
theorem Th22:
  for X be non empty set, f,g be PartFunc of X,ExtREAL st f is
  nonnegative & g is nonnegative holds dom(f+g)=dom f /\ dom g & f+g is
  nonnegative
proof
  let X be non empty set, f,g be PartFunc of X,ExtREAL;
  assume that
A1: f is nonnegative and
A2: g is nonnegative;
  thus
A3: dom(f+g)=dom f /\ dom g by A1,A2,Th16;
  now
    let x be object;
    assume
A4: x in dom f /\ dom g;
A5: 0 <= g.x by A2,SUPINF_2:51;
    0 <= f.x by A1,SUPINF_2:51;
    then 0 <= f.x +g.x by A5;
    hence 0 <= (f+g).x by A3,A4,MESFUNC1:def 3;
  end;
  hence thesis by A3,SUPINF_2:52;
end;
