  In this paper we prove an analogue of the uniqueness theorems of F\"uhrer and
Amann and Weiss to cover the degree of Fredholm operators of index zero of
Fitzpatrick, Pejsachowicz and Rabier, whose range of applicability is
substantially wider than for the most classical degrees of Brouwer and
Leray-Schauder. A crucial step towards the axiomatization of the
Fitzpatrick-Pejsachowicz-Rabier degree is provided by the generalized algebraic
multiplicity of Esquinas and L\'{o}pez-G\'{o}mez, $\chi$, and the
axiomatization theorem of Mora-Corral. The latest result facilitates the
axiomatization of the parity of Fitzpatrick and Pejsachowicz,
$\sigma(\cdot,[a,b])$, which provides the key step for establishing the
uniqueness of the degree for Fredholm maps.
