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 Th40:
  x in dom |.f+g.| implies (|.f+g.|).x <= (|.f.|+|.g.|).x
proof
A1: |. f.x + g.x .| <= |. f.x .| + |. g.x .| by COMPLEX1:56;
  assume
A2: x in dom |.f+g.|;
  then x in dom (f+g) by VALUED_1:def 11;
  then
A3: |. (f+g).x .| <= |. f.x .| + |. g.x .| by A1,VALUED_1:def 1;
A4: dom |.f+g.| c= dom |.g.| by Th38;
  then
A5: |. g.x .| = |.g.| .x by A2,VALUED_1:def 11;
  x in dom |.g.| by A2,A4;
  then
A6: x in dom g by VALUED_1:def 11;
A7: dom |.f+g.| c= dom |.f.| by Th38;
  then x in dom |.f.| by A2;
  then x in dom f by VALUED_1:def 11;
  then x in dom f /\ dom g by A6,XBOOLE_0:def 4;
  then
A8: x in dom(|.f.| + |.g.|) by Th38;
  |. f.x .| = |.f.| .x by A2,A7,VALUED_1:def 11;
  then |. f.x .| + |. g.x .| = (|.f.| + |.g.|).x by A5,A8,VALUED_1:def 1;
  hence thesis by A2,A3,VALUED_1:def 11;
end;
