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 Th62:
  dom(max+(f+g) + max-f) = dom f /\ dom g & dom(max-(f+g) + max+f)
= dom f /\ dom g & dom(max+(f+g) + max-f + max-g) = dom f /\ dom g & dom(max-(f
+g) + max+f + max+g) = dom f /\ dom g & max+(f+g) + max-f is nonnegative & max-
  (f+g) + max+f is nonnegative
proof
A1: dom max-f = dom f & dom(max+(f+g) + max-f) = dom max+(f+g) /\ dom max-f
  by RFUNCT_3:def 11,VALUED_1:def 1;
A2: dom max+f = dom f & dom(max-(f+g) + max+f) = dom max-(f+g) /\ dom max+f
  by RFUNCT_3:def 10,VALUED_1:def 1;
A3: dom(f+g) = dom f /\ dom g by VALUED_1:def 1;
  then
A4: dom(max-(f+g)) = dom f /\ dom g by RFUNCT_3:def 11;
  dom(max+(f+g)) = dom f /\ dom g by A3,RFUNCT_3:def 10;
  then
A5: dom(max+(f+g) + max-f) = dom g /\ (dom f /\ dom f) by A1,XBOOLE_1:16;
  hence dom(max+(f+g) + max-f) = dom f /\ dom g & dom(max-(f+g) + max+f) = dom
  f /\ dom g by A4,A2,XBOOLE_1:16;
  dom max-g = dom g by RFUNCT_3:def 11;
  then dom(max+(f+g) + max-f + max-g) = dom f /\ dom g /\ dom g by A5,
VALUED_1:def 1
    .= dom f /\ (dom g /\ dom g) by XBOOLE_1:16;
  hence dom(max+(f+g) + max-f + max-g) = dom f /\ dom g;
  dom max+g = dom g & dom(max-(f+g) + max+f) = dom g /\ (dom f /\ dom f)
  by A4,A2,RFUNCT_3:def 10,XBOOLE_1:16;
  then dom(max-(f+g) + max+ f + max+ g) = dom f /\ dom g /\ dom g by
VALUED_1:def 1;
  then dom(max-(f+g) + max+f + max+g) = dom f /\ (dom g /\ dom g) by
XBOOLE_1:16;
  hence dom(max-(f+g) + max+f + max+g) = dom f /\ dom g;
  now
    let x be object;
    assume
A6: x in dom(max+(f+g) + max-f);
    then 0 <= max+(f+g).x & 0 <= max-f.x by RFUNCT_3:37,40;
    then 0+0 <= max+(f+g).x + max-f.x;
    hence 0 <= (max+(f+g) + max-f).x by A6,VALUED_1:def 1;
  end;
  hence max+(f+g) + max-f is nonnegative by Th52;
  now
    let x be object;
    assume
A7: x in dom(max-(f+g) + max+f);
    then 0 <= max-(f+g).x & 0 <= max+f.x by RFUNCT_3:37,40;
    then 0+0 <= max-(f+g).x + max+f.x;
    hence 0 <= (max-(f+g) + max+f).x by A7,VALUED_1:def 1;
  end;
  hence thesis by Th52;
end;
