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;
reserve T for BinHuffmanTree of p;

theorem Th27:
 Tseq,q,p is_constructingBinHuffmanTree implies
 for i be Nat
 for s,t be finite binary DecoratedTree of IndexedREAL
 for X be non empty finite Subset of BinFinTrees IndexedREAL
 st X=Tseq.i & s in X & t in X
 for z be finite binary DecoratedTree of IndexedREAL st z in X holds
 not [(MaxVl(X) + 1),(Vrootr t) +(Vrootr s)] in rng z
proof
assume A1:Tseq,q,p is_constructingBinHuffmanTree;
 let i be Nat;
 let s,t be finite binary DecoratedTree of IndexedREAL;
 let X be non empty finite Subset of BinFinTrees IndexedREAL;
 assume A2: X=Tseq.i & s in X & t in X;
 let z be finite binary DecoratedTree of IndexedREAL;
 assume A3: z in X;

assume [(MaxVl(X) + 1),(Vrootr t) +(Vrootr s)] in rng z;
 then consider p0 be object such that
 A4: p0 in dom z
 & [(MaxVl(X) + 1),(Vrootr t) +(Vrootr s)] = z.p0 by FUNCT_1:def 3;
reconsider p0 as Element of (dom z) by A4;
ex x,y be object
st x in NAT & y in REAL
 & z.p0 = [x,y] by ZFMISC_1:def 2;
then reconsider r = (z.p0) `1 as Element of NAT;
r = MaxVl(X) + 1 by A4;
hence contradiction by NAT_1:16,A3,Th26,A1,A2;
end;
