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 Th8:
 for X,Y be set
 holds LeavesSet(X \/ Y ) = LeavesSet(X) \/ LeavesSet(Y)
proof
 let X,Y be set;
for x be object
 holds x in LeavesSet(X \/ Y) iff x in LeavesSet(X) \/ LeavesSet(Y)
 proof
 let x be object;
 hereby assume x in LeavesSet(X \/ Y); then
 consider p be Element of BinFinTrees IndexedREAL such that
 A1: x = Leaves p & p in X \/ Y;
 p in X or p in Y by A1,XBOOLE_0:def 3;
 then x in LeavesSet(X) or x in LeavesSet(Y) by A1;
 hence x in LeavesSet(X) \/ LeavesSet(Y) by XBOOLE_0:def 3;
 end;
 assume A2: x in LeavesSet(X) \/ LeavesSet(Y);
 per cases by A2,XBOOLE_0:def 3;
 suppose x in LeavesSet(X); then
 consider p be Element of BinFinTrees IndexedREAL such that
 A3: x = Leaves p & p in X;
 p in X \/ Y by TARSKI:def 3,XBOOLE_1:7,A3;
 hence x in LeavesSet(X \/ Y) by A3;
 end;
 suppose x in LeavesSet(Y); then
 consider p be Element of BinFinTrees IndexedREAL such that
 A4: x = Leaves p & p in Y;
 p in X \/ Y by TARSKI:def 3,XBOOLE_1:7,A4;
 hence x in LeavesSet(X \/ Y) by A4;
 end;
 end;
hence thesis by TARSKI:2;
end;
