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 Th5:
  for X being BCK-algebra,x,y being Element of X st m>=n
   holds (x,y) to_power m <= (x,y) to_power n
proof
  let X be BCK-algebra;
  let x,y be Element of X;
  assume m>=n;
  then m - n is Element of NAT by NAT_1:21;
  then consider k being Element of NAT such that
A1: k=m-n;
  ((x,y) to_power k)\x = (x\x,y) to_power k by BCIALG_2:7
    .= (0.X,y) to_power k by BCIALG_1:def 5
    .= 0.X by Th4;
  then (x,y) to_power k <= x;
  then ((x,y) to_power k,y) to_power n <= (x,y) to_power n by BCIALG_2:19;
  then (x,y) to_power (k+n) <= (x,y) to_power n by BCIALG_2:10;
  hence thesis by A1;
end;
