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 holds ( i <= m & i < n implies X is
  BCK-algebra of i,j,i,i+1 )
proof
  let X be BCK-algebra of i,j,m,n;
  assume that
A1: i <= m and
A2: i < n;
  for x,y being Element of X holds Polynom (i,j,x,y) = Polynom (i,i+1,y,x)
  proof
    n - i is Element of NAT & n-i>i-i by A2,NAT_1:21,XREAL_1:9;
    then n-i >=1 by NAT_1:14;
    then
A3: n-i+i >= 1+i by XREAL_1:6;
    let x,y be Element of X;
A4: i+1 < n+1 by A2,XREAL_1:6;
A5: Polynom (i,j,x,y) = Polynom (m,n,y,x) & (((y,(y\x)) to_power (m+1)),(x
\y)) to_power (i+1) = (((y,(y\x)) to_power (m+1)),(x\y)) to_power (n+1) by Def3
,Th19;
    (y,(y\x)) to_power (i+1) = (y,(y\x)) to_power (n+1) & m+1 >= i+1 by A1,Th19
,XREAL_1:6;
    then (y,(y\x)) to_power (i+1) = (y,(y\x)) to_power (m+1) by A4,Th6;
    hence thesis by A4,A5,A3,Th6;
  end;
  hence thesis by Def3;
end;
