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 ( m = 0 & j > 0 implies X is
  BCK-algebra of 0,1,0,1 )
proof
  let X be BCK-algebra of i,j,m,n;
  reconsider X as BCK-algebra of i+1,j,m,n+1 by Th17;
  assume that
A1: m = 0 and
A2: j > 0;
  for x,y being Element of X holds Polynom (0,1,x,y) = Polynom (0,1,y,x)
  proof
    let x,y be Element of X;
A3: (i+1)+1 > (m+1)+0 by A1,XREAL_1:8;
A4: (((x,(x\y)) to_power (0+1)),(y\x)) to_power (j+1) = (((x,(x\y))
    to_power (0+1)),(y\x)) to_power (m+1) by Th20;
A5: j+1 > m+1 by A1,A2,XREAL_1:6;
    n+1 >= m+1 & (((y,(y\x)) to_power (0+1)),(x\y)) to_power (j+1) = (((y,
    (y\x)) to_power (0+1)),(x\y)) to_power (m+1) by A1,Th20,XREAL_1:6;
    then
A6: (((y,(y\x)) to_power (0+1)),(x\y)) to_power (0+1) = (((y,(y\x))
    to_power (0+1)),(x\y)) to_power (n+1) by A1,A5,Th6;
    Polynom (i+1,j,x,y) = Polynom (m,n+1,y,x) & (x,(x\y)) to_power (j+1) =
    (x,(x\ y)) to_power (m+1) by Def3,Th20;
    then
    (((x,(x\y)) to_power (0+1)),(y\x)) to_power j = (((y,(y\x)) to_power (0
    +1)),(x\y)) to_power (n+1) by A1,A5,A3,Th6;
    hence thesis by A1,A5,A6,A4,Th6,NAT_1:14;
  end;
  hence thesis by Def3;
end;
