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 Th11:
 for s,t be DecoratedTree, x be object,
 q being FinSequence of NAT st q in dom t holds
 (x-tree (t,s)). (<* 0 *>^q) = t.q
proof
 let s,t be DecoratedTree, x be object, q be FinSequence of NAT;
 assume A1: q in dom t;
 set r = <*t, s*>;
 0 < len r; then
 A2: (x-tree (t,s)) | <* 0 *> = r . (0 + 1) by TREES_4:def 4
 .= t;
 dom ( (x-tree (t,s)) | <* 0 *> ) = dom (x-tree (t,s)) | <* 0 *>
 by TREES_2:def 10;
hence thesis by A1,A2,TREES_2:def 10;
end;
