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 = n & m < j holds X is
  BCK-algebra of i,m+1,m,i
proof
  let X be BCK-algebra of i,j,m,n;
  assume that
A1: i = n and
A2: m < j;
  for x,y being Element of X holds Polynom (i,m+1,x,y) = Polynom (m,i,y,x)
  proof
    j - m is Element of NAT & j-m>m-m by A2,NAT_1:21,XREAL_1:9;
    then j-m >=1 by NAT_1:14;
    then
A3: j-m+m >= 1+m by XREAL_1:6;
    let x,y be Element of X;
A4: m+1 < j+1 by A2,XREAL_1:6;
    Polynom (i,j,x,y) = Polynom (m,n,y,x) & (((x,(x\y)) to_power (i+1)),(y
\x)) to_power (j+1) = (((x,(x\y)) to_power (i+1)),(y\x)) to_power (m+1) by Def3
,Th20;
    hence thesis by A1,A4,A3,Th6;
  end;
  hence thesis by Def3;
end;
