reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem
  for X being BCI-Algebra_with_Condition(S) holds for x,y being Element
  of X holds ex a be Element of Condition_S(x,y) st for z being Element of
  Condition_S(x,y) holds z <= a
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,y be Element of X;
  ((x*y)\x)\y = (x*y)\(x*y) by Def2
    .= 0.X by BCIALG_1:def 5;
  then (x*y)\x <= y;
  then (x*y) in {t where t is Element of X: t\x <= y};
  then consider u being Element of Condition_S(x,y) such that
A1: u =x * y;
  take u;
  for z being Element of Condition_S(x,y) holds z <= u
  proof
    let z be Element of Condition_S(x,y);
    z in {t where t is Element of X: t\x <= y};
    then consider z1 being Element of X such that
A2: z=z1 and
A3: z1\x <= y;
    z\u = (z1\x)\y by A1,A2,Def2
      .= 0.X by A3;
    hence thesis;
  end;
  hence thesis;
end;
