reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;

theorem
  for x,a,b being Element of AtomSet(X) holds x is Element of BranchV(b)
  implies a\x=a\b
proof
  set P = AtomSet(X);
  let x,a,b be Element of P;
  set B = BranchV(b);
  assume x is Element of B;
  then x in B;
  then ex x3 being Element of X st x=x3 & b<=x3;
  then
A1: b\x=0.X;
  x in {x1 where x1 is Element of X:x1 is minimal};
  then ex x1 being Element of X st x=x1 & x1 is minimal;
  hence thesis by A1;
end;
