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 Th12:
 for s,t be DecoratedTree, x be object,
 q being FinSequence of NAT st q in dom s holds
 (x-tree (t,s)). (<* 1 *>^q) = s.q
proof
let s,t be DecoratedTree, x be object,
 q being FinSequence of NAT;
 assume A1: q in dom s;
 set r = <*t, s*>;
 A2: len r = 2 by FINSEQ_1:44;
 A3: (x-tree (t,s)) | <* 1 *> = r . (1 + 1) by A2,TREES_4:def 4
 .= s;
 dom ( (x-tree (t,s)) | <* 1 *> ) = dom (x-tree (t,s)) | <* 1 *>
 by TREES_2:def 10;
hence thesis by A1,A3,TREES_2:def 10;
end;
