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
  for X being BCK-algebra of i,j,m,n st i <= m & j <= n holds X is
  BCK-algebra of i,j,i,j
proof
  let X be BCK-algebra of i,j,m,n;
  assume that
A1: i <= m and
A2: j <= n;
A3: for x,y being Element of X holds Polynom (i,j,x,y) <= Polynom (i,j,y,x)
  proof
    let x,y be Element of X;
    i+1 <= m+1 by A1,XREAL_1:6;
    then
A4: ((y,(y\x)) to_power (m+1),(x\y)) to_power n <= ((y,(y\x)) to_power (i+1
    ),(x\y)) to_power n by Th5,BCIALG_2:19;
    Polynom (i,j,x,y) = Polynom (m,n,y,x) & ((y,(y\x)) to_power (i+1),(x\y
)) to_power n <= ((y,(y\x)) to_power (i+1),(x\y)) to_power j by A2,Def3,Th5;
    hence thesis by A4,Th1;
  end;
  for x,y being Element of X holds Polynom (i,j,y,x) = Polynom (i,j,x,y)
  proof
    let x,y be Element of X;
    Polynom (i,j,x,y) <= Polynom (i,j,y,x) by A3;
    then
A5: Polynom (i,j,x,y)\Polynom (i,j,y,x)=0.X;
    Polynom (i,j,y,x) <= Polynom (i,j,x,y) by A3;
    then Polynom (i,j,y,x)\Polynom (i,j,x,y)=0.X;
    hence thesis by A5,BCIALG_1:def 7;
  end;
  hence thesis by Def3;
end;
