reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;

theorem
  Y c= dom(f+g) implies dom((f+g)|Y) = Y & dom(f|Y+g|Y)=Y & (f+g)|Y = f|Y +g|Y
proof
  assume
A1: Y c= dom(f+g);
  dom(f|Y) /\ dom(g|Y) = dom f /\ Y /\ dom(g|Y) by RELAT_1:61
    .= dom f /\ Y /\ (dom g /\ Y) by RELAT_1:61
    .= dom f /\ Y /\ dom g /\ Y by XBOOLE_1:16
    .= dom f /\ dom g /\ Y /\ Y by XBOOLE_1:16
    .= dom f /\ dom g /\ (Y /\ Y) by XBOOLE_1:16;
  then
A2: dom(f|Y+g|Y) = dom f /\ dom g /\ Y by VALUED_1:def 1
    .= dom(f+g) /\ Y by VALUED_1:def 1
    .= Y by A1,XBOOLE_1:28;
A3: dom(f+g) = dom f /\ dom g by VALUED_1:def 1;
  dom(g|Y) = dom g /\ Y by RELAT_1:61;
  then
A4: dom(g|Y) = Y by A1,A3,XBOOLE_1:18,28;
A5: dom((f+g)|Y) =dom(f+g) /\ Y by RELAT_1:61;
  then
A6: dom((f+g)|Y) = Y by A1,XBOOLE_1:28;
  dom(f|Y) = dom f /\ Y by RELAT_1:61;
  then
A7: dom(f|Y) = Y by A1,A3,XBOOLE_1:18,28;
  now
    let x be object;
    assume
A8: x in dom((f+g)|Y);
    hence ((f+g)|Y).x = (f+g).x by FUNCT_1:47
      .=f.x+g.x by A1,A6,A8,VALUED_1:def 1
      .=(f|Y).x + g.x by A6,A7,A8,FUNCT_1:47
      .=(f|Y).x + (g|Y).x by A6,A4,A8,FUNCT_1:47
      .= ((f|Y)+(g|Y)).x by A6,A2,A8,VALUED_1:def 1;
  end;
  hence thesis by A1,A5,A2,FUNCT_1:2,XBOOLE_1:28;
end;
