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,Z be non empty finite Subset of BinFinTrees IndexedREAL
for Y be set st Z = X \ Y holds MaxVl(Z) <= MaxVl(X)
proof
let X,Z be non empty finite Subset of BinFinTrees IndexedREAL;
let Y be set;
assume A1:Z = X \ Y;
per cases;
suppose X misses Y;
hence thesis by XBOOLE_1:83,A1;
end;
suppose X meets Y;
 A2:X = Z \/ (X \ Z) by XBOOLE_1:45,A1,XBOOLE_1:36;
 per cases;
 suppose X=Z; hence thesis; end;
 suppose X <> Z;then
 Z c< X by A1,XBOOLE_1:36,XBOOLE_0:def 8;then
 reconsider W = X \ Z as non empty
 finite Subset of BinFinTrees IndexedREAL by XBOOLE_1:105;
 MaxVl(X) = max (MaxVl(Z), MaxVl(W)) by Th2,A2;
 hence thesis by XXREAL_0:25;
 end;
 end;
end;
