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