let X be non empty set ; :: thesis: for f, g being PartFunc of X,ExtREAL st f is without+infty & g is without+infty holds
dom (f + g) = (dom f) /\ (dom g)
let f, g be PartFunc of X,ExtREAL; :: thesis: ( f is without+infty & g is without+infty implies dom (f + g) = (dom f) /\ (dom g) )
assume that
A1: f is without+infty and
A2: g is without+infty ; :: thesis: dom (f + g) = (dom f) /\ (dom g)
not +infty in rng g by A2;
then A3: g " {+infty} = {} by FUNCT_1:72;
not +infty in rng f by A1;
then f " {+infty} = {} by FUNCT_1:72;
then ((f " {+infty}) /\ (g " {-infty})) \/ ((f " {-infty}) /\ (g " {+infty})) = {} by A3;
then dom (f + g) = ((dom f) /\ (dom g)) \ {} by MESFUNC1:def 3;
hence dom (f + g) = (dom f) /\ (dom g) ; :: thesis: verum