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,v being Element of X holds (x\(y\(z\u)))\(x\(y\(z\v)))<=v\ u
proof
  let x,y,z,u,v be Element of X;
  (x\(y\(z\u)))\(x\(y\(z\v)))\((y\(z\v))\(y\(z\u)))=0.X by BCIALG_1:1;
  then (x\(y\(z\u)))\(x\(y\(z\v))) <=((y\(z\v))\(y\(z\u)));
  then
A1: (x\(y\(z\u)))\(x\(y\(z\v)))\((z\u)\(z\v)) <=((y\(z\v))\(y\(z\u)))\((z\u)\
  (z\v)) by BCIALG_1:5;
  ((y\(z\v))\(y\(z\u)))\((z\u)\(z\v))=((y\(z\v))\((z\u)\(z\v)))\(y\(z\u))
  by BCIALG_1:7;
  then (x\(y\(z\u)))\(x\(y\(z\v)))\((z\u)\(z\v))<=0.X by A1,BCIALG_1:def 3;
  then (x\(y\(z\u)))\(x\(y\(z\v)))\((z\u)\(z\v))\0.X=0.X;
  then (x\(y\(z\u)))\(x\(y\(z\v)))\((z\u)\(z\v))=0.X by BCIALG_1:2;
  then (x\(y\(z\u)))\(x\(y\(z\v))) <= (z\u)\(z\v);
  then
A2: (x\(y\(z\u)))\(x\(y\(z\v)))\(v\u)<= ((z\u)\(z\v))\(v\u) by BCIALG_1:5;
  ((z\u)\(z\v))\(v\u)=((z\u)\(v\u))\(z\v) by BCIALG_1:7
    .=0.X by BCIALG_1:def 3;
  then (x\(y\(z\u)))\(x\(y\(z\v)))\(v\u)\0.X=0.X by A2;
  then (x\(y\(z\u)))\(x\(y\(z\v)))\(v\u)=0.X by BCIALG_1:2;
  hence thesis;
end;
