reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;

theorem Th3:
  for X being BCK-algebra,x,y being Element of X holds x\y <= x & (
  x,y) to_power (n+1) <= (x,y) to_power n
proof
  let X be BCK-algebra;
  let x,y be Element of X;
  ((x,y) to_power n\y)\(x,y) to_power n = ((x,y) to_power n\(x,y) to_power
  n)\y by BCIALG_1:7
    .= y` by BCIALG_1:def 5
    .= 0.X by BCIALG_1:def 8;
  then
A1: (x,y) to_power n\y <= (x,y) to_power n;
  (x\y)\x = (x\x)\y by BCIALG_1:7
    .= y` by BCIALG_1:def 5
    .= 0.X by BCIALG_1:def 8;
  hence thesis by A1,BCIALG_2:4;
end;
