
theorem Th23:
for X be non empty set, f,g be PartFunc of X,ExtREAL st
  f is without-infty without+infty holds
    dom(f+g)=dom f /\ dom g & dom(f-g)=dom f /\ dom g
  & dom(g-f)=dom f /\ dom g
proof
    let X be non empty set, f,g be PartFunc of X,ExtREAL;
    assume A1: f is without-infty without+infty; then
    not -infty in rng f by MESFUNC5:def 3; then
A2: f"{-infty} = {} by FUNCT_1:72;
    not +infty in rng f by A1,MESFUNC5:def 4; then
A3: f"{+infty} = {} by FUNCT_1:72; then
    f"{+infty} /\ g"{-infty} \/ f"{-infty} /\ g"{+infty} = {} by A2; then
    dom(f+g) = (dom f /\ dom g)\{} by MESFUNC1:def 3;
    hence dom(f+g)=dom f /\ dom g;
A4: f"{+infty} /\ g"{+infty} \/ f"{-infty} /\ g"{-infty} = {} by A2,A3; then
    dom(f-g) = (dom f /\ dom g)\{} by MESFUNC1:def 4;
    hence dom(f-g)=dom f /\ dom g;
    dom(g-f) = (dom f /\ dom g)\{} by A4,MESFUNC1:def 4;
    hence dom(g-f)=dom f /\ dom g;
end;
