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 Th39:
  (|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|) = (|.f.|+|.g.|)|( dom |.f+g.|)
proof
A1: dom |.f+g.| c= dom |.g.| by Th38;
  then
A2: dom |.f+g.| c= dom g by VALUED_1:def 11;
  dom(g|(dom |.f+g.|)) = dom g /\ dom |.f+g.| by RELAT_1:61;
  then
A3: dom(g|(dom |.f+g.|)) = dom |.f+g.| by A2,XBOOLE_1:28;
  then
A4: dom |.(g|(dom |.f+g.|)).| = dom |.f+g.| by VALUED_1:def 11;
A5: dom |.f+g.| c= dom |.f.| by Th38;
  then
A6: dom |.f+g.| c= dom f by VALUED_1:def 11;
  then dom |.f+g.| /\ dom |.f+g.| c= dom f /\ dom g by A2,XBOOLE_1:27;
  then
A7: dom |.f+g.| c= dom(|.f.|+|.g.|) by Th38;
  then
A8: dom((|.f.|+|.g.|)|(dom |.f+g.|)) = dom |.f+g.| by RELAT_1:62;
  dom(f|(dom |.f+g.|)) = dom f /\ dom |.f+g.| by RELAT_1:61;
  then
A9: dom(f|(dom |.f+g.|)) = dom |.f+g.| by A6,XBOOLE_1:28;
A10: (|.f.|)|(dom |.f+g.|) = |.(f|(dom |.f+g.|)).| & (|.g.|)|(dom |.f+g.|) =
  |.(g |(dom |.f+g.|)).| by Th37;
  then
A11: dom((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)) = dom (f|(dom |.f+g.|
  )) /\ dom (g|(dom |.f+g.|)) by Th38
    .= dom |.f+g.| by A9,A3;
A12: dom |.(f|(dom |.f+g.|)).| = dom |.f+g.| by A9,VALUED_1:def 11;
  now
    let x be Element of X;
    assume
A13: x in dom((|.f.|+|.g.|)|(dom |.f+g.|));
    then
A14: ((|.f.|+|.g.|)|(dom |.f+g.|)).x = (|.f.|+|.g.|).x by FUNCT_1:47
      .= (|.f.|).x + (|.g.|).x by A7,A8,A13,VALUED_1:def 1
      .= |. f.x .| + (|.g.|).x by A5,A8,A13,VALUED_1:def 11;
A15: x in dom |.f+g.| by A7,A13,RELAT_1:62;
    then
    ((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f
    +g.|)).x + ((|.g.|)|(dom |.f+g.|)).x by A11,VALUED_1:def 1
      .= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).|.x by A12,A10,A15,
VALUED_1:def 11
      .= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).x .| by A4,A15,
VALUED_1:def 11
      .= |. f.x .| + |.(g|(dom |.f+g.|)).x .| by A15,FUNCT_1:49
      .= |. f.x .| + |. g.x .| by A15,FUNCT_1:49;
    hence ((|.f.|+|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f+g.|) + (|.g.|)|(
    dom |.f+g.|)).x by A1,A8,A13,A14,VALUED_1:def 11;
  end;
  hence thesis by A11,A7,PARTFUN1:5,RELAT_1:62;
end;
