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 Th20:
  (|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|) = (|.f.|+|.g.|)|( dom |.f+g.|)
proof
A1: (|.g.|)|(dom |.f+g.|) = |.(g|(dom |.f+g.|)).| by Th18;
A2: dom |.f+g.| c= dom |.g.| by Th19;
  then
A3: dom |.f+g.| c= dom g by MESFUNC1:def 10;
  dom(g|(dom |.f+g.|)) = dom g /\ dom |.f+g.| by RELAT_1:61;
  then
A4: dom(g|(dom |.f+g.|)) = dom |.f+g.| by A3,XBOOLE_1:28;
  then
A5: dom |.(g|(dom |.f+g.|)).| = dom |.f+g.| by MESFUNC1:def 10;
A6: dom |.f+g.| c= dom |.f.| by Th19;
  then
A7: dom |.f+g.| c= dom f by MESFUNC1:def 10;
  then dom |.f+g.| /\ dom |.f+g.| c= dom f /\ dom g by A3,XBOOLE_1:27;
  then
A8: dom |.f+g.| c= dom(|.f.|+|.g.|) by Th19;
  then
A9: 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
A10: dom(f|(dom |.f+g.|)) = dom |.f+g.| by A7,XBOOLE_1:28;
A11: (|.f.|)|(dom |.f+g.|) = |.(f|(dom |.f+g.|)).| by Th18;
  then
A12: dom((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)) = dom (f|(dom |.f+g
  .|)) /\ dom (g|(dom |.f+g.|)) by A1,Th19
    .= dom |.f+g.| by A10,A4;
A13: dom |.(f|(dom |.f+g.|)).| = dom |.f+g.| by A10,MESFUNC1:def 10;
  now
    let x be Element of X;
    assume
A14: x in dom((|.f.|+|.g.|)|(dom |.f+g.|));
    then
A15: ((|.f.|+|.g.|)|(dom |.f+g.|)).x = (|.f.|+|.g.|).x by FUNCT_1:47
      .= (|.f.|).x + (|.g.|).x by A8,A9,A14,MESFUNC1:def 3
      .= |. f.x .| + (|.g.|).x by A6,A9,A14,MESFUNC1:def 10;
A16: x in dom |.f+g.| by A8,A14,RELAT_1:62;
    then
    ((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f
    +g.|)).x + ((|.g.|)|(dom |.f+g.|)).x by A12,MESFUNC1:def 3
      .= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).|.x by A13,A11,A1,A16,
MESFUNC1:def 10
      .= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).x .| by A5,A16,
MESFUNC1:def 10
      .= |. f.x .| + |.(g|(dom |.f+g.|)).x .| by A16,FUNCT_1:49
      .= |. f.x .| + |. g.x .| by A16,FUNCT_1:49;
    hence ((|.f.|+|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f+g.|) + (|.g.|)|(
    dom |.f+g.|)).x by A2,A9,A14,A15,MESFUNC1:def 10;
  end;
  hence thesis by A12,A8,PARTFUN1:5,RELAT_1:62;
end;
