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,
  c for Complex,
  E,A,B for Element of S;

theorem
  Y c= dom(f+g) implies dom(f|Y+g|Y)=Y & (f+g)|Y = f|Y+g|Y
proof
A1: dom(f+g) = dom f /\ dom g by VALUED_1:def 1;
  assume
A2: Y c= dom(f+g);
  then
A3: dom((f+g)|Y) = Y by RELAT_1:62;
  dom(g|Y) = dom g /\ Y by RELAT_1:61;
  then
A4: dom(g|Y) = Y by A2,A1,XBOOLE_1:18,28;
  dom(f|Y) = dom f /\ Y by RELAT_1:61;
  then
A5: dom(f|Y) = Y by A2,A1,XBOOLE_1:18,28;
  then
A6: dom(f|Y+g|Y) = Y /\ Y by A4,VALUED_1:def 1;
  now
    let x be object;
    assume
A7: x in dom((f+g)|Y);
    hence ((f+g)|Y).x = (f+g).x by FUNCT_1:47
      .=f.x+g.x by A2,A3,A7,VALUED_1:def 1
      .=(f|Y).x + g.x by A3,A5,A7,FUNCT_1:47
      .=(f|Y).x + (g|Y).x by A3,A4,A7,FUNCT_1:47
      .= ((f|Y)+(g|Y)).x by A3,A6,A7,VALUED_1:def 1;
  end;
  hence thesis by A2,A6,FUNCT_1:2,RELAT_1:62;
end;
