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),x being Element of BranchV(a), y
  being Element of BranchV(b) st a<>b holds not x\y in BCK-part(X)
proof
  let a,b be Element of AtomSet(X),x be Element of BranchV(a), y be Element of
  BranchV(b);
  assume
A1: a<>b;
  x\y in BranchV(a\b) by Th39;
  then ex xy being Element of X st x\y=xy & a\b <= xy;
  then (a\b)\(x\y) =0.X;
  then (a\(x\y))\b =0.X by Th7;
  then (a\(x\y))\((a\(x\y))\b) =a\(x\y) by Th2;
  then
A2: b = a\(x\y) by Th24;
  assume x\y in BCK-part(X);
  hence contradiction by A1,A2,Th38;
end;
