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 Th27:
  for X being BCK-algebra of i,j,m,n st i = min(i,j,m,n) holds ( i
  = j implies X is BCK-algebra of i,i,i,i )
proof
  let X be BCK-algebra of i,j,m,n;
  assume
A1: i = min(i,j,m,n);
  assume
A2: i = j;
A3: for x,y being Element of X holds Polynom (i,i,x,y) <= Polynom (i,i,y,x)
  proof
    let x,y be Element of X;
    i<= n by A1,Th25;
    then
A4: ((y,(y\x)) to_power (i+1),(x\y)) to_power n <= ((y,(y\x)) to_power (i+
    1),(x\y)) to_power i by Th5;
    i<= m by A1,Th25;
    then i+1 <= m+1 by XREAL_1:6;
    then
A5: ((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) by Def3;
    hence thesis by A2,A5,A4,Th1;
  end;
  for x,y being Element of X holds Polynom (i,i,y,x) = Polynom (i,i,x,y)
  proof
    let x,y be Element of X;
    Polynom (i,i,x,y) <= Polynom (i,i,y,x) by A3;
    then
A6: Polynom (i,i,x,y)\Polynom (i,i,y,x)=0.X;
    Polynom (i,i,y,x) <= Polynom (i,i,x,y) by A3;
    then Polynom (i,i,y,x)\Polynom (i,i,x,y)=0.X;
    hence thesis by A6,BCIALG_1:def 7;
  end;
  hence thesis by Def3;
end;
