reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem
  for a,b being Element of AtomSet(X) st a<>b holds BranchV(a) /\
  BranchV(b) = {}
proof
  let a,b be Element of AtomSet(X);
  assume
A1: a<>b;
  assume BranchV(a) /\ BranchV(b) <> {};
  then consider c being object such that
A2: c in BranchV(a) /\ BranchV(b) by XBOOLE_0:def 1;
  reconsider z2 = c as Element of BranchV(b) by A2,XBOOLE_0:def 4;
  reconsider z1 = c as Element of BranchV(a) by A2,XBOOLE_0:def 4;
  z1 \ z2 in BranchV(a\b) by Th39;
  then 0.X in {x3 where x3 is Element of X:a\b<=x3}by Def5;
  then ex x3 being Element of X st 0.X = x3 & a\b <= x3;
  then (a\b)\0.X = 0.X;
  then
A3: a\b = 0.X by Th2;
  b in {xx where xx is Element of X:xx is atom};
  then ex xx being Element of X st b=xx & xx is atom;
  hence contradiction by A1,A3;
end;
