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 Th6:
  for X being BCK-algebra, x,y being Element of X st m>n & (x,y)
to_power n = (x,y) to_power m holds for k being Nat st k >=n holds (
  x,y) to_power n = (x,y) to_power k
proof
  let X be BCK-algebra;
  let x,y be Element of X;
  assume that
A1: m>n and
A2: (x,y) to_power n = (x,y) to_power m;
  m - n is Element of NAT & m-n>n-n by A1,NAT_1:21,XREAL_1:9;
  then m-n >=1 by NAT_1:14;
  then m-n+n >= 1+n by XREAL_1:6;
  then
A3: (x,y) to_power n <= (x,y) to_power (n+1) by A2,Th5;
A4: (x,y) to_power (n+1) <= (x,y) to_power n by Th3;
  for k being Nat st k >=n holds (x,y) to_power n = (x,y) to_power k
  proof
    let k be Nat;
    assume k >=n;
    then k - n is Element of NAT by NAT_1:21;
    then consider k1 being Element of NAT such that
A5: k1=k-n;
    (x,y) to_power n = ((x,y) to_power n,y) to_power k1
    proof
      defpred P[Nat] means
$1<= k1 implies (x,y) to_power n = ((x,y
      ) to_power n,y) to_power $1;
      now
        let k;
        assume
A6:     k<= k1 implies (x,y) to_power n = ((x,y) to_power n,y) to_power k;
        set m=k+1;
A7:     ((x,y) to_power n,y) to_power m = ((x,y) to_power n,y) to_power k
        \y by BCIALG_2:4
          .= ((x,y) to_power n\y,y) to_power k by BCIALG_2:7
          .= ((x,y) to_power (n+1),y) to_power k by BCIALG_2:4
          .= ((x,y) to_power n,y) to_power k by A4,A3,Th2;
        assume m<=k1;
        hence (x,y) to_power n = ((x,y) to_power n,y) to_power m by A6,A7,
NAT_1:13;
      end;
      then
A8:   for k st P[k] holds P[k+1];
A9:   P[0] by BCIALG_2:1;
      for n holds P[n] from NAT_1:sch 2(A9,A8);
      hence thesis;
    end;
    then (x,y) to_power n = (x,y) to_power (n+k1) by BCIALG_2:10;
    hence thesis by A5;
  end;
  hence thesis;
end;
