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 Th10:
 for s,t be Tree holds Leaves (tree (t,s)) =
 {<* 0 *>^p where p is Element of t : p in Leaves t}
 \/ {<* 1 *>^p where p is Element of s : p in Leaves s}
proof
 let s,t be Tree;
 set L = {<* 0 *>^p where p is Element of t:p in Leaves t };
 set R = {<*1*>^p where p is Element of s:p in Leaves s };
 set H = Leaves tree(t,s);
 set q = <*t,s*>;
 A1: len q = 2 by FINSEQ_1:44;
 for x be object holds x in H iff x in L \/ R
 proof
 let x be object;
 hereby assume A4: x in H; then
 x = {} or ex n being Nat, r being FinSequence st
 ( n < len q & r in q . (n + 1) & x = <*n*> ^ r ) by TREES_3:def 15; then
 consider n being Nat, r being FinSequence such that
 A5: n < len q & r in q . (n + 1) & x = <*n*> ^ r by A4,Th9;
 per cases by NAT_1:23,A1,A5;
 suppose A6: n = 0; then
 r in Leaves t by BINTREE1:6,A4,A5; then
 x in L by A6,A5;
 hence x in L \/ R by XBOOLE_0:def 3;
 end;
 suppose A7:n = 1;
 then r in Leaves s by BINTREE1:6,A4,A5;
 then x in R by A7,A5;
 hence x in L \/ R by XBOOLE_0:def 3;
 end;
 end;
assume
 A8: x in L \/ R;
 per cases by A8,XBOOLE_0:def 3;
 suppose x in L; then
 consider p be Element of t
 such that A9: x = <* 0 *>^p & p in Leaves t;
 0 < len q & p in q . (0 + 1); then
 x in tree(t,s) by A9,TREES_3:def 15;
 hence x in H by BINTREE1:6,A9;
 end;
 suppose x in R; then
 consider p be Element of s such that A10: x = <* 1 *>^p & p in Leaves s;
 1 < len q & p in q . (1 + 1) by A1; then
 x in tree(t,s) by A10,TREES_3:def 15;
 hence x in H by BINTREE1:6,A10;
 end;
end;
hence thesis by TARSKI:2;
end;
