
theorem Th23:
  for X be non empty set, f,g,h be PartFunc of X,ExtREAL st f is
  nonnegative & g is nonnegative & h is nonnegative holds dom(f+g+h) = dom f /\
dom g /\ dom h & f+g+h is nonnegative & for x be set st x in dom f /\ dom g /\
  dom h holds (f+g+h).x=f.x+g.x+h.x
proof
  let X be non empty set;
  let f,g,h be PartFunc of X,ExtREAL;
  assume that
A1: f is nonnegative and
A2: g is nonnegative and
A3: h is nonnegative;
A4: f+g is nonnegative by A1,A2,Th22;
  then
A5: dom(f+g+h) = dom(f+g) /\ dom h by A3,Th22;
  hence dom(f+g+h) = dom f /\ dom g /\ dom h by A1,A2,Th22;
  thus f+g+h is nonnegative by A3,A4,Th22;
  hereby
    let x be set;
    assume x in dom f /\ dom g /\ dom h;
    then
A6: x in dom(f+g) /\ dom h by A1,A2,Th22;
    then
A7: x in dom(f+g) by XBOOLE_0:def 4;
    thus (f+g+h).x = (f+g).x + h.x by A5,A6,MESFUNC1:def 3
      .= f.x + g.x + h.x by A7,MESFUNC1:def 3;
  end;
end;
