infty by MESFUNC5:12;
  then not -infty in rng G;
  then
A2: G"{-infty} = {} by FUNCT_1:72;
  dom(|.f.|+|.g.|) = (dom F /\ dom G)\((F"{-infty} /\ G"{+infty}) \/ (F"{
  +infty} /\ G"{-infty})) by MESFUNC1:def 3;
  then dom(|.f.|+|.g.|) = dom f /\ dom G by A1,A2,MESFUNC1:def 10;
  hence dom(|.f.|+|.g.|) = dom f /\ dom g by MESFUNC1:def 10;
  dom |.f+g.| = dom(f+g) by MESFUNC1:def 10
    .= (dom f /\ dom g) \((f"{-infty} /\ g"{+infty}) \/ (f"{+infty} /\ g"{
  -infty})) by MESFUNC1:def 3
    .= dom f /\ (dom g \((f"{-infty} /\ g"{+infty}) \/ (f"{+infty} /\ g"{
  -infty}))) by XBOOLE_1:49;
  then dom |.f+g.| c= dom f by XBOOLE_1:17;
  hence thesis by MESFUNC1:def 10;
end;
