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
  Polynom (n,n+1,x,y) <= Polynom (n,n,y,x)
proof
  (x\y)\(x\y) = 0.X by BCIALG_1:def 5;
  then (x\(x\y))\y = 0.X by BCIALG_1:7;
  then (x\(x\y)) <= y;
  then ((x\(x\y)),(x\y)) to_power n <= (y,(x\y)) to_power n by BCIALG_2:19;
  then (((x\(x\y)),(x\y)) to_power n,(y\x)) to_power (n+1) <= ((y,(x\y))
  to_power n,(y\x)) to_power (n+1) by BCIALG_2:19;
  then
  ((((x,(x\y)) to_power 1),(x\y)) to_power n,(y\x)) to_power (n+1) <= ((y,
  (x\y)) to_power n,(y\x)) to_power (n+1) by BCIALG_2:2;
  then
  ((x,(x\y)) to_power (n+1),(y\x)) to_power (n+1) <= ((y,(x\y)) to_power n
  ,(y\x)) to_power (n+1) by BCIALG_2:10;
  hence thesis by BCIALG_2:11;
end;
