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;
reserve X for BCK-algebra;
reserve X for BCI-algebra;

theorem
  (for X being BCI-algebra,x,y,z being Element of X holds (x\y)\y=(x\z)\
  (y\z)) implies the carrier of X = BCK-part(X)
proof
  assume
  for X being BCI-algebra,x,y,z being Element of X holds (x\y)\y=(x\z) \(y\z);
  then X is BCK-algebra by BCIALG_1:15;
  hence thesis by Th25;
end;
