Theorem: Function that Satisfies Axioms of Uncertainty

Let $n \in \N$ be a natural number.
Let $p_1, p_2, \dotsc, p_n$ be real numbers such that:
:$\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$
:$\ds \sum_{i \mathop = 1}^n p_i = 1$
Let $\map H {p_1, p_2, \ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty.
Then:
:$\ds \map H {p_1, p_2, \ldots, p_n} = -\lambda \sum_{i \mathop = 1}^n p_i \log_b p_i$
where:
:$\lambda \in \R_{>0}$
:$b \in \R_{>1}$

Proof:
Let $g: \Z \to \R$ be the mapping defined as:
:$(1): \quad \map g n = \map H {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n}$
Let $k \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = \map g {n^k}
      | r = \map g {n \times n^{k - 1} }
      | c = 
}}
{{eqn | r = \map g n + \map g {n^{k - 1} }
      | c = Axiom 7
}}
{{eqn | r = \map g n + \map g n + \map g {n^{k - 2} }
      | c = Axiom 7 again
}}
{{eqn | r = \underbrace {\map g n + \map g n + \dotsb + \map g n}_{\text {$k - 1$ times} } + \map g {n^{k - \paren {k - 1} } }
      | c = after $k - 1$ applications of Axiom 7
}}
{{eqn | r = \underbrace {\map g n + \map g n + \dotsb + \map g n}_{\text {$k - 1$ times} } + \map g n
      | c = 
}}
{{eqn | n = 2
      | r = k \map g n
      | c = 
}}
{{end-eqn}}
Let $r, s \in \R$ and $n \in \Z_{>0}$.
Let $m$ satisfy:
:$(3): \quad r^m \le s^n \le r^{m + 1}$
From $(2)$ and Axiom 5:
:$\map g {r^m} \le \map g {s^n} \le \map g {r^{m + 1} }$
and so:
:$m \map g r \le n \map g s \le \paren {m + 1} \map g r$
Taking logarithms of $(3)$:
:$m \ln r \le n \ln s \le \paren {m + 1} \ln r$
which holds because Logarithm is Strictly Increasing.
Hence:
:$\size {\dfrac {\map g s} {\map g r} - \dfrac {\map \ln s} {\map \ln r} } \le \dfrac 1 n$
We have that $n$ is an arbitrary positive integer.
Hence the {{RHS}} of the above can be made arbitrarily small.
Hence:
:$\size {\dfrac {\map g s} {\map g r} - \dfrac {\map \ln s} {\map \ln r} } = 0$
and so:
:$(4): \quad \dfrac {\map g s} {\map \ln s} = \dfrac {\map g r} {\map \ln r} = A$
for some constant $A$.
That is:
:$(5): \quad \map g s = A \map \ln s$
for $s \in \Z$.
Let $p$ be a rational number such that $0 < p < 1$.
That is:
:$p = \dfrac t n$
for some integers $t$ and $n$.
Thus we can set:
:$q = 1 = p = \dfrac {n - t} n$
Axiom 8 gives:
:$\map g n = \map H {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n} = \map H {\dfrac 1 n, \dfrac {n - t} n} + \dfrac t n \map g t + \dfrac {n - t} n \map g {n - t}$
Using $(5)$ and gathering up terms:
:$\map H {\dfrac 1 n, \dfrac {n - t} n} = A \paren {\dfrac t n} \ln \dfrac t n + A \paren {\dfrac {n - t} n} \ln \dfrac {n - t} n$
which gives:
:$(6): \quad \map H {p, 1 - p} = A p \ln p + A \paren {1 - p} \, \map \ln {1 - p}$
for rational $p$.
Axiom 6 gives that $H$ is continuous.
Thus $(6)$ extends to all real $p$ such that $0 < p < 1$.
It remains to be demonstrated that:
:$(7): \quad \map H {p_1, p_2, \ldots, p_N} = A \displaystyle \sum_{i \mathop = 1}^N p_i \ln p_i$
where:
:$0 < p_i < 1$ for all $p_i$
:$p_1 + p_2 + \dotsb + p_N$
for all $N \in \Z_{\ge 2}$.
We have demonstrated that $(7)$ is true for $N = 2$.
Assume the induction hypothesis that $(7)$ holds for $N = k$.
Consider:
:$\map H {p_1, p_2, \ldots, p_k, p_{k + 1} }$
Let:
:$p = p_1 + p_2 + \dotsb + p_k$
:$q = p_{k + 1}$
and apply Axiom 8:
{{begin-eqn}}
{{eqn | l = \map H {p_1, p_2, \ldots, p_k, p_{k + 1} }
      | r = \map H {p, q} + p \map H {\dfrac {p_1} p, \dfrac {p_2} p, \dotsc, \dfrac {p_k} p} + q \map H 1
      | c = 
}}
{{eqn | r = A p \ln p + A q \ln q + p A \sum_{i \mathop = 1}^k \dfrac {p_i} p \ln \dfrac {p_i} p
      | c = from the induction hypothesis
}}
{{eqn | r = A p \ln p + A p_{k + 1} \ln p_{k + 1} + A \sum_{i \mathop = 1}^k p_i \paren {\ln p_i - \ln p}
      | c = Difference of Logarithms
}}
{{eqn | r = A p \ln p + A p_{k + 1} \ln p_{k + 1} + A \sum_{i \mathop = 1}^k p_i \ln p_i - A p \ln p
      | c = as $A \displaystyle \sum_{i \mathop = 1}^k p_i = p$
}}
{{eqn | ll= \leadsto
      | l = \map H {p_1, p_2, \ldots, p_k, p_{k + 1} }
      | r = A \sum_{i \mathop = 1}^{k + 1} p_i \ln p_i
      | c = 
}}
{{end-eqn}}
Hence the result from the Principle of Mathematical Induction.
{{qed}}