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
for X be non empty finite Subset of BinFinTrees IndexedREAL,
    w be finite binary DecoratedTree of IndexedREAL st X = {w}
holds MaxVl X = Vrootl w
proof
let X be non empty finite Subset of BinFinTrees IndexedREAL,
w be finite binary DecoratedTree of IndexedREAL;
assume A1:X ={w};
 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;
A3:for n be object st n in L holds n = Vrootl w
proof
let n be object;
assume n in L;
then ex p be Element of BinFinTrees IndexedREAL st n = Vrootl p &
 p in X by A2;
hence thesis by TARSKI:def 1,A1;
end;
for n be object st n = Vrootl w holds n in L
proof
let n be object;
assume A4: n = Vrootl w;
consider y be object such that
A5: y in L by XBOOLE_0:def 1;
ex p be Element of BinFinTrees IndexedREAL st y = Vrootl p &
 p in X by A2,A5;
hence thesis by A5,A4,TARSKI:def 1,A1;
end; then
L = {Vrootl w} by TARSKI:def 1,A3;
hence thesis by A2,XXREAL_2:11;
end;
