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 Th2:
for X,Y,Z be non empty finite Subset of BinFinTrees IndexedREAL
st Z = X \/ Y holds
 MaxVl(Z) = max (MaxVl(X), MaxVl(Y))
proof
let X,Y,Z be non empty finite Subset of BinFinTrees IndexedREAL;
assume A1: Z = X \/ Y;
set mz = MaxVl(Z);
 consider L be non empty finite Subset of NAT such that A2:
L = {Vrootl p where p is Element of BinFinTrees IndexedREAL: p in Z }
 & MaxVl Z = max L by Def9;
mz in L & for b be Nat st b in L holds b <= mz by XXREAL_2:def 8,A2;then
consider p be Element of BinFinTrees IndexedREAL
such that A3: mz = Vrootl p & p in Z by A2;
 consider LX be non empty finite Subset of NAT such that A4:
LX = {Vrootl p where p
 is Element of BinFinTrees IndexedREAL: p in X }
 & MaxVl X = max LX by Def9;
 max LX in LX & for x be Nat st x in LX holds x <= max LX by XXREAL_2:def 8;
then consider px be Element of BinFinTrees IndexedREAL such that
A5: max LX = Vrootl px & px in X by A4;
px in Z by A5,A1,XBOOLE_0:def 3;then
Vrootl px in L by A2;then
A6: max LX <= mz by A5,XXREAL_2:def 8,A2;
 consider LY be non empty finite Subset of NAT such that A7:
LY = {Vrootl p where p is Element of BinFinTrees IndexedREAL: p in Y }
 & MaxVl Y = max LY by Def9;
 max LY in LY & for y be Nat st y in LY holds y <= max LY
by XXREAL_2:def 8; then
consider py be Element of BinFinTrees IndexedREAL such that
A8: max LY = Vrootl py & py in Y by A7;
py in Z by A8,A1,XBOOLE_0:def 3; then
Vrootl py in L by A2;then
A9: max LY <= mz by A8,XXREAL_2:def 8,A2;
per cases by XBOOLE_0:def 3,A3,A1;
suppose p in X;then
Vrootl p in LX by A4;then
mz <= max LX by XXREAL_2:def 8,A3;then
mz = max LX by A6,XXREAL_0:1;
hence thesis by A4,A7,A9,XXREAL_0:def 10;
end;
suppose p in Y;then
Vrootl p in LY by A7;then
mz <= max LY by XXREAL_2:def 8,A3;then
mz = max LY by A9,XXREAL_0:1;
hence thesis by A4,A7,A6,XXREAL_0:def 10;
end;
end;
