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 Th6:
 for X be non empty finite Subset of BinFinTrees IndexedREAL,
 s,t be finite binary DecoratedTree of IndexedREAL holds
 not MakeTree (t,s,(MaxVl(X) + 1)) in X
 proof
 let X be non empty finite Subset of BinFinTrees IndexedREAL,
 s,t be finite binary DecoratedTree of IndexedREAL;
 assume A1:MakeTree (t,s,(MaxVl(X) + 1)) in X;
set px = MakeTree (t,s,(MaxVl(X) + 1));
consider L be non empty finite Subset of NAT such that
 A2: L = {Vrootl p where p
 is Element of BinFinTrees IndexedREAL: p in X }
 & MaxVl(X) = max L by Def9;
 dom px is finite & dom px is binary by BINTREE1:def 3; then
 reconsider px as Element of BinFinTrees IndexedREAL by Def2;
 Vrootl px in L by A1,A2; then
 A3: Vrootl px <= MaxVl(X) by A2,XXREAL_2:def 8;
 px.{} = [(MaxVl(X) + 1),(Vrootr t) +(Vrootr s)] by TREES_4:def 4;
 hence contradiction by NAT_1:13,A3;
 end;
