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 Th9:
  Polynom (m+1,n,x,y) = Polynom (m,n,x,y) \ (x\y)
proof
  Polynom (m+1,n,x,y) = (((x,(y\x)) to_power n),(x\y)) to_power (m+1+1) by
BCIALG_2:11
    .= ((((x,(y\x)) to_power n),(x\y)) to_power (m+1)) \ (x\y) by BCIALG_2:4
    .= (((x,(x\y)) to_power (m+1)),(y\x)) to_power n \ (x\y) by BCIALG_2:11;
  hence thesis;
end;
