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,y,z,u being Element of X holds (x\(y\z))\(x\(y\u))<=z\u
proof
  let x,y,z,u be Element of X;
  ((x\(y\z))\(x\(y\u)))\((y\u)\(y\z))=0.X by BCIALG_1:1;
  then ((x\(y\z))\(x\(y\u)))<=((y\u)\(y\z));
  then
A1: ((x\(y\z))\(x\(y\u)))\(z\u)<=((y\u)\(y\z))\(z\u) by BCIALG_1:5;
  ((y\u)\(y\z))\(z\u)=((y\u)\(z\u))\(y\z) by BCIALG_1:7;
  then ((x\(y\z))\(x\(y\u)))\(z\u)<=0.X by A1,BCIALG_1:def 3;
  then ((x\(y\z))\(x\(y\u)))\(z\u)\0.X=0.X;
  then ((x\(y\z))\(x\(y\u)))\(z\u)=0.X by BCIALG_1:2;
  hence thesis;
end;
