
theorem Th64:
  for F,G be FinSequence of ExtREAL holds
   (F is without-infty & G is without-infty implies F^G is without-infty)
 & (F is without+infty & G is without+infty implies F^G is without+infty)
proof
    let F,G be FinSequence of ExtREAL;
    hereby assume F is without-infty & G is without-infty; then
A1:  not -infty in rng F & not -infty in rng G by MESFUNC5:def 3;
     rng(F^G) = rng F \/ rng G by FINSEQ_1:31; then
     not -infty in rng(F^G) by A1,XBOOLE_0:def 3;
     hence F^G is without-infty by MESFUNC5:def 3;
    end;
    assume F is without+infty & G is without+infty; then
A2: not +infty in rng F & not +infty in rng G by MESFUNC5:def 4;
    rng(F^G) = rng F \/ rng G by FINSEQ_1:31; then
    not +infty in rng(F^G) by A2,XBOOLE_0:def 3;
    hence F^G is without+infty by MESFUNC5:def 4;
end;
