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 BCK-algebra of i,j,m,n holds ( j = 0 & m > 0 implies X is
  BCK-algebra of 0,0,0,0 )
proof
  let X be BCK-algebra of i,j,m,n;
  assume that
A1: j = 0 and
A2: m > 0;
  for x,y being Element of X holds Polynom (0,0,x,y) = Polynom (0,n,y,x)
  proof
    let x,y be Element of X;
A3: Polynom (i,j,x,y) = Polynom (m,n,y,x) by Def3;
A4: i+1 >= j+1 by A1,XREAL_1:6;
    (x,(x\y)) to_power (j+1) = (x,(x\y)) to_power (m+1) & j+1 < m+1 by A1,A2
,Th20,XREAL_1:6;
    then (x,(x\y)) to_power (i+1) = (x,(x\y)) to_power (0+1) by A1,A4,Th6;
    hence thesis by A1,A3,Th20;
  end;
  then reconsider X as BCK-algebra of 0,0,0,n by Def3;
A5: for x,y being Element of X holds Polynom (0,0,x,y) <= Polynom (0,0,y,x)
  proof
    let x,y be Element of X;
    Polynom (0,0,x,y) = Polynom (0,n,y,x) by Def3;
    hence thesis by Th5;
  end;
  for x,y being Element of X holds Polynom (0,0,y,x) = Polynom (0,0,x,y)
  proof
    let x,y be Element of X;
    Polynom (0,0,x,y) <= Polynom (0,0,y,x) by A5;
    then
A6: Polynom (0,0,x,y)\Polynom (0,0,y,x)=0.X;
    Polynom (0,0,y,x) <= Polynom (0,0,x,y) by A5;
    then Polynom (0,0,y,x)\Polynom (0,0,x,y)=0.X;
    hence thesis by A6,BCIALG_1:def 7;
  end;
  hence thesis by Def3;
end;
