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 Th13:
 for s,t be DecoratedTree, x be object holds
 Leaves (x-tree (t,s)) = (Leaves t) \/ Leaves s
proof
 let s,t be DecoratedTree, x be object;
 set q = <*dom t,dom s*>;
 A1: len q = 2 by FINSEQ_1:44;
 A4: dom (x -tree (t,s)) = tree ((dom t),(dom s)) by TREES_4:14;
 A5: Leaves (tree ((dom t),(dom s))) =
 {<* 0 *>^p where p is Element of (dom t) : p in Leaves (dom t)}
 \/ {<* 1 *>^p where p is Element of (dom s)
 : p in Leaves (dom s)} by Th10;
set L = {<* 0 *>^p where p is Element of (dom t) : p in Leaves (dom t) };
set R = {<* 1 *>^p where p is Element of (dom s) : p in Leaves (dom s) };
A6: Leaves (x-tree(t,s))
 = ((x-tree(t,s)) .: L) \/ ((x-tree(t,s)) .: R) by RELAT_1:120,A5,A4;
for z be object holds
 z in ((x-tree(t,s)) .: L) iff z in t .: (Leaves dom t)
proof
let z be object;
 hereby assume z in ((x-tree(t,s)) .: L);
 then consider q be object such that
 A7: q in dom (x-tree(t,s)) & q in L & z = (x-tree(t,s)).q
 by FUNCT_1:def 6;
 consider p be Element of dom t such that
 A8: q=<* 0 *>^p & p in Leaves dom t by A7;
 z = t.p by A7,Th11,A8;
 hence z in t .: (Leaves dom t) by A8,FUNCT_1:def 6;
 end;
assume z in t .: (Leaves dom t); then
 consider p0 being object such that
 A9: p0 in dom t & p0 in Leaves dom t
 & z = t.p0 by FUNCT_1:def 6;
 reconsider p=p0 as Element of dom t by A9;
 A10: (x-tree (t,s)). (<* 0 *>^p) = t.p by Th11;
 0 < len q & p in q . (0 + 1); then
 A11: (<* 0 *>^p) in dom (x -tree (t,s)) by TREES_3:def 15,A4;
 (<* 0 *>^p) in L by A9;
 hence z in ((x-tree(t,s)) .: L) by A10,A9,FUNCT_1:def 6,A11;
 end;
then
A12: ((x-tree(t,s)) .: L) = t .: (Leaves (dom t )) by TARSKI:2;
for z be object holds
 z in ((x-tree(t,s)) .: R) iff z in s .: (Leaves (dom s ))
proof
let z be object;
 hereby assume z in ((x-tree(t,s)) .: R);
 then consider q be object such that
 A13: q in dom (x-tree(t,s)) & q in R & z = (x-tree(t,s)).q
 by FUNCT_1:def 6;
 consider p be Element of dom s such that
 A14: q=<* 1 *>^p & p in Leaves dom s by A13;
 (x-tree(t,s)).(<* 1 *>^p ) = s.p by Th12;
 hence z in s .: (Leaves dom s) by A14,FUNCT_1:def 6,A13;
 end;
assume z in s .: (Leaves dom s); then
 consider p0 being object such that
 A15: p0 in dom s & p0 in Leaves dom s
 & z = s.p0 by FUNCT_1:def 6;
 reconsider p=p0 as Element of dom s by A15;
 A16: (x-tree (t,s)). (<* 1 *>^p) = s.p by Th12;
 1 < len q & p in q . (1 + 1) by A1; then
 A17: (<* 1 *>^p) in dom (x -tree (t,s)) by TREES_3:def 15,A4;
 (<* 1 *>^p) in R by A15;
 hence z in ((x-tree(t,s)) .: R) by A16,A15,FUNCT_1:def 6,A17;
 end;
hence thesis by A6,A12,TARSKI:2;
end;
