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(b)
  holds a\x =a\b
proof
  let a,b be Element of AtomSet(X),x be Element of BranchV(b);
  a\b in {x1 where x1 is Element of X:x1 is atom};
  then
A1: ex x1 being Element of X st a\b=x1 & x1 is atom;
  x in {yy where yy is Element of X:b<=yy};
  then
A2: ex yy being Element of X st x=yy & b<= yy;
  (a\x)\(a\b)=(a\(a\b))\x by Th7;
  then (a\x)\(a\b)=b\x by Th24;
  then (a\x)\(a\b)=0.X by A2;
  hence thesis by A1;
end;
