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 = min(i,j,m,n) holds ( i < j &
  i < n implies X is BCK-algebra of i,i+1,i,i+1 )
proof
  let X be BCK-algebra of i,j,m,n;
  assume
A1: i = min(i,j,m,n);
  assume that
A2: i < j and
A3: i < n;
  for x,y being Element of X holds Polynom (i,i+1,x,y) = Polynom (i,i+1,y, x)
  proof
    n - i is Element of NAT & n-i>i-i by A3,NAT_1:21,XREAL_1:9;
    then n-i >=1 by NAT_1:14;
    then
A4: n-i+i >= 1+i by XREAL_1:6;
    m >= i by A1,Th25;
    then
A5: m+1 >= i+1 by XREAL_1:6;
    j - i is Element of NAT & j-i>i-i by A2,NAT_1:21,XREAL_1:9;
    then j-i >=1 by NAT_1:14;
    then
A6: j-i+i >= 1+i by XREAL_1:6;
    let x,y be Element of X;
A7: i+1 < n+1 by A3,XREAL_1:6;
    (y,(y\x)) to_power (i+1) = (y,(y\x)) to_power (n+1) by Th19;
    then
A8: (y,(y\x)) to_power (i+1) = (y,(y\x)) to_power (m+1) by A7,A5,Th6;
A9: Polynom (i,j,x,y) = Polynom (m,n,y,x) & (((y,(y\x)) to_power (i+1)),(
    x\y)) to_power (i+1) = (((y,(y\x)) to_power (i+1)),(x\y)) to_power (n+1)
by Def3,Th19;
    (((x,(x\y)) to_power (i+1)),(y\x)) to_power (i+1) = (((x,(x\y))
    to_power (i+1)),(y\x)) to_power (n+1) by Th19;
    then (((x,(x\y)) to_power (i+1)),(y\x)) to_power (i+1) = (((x,(x\y))
    to_power (i+1)),(y\x)) to_power j by A7,A6,Th6;
    hence thesis by A7,A8,A9,A4,Th6;
  end;
  hence thesis by Def3;
end;
