
theorem Th52:
for X be non empty set holds
 <* X --> 0 *> is FinSequence of Funcs(X,ExtREAL) &
 (for n be Nat st n in dom <* X --> 0 *>
    holds <* X --> 0 *>.n is without+infty) &
 (for n be Nat st n in dom <* X --> 0 *>
    holds <* X --> 0 *>.n is without-infty)
proof
   let X be non empty set;
   X --> 0 is Function of X,ExtREAL by XXREAL_0:def 1,FUNCOP_1:45; then
   reconsider f = X --> 0 as Element of Funcs(X,ExtREAL) by FUNCT_2:8;
   <* f *> is FinSequence of Funcs(X,ExtREAL);
   hence <* X --> 0 *> is FinSequence of Funcs(X,ExtREAL);
   hereby let n be Nat;
    assume n in dom <* X --> 0 *>; then
    n in Seg 1 by FINSEQ_1:38; then
    n = 1 by FINSEQ_1:2,TARSKI:def 1; then
A1: <* X--> 0 *>.n = X --> 0;
    not +infty in rng(X --> 0);
    hence <* X --> 0 *>.n is without+infty by A1,MESFUNC5:def 4;
   end;
    let n be Nat;
    assume n in dom <* X --> 0 *>; then
    n in Seg 1 by FINSEQ_1:38; then
    n = 1 by FINSEQ_1:2,TARSKI:def 1; then
A2: <* X--> 0 *>.n = X --> 0;
    not -infty in rng(X --> 0);
    hence <* X --> 0 *>.n is without-infty by A2,MESFUNC5:def 3;
end;
