
theorem Th24:
  for X be non empty set, f,g being PartFunc of X,ExtREAL st f is
without-infty & g is without-infty holds 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
  let X be non empty set;
  let f,g be PartFunc of X,ExtREAL;
  assume that
A1: f is without-infty and
A2: g is without-infty;
A3: dom(f+g) = dom f /\ dom g by A1,A2,Th16;
  then
A4: dom(max-(f+g)) = dom f /\ dom g by MESFUNC2:def 3;
A5: for x be set holds (x in dom(max- f) implies -infty < (max- f).x) & (x
in dom(max+ f) implies -infty < (max+ f).x) & (x in dom(max+ g) implies -infty
< (max+ g).x) & (x in dom(max- g) implies -infty < (max- g).x) by MESFUNC2:12
,13;
  then
A6: max+f is without-infty by Th10;
A7: max-f is without-infty by A5,Th10;
A8: for x be set holds (x in dom max+(f+g) implies -infty < (max+(f+g)).x) &
  (x in dom max-(f+g) implies -infty < (max-(f+g)).x) by MESFUNC2:12,13;
  then max+(f+g) is without-infty by Th10;
  then
A9: dom(max+(f+g) + max- f) = dom(max+(f+g)) /\ dom(max- f) by A7,Th16;
  max-(f+g) is without-infty by A8,Th10;
  then
A10: dom(max-(f+g) + max+ f) = dom(max-(f+g)) /\ dom(max+ f) by A6,Th16;
A11: max-g is without-infty by A5,Th10;
A12: dom(max- f) = dom f by MESFUNC2:def 3;
A13: max+g is without-infty by A5,Th10;
A14: dom(max- g) = dom g by MESFUNC2:def 3;
A15: dom(max+ f) = dom f by MESFUNC2:def 2;
  then
A16: dom(max-(f+g) + max+ f) = dom g /\ (dom f /\ dom f) by A4,A10,XBOOLE_1:16;
  dom(max+(f+g)) = dom f /\ dom g by A3,MESFUNC2:def 2;
  then
A17: dom(max+(f+g) + max- f) = dom g /\ (dom f /\ dom f) by A12,A9,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,A15,A10,XBOOLE_1:16;
A18: dom(max+ g) = dom g by MESFUNC2:def 2;
A19: for x be object
  holds ( x in dom(max+(f+g) + max-f) implies 0 <= (max+(f+g
) + max-f).x ) & ( x in dom(max-(f+g) + max+f) implies 0 <= (max-(f+g) + max+f)
  .x )
  proof
    let x be object;
    hereby
      assume
A20:  x in dom(max+(f+g) + max- f);
      then
A21:  0 <= (max- f).x by MESFUNC2:13;
      0 <= (max+(f+g)).x by A20,MESFUNC2:12;
      then 0 <= (max+(f+g)).x + (max- f).x by A21;
      hence 0 <= (max+(f+g) + max- f).x by A20,MESFUNC1:def 3;
    end;
    assume
A22: x in dom(max-(f+g) + max+ f);
    then
A23: 0 <= (max+ f).x by MESFUNC2:12;
    0 <= (max-(f+g)).x by A22,MESFUNC2:13;
    then 0 <= (max-(f+g)).x + (max+ f).x by A23;
    hence thesis by A22,MESFUNC1:def 3;
  end;
  then
A24: for x be set holds (x in dom(max+(f+g) + max-f) implies -infty < (max+(
f+g) + max-f).x ) & (x in dom(max-(f+g) + max+f) implies -infty < (max-(f+g) +
  max+f).x );
  then max+(f+g) + max-f is without-infty by Th10;
  then
  dom(max+(f+g) + max-f + max-g) = dom f /\ dom g /\ dom g by A14,A11,A17,Th16
    .= dom f /\ (dom g /\ dom g) by XBOOLE_1:16;
  hence dom(max+(f+g) + max- f + max- g) = dom f /\ dom g;
  max-(f+g) + max+f is without-infty by A24,Th10;
  then dom(max-(f+g) + max+ f + max+ g) = dom f /\ dom g /\ dom g by A18,A13
,A16,Th16;
  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;
  thus thesis by A19,SUPINF_2:52;
end;
