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 be BCI-Algebra_with_Condition(S),x,y,u,v being Element of X st u
  in Condition_S(x,y) & v <= u holds v in Condition_S(x,y)
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,y,u,v be Element of X;
  assume that
A1: u in Condition_S(x,y) and
A2: v <= u;
  v\x <= u\x by A2,BCIALG_1:5;
  then
A3: (v\x)\(u\x) = 0.X;
  ex u1 being Element of X st u=u1 & u1\x <= y by A1;
  then (u\x)\y = 0.X;
  then (v\x)\y = 0.X by A3,BCIALG_1:3;
  then v\x <= y;
  hence thesis;
end;
