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;
reserve B,P for non empty Subset of X;

theorem
  for X being quasi-commutative BCK-algebra holds (X is BCK-algebra of n
  ,n+1,n,n+1 iff for x,y being Element of X holds (x,y) to_power (n+1) = (x,y)
  to_power (n+2) )
proof
  let X be quasi-commutative BCK-algebra;
  thus X is BCK-algebra of n,n+1,n,n+1 implies for x,y being Element of X
  holds (x,y) to_power (n+1) = (x,y) to_power (n+2)
  proof
    assume
A1: X is BCK-algebra of n,n+1,n,n+1;
    for x,y being Element of X holds (x,y) to_power (n+1) = (x,y) to_power
    (n+2)
    proof
      let x,y be Element of X;
A2:   (x\y)\x = (x\x)\y by BCIALG_1:7
        .= y` by BCIALG_1:def 5
        .= 0.X by BCIALG_1:def 8;
      then
A3:   ((x,(x\(x\y))) to_power (n+1),((x\y)\x)) to_power (n+1) = ((x,y)
      to_power (n+1),0.X) to_power (n+1) by BCIALG_2:8
        .= (x,y) to_power (n+1) by BCIALG_2:5;
A4:   (((x\y),((x\y)\x)) to_power (n+1),(x\(x\y))) to_power (n+1) = ((x\y)
      ,(x\(x\y))) to_power (n+1) by A2,BCIALG_2:5
        .= (x,(x\(x\y))) to_power (n+1) \ y by BCIALG_2:7
        .= (x,y) to_power (n+1) \ y by BCIALG_2:8
        .= (x,y) to_power ((n+1)+1) by BCIALG_2:4;
      Polynom (n,n+1,x,(x\y)) = Polynom (n,n+1,(x\y),x) by A1,Def3;
      hence thesis by A3,A4;
    end;
    hence thesis;
  end;
  assume
A5: for x,y being Element of X holds (x,y) to_power (n+1) = (x,y)
  to_power (n+2);
  for x,y being Element of X holds Polynom (n,n+1,x,y) = Polynom (n,n+1,y, x)
  proof
    let x,y be Element of X;
    (x\y)\(x\y) = 0.X by BCIALG_1:def 5;
    then (x\(x\y))\y = 0.X by BCIALG_1:7;
    then x\(x\y) <= y;
    then ((x\(x\y)),(x\y)) to_power (n+1) <= (y,(x\y)) to_power (n+1) by
BCIALG_2:19;
    then
    ((x,(x\y)) to_power 1,(x\y)) to_power (n+1) <= (y,(x\y)) to_power (n+
    1) by BCIALG_2:2;
    then
A6: (x,(x\y)) to_power (1+(n+1)) <= (y,(x\y)) to_power (n+1) by BCIALG_2:10;
    (y\x)\(y\x) = 0.X by BCIALG_1:def 5;
    then (y\(y\x))\x = 0.X by BCIALG_1:7;
    then y\(y\x) <= x;
    then ((y\(y\x)),(y\x)) to_power (n+1) <= (x,(y\x)) to_power (n+1) by
BCIALG_2:19;
    then
    ((y,(y\x)) to_power 1,(y\x)) to_power (n+1) <= (x,(y\x)) to_power (n+
    1) by BCIALG_2:2;
    then
A7: (y,(y\x)) to_power (1+(n+1)) <= (x,(y\x)) to_power (n+1) by BCIALG_2:10;
    (y,(y\x)) to_power (n+1) = (y,(y\x)) to_power (n+2) by A5;
    then ((y,(y\x)) to_power (n+1),(x\y)) to_power (n+1) <= ((x,(y\x))
    to_power (n+1),(x\y)) to_power (n+1) by A7,BCIALG_2:19;
    then
A8: ((y,(y\x)) to_power (n+1),(x\y)) to_power (n+1) <= ((x,(x\y))
    to_power (n+1),(y\x)) to_power (n+1) by BCIALG_2:11;
    (x,(x\y)) to_power (n+1) = (x,(x\y)) to_power (n+2) by A5;
    then ((x,(x\y)) to_power (n+1),(y\x)) to_power (n+1) <= ((y,(x\y))
    to_power (n+1),(y\x)) to_power (n+1) by A6,BCIALG_2:19;
    then
    ((x,(x\y)) to_power (n+1),(y\x)) to_power (n+1) <= ((y,(y\x)) to_power
    (n+1),(x\y)) to_power (n+1) by BCIALG_2:11;
    hence thesis by A8,Th2;
  end;
  hence thesis by Def3;
end;
