reserve SOURCE for non empty finite set,
 p for Probability of Trivial-SigmaField SOURCE,
 Tseq for FinSequence of BoolBinFinTrees IndexedREAL,
 q for FinSequence of NAT;

theorem Th9:
 for s,t be Tree holds not {} in Leaves (tree (t,s))
proof
 let s,t be Tree;
 assume A1: {} in Leaves (tree (t,s));
 set q = <*t,s*>;
 0 < len q & {} in q . (0 + 1) by TREES_1:22; then
 <* 0 *>^{} in tree(t,s) by TREES_3:def 15; then
 A2: <* 0 *> in tree(t,s) by FINSEQ_1:34;
 for p be object holds p in tree (t,s) iff p in elementary_tree 0
 proof
 let p0 be object;
 hereby assume that
 A3: p0 in tree (t,s) and
A4: not p0 in elementary_tree 0;
 reconsider p=p0 as FinSequence of NAT by A3,TREES_1:19;
 p <> {} by A4, TARSKI:def 1, TREES_1:29;
 then consider q being FinSequence of NAT, n being Element of NAT such that
A5: p = <*n*> ^ q by FINSEQ_2:130;
 {} ^ <*n*> = <*n*> by FINSEQ_1:34;
hence contradiction by A1,TREES_1:55,A3, A5, TREES_1:21;
end;
assume p0 in elementary_tree 0;
then p0 = {} by TARSKI:def 1, TREES_1:29;
hence p0 in tree (t,s) by TREES_1:22;
end;
 then <* 0 *> in elementary_tree 0 by A2;
 hence contradiction by TARSKI:def 1,TREES_1:29;
end;
