Q:

Proof of $p\rightarrow(q\rightarrow r)\vdash_{HR} (p\rightarrow q)\rightarrow r$

Let HR be an Hilbert style proof system:  
Inference rule: MP
Axiom schemes:

$A\rightarrow A$
$(A\rightarrow B)\rightarrow ((B\rightarrow C)\rightarrow(A\rightarrow C))$
$(A\rightarrow(B\rightarrow C))\rightarrow(B\rightarrow(A\rightarrow C))$
$(A\rightarrow(A\rightarrow B))\rightarrow B$

Prove formally that $p\rightarrow(q\rightarrow r)\vdash_{HR} (p\rightarrow q)\rightarrow r$
I've seen this similar question:
Proof for $\{p,p\rightarrow (q\rightarrow r)\}\vdash (p\rightarrow q)\rightarrow r$ in HR
However, they use $p$ as an additional assumption, and I don't know how to prove it without that assumption

A:

$(p\rightarrow q)\rightarrow ((q\rightarrow r)\rightarrow(p\rightarrow r))$ - Axiom 2
$(q\rightarrow r)\rightarrow ((p\rightarrow q)\rightarrow(p\rightarrow r))$ - (1) and axiom 3, MP
$p\rightarrow(q\rightarrow r)$ - Assumption
$p\rightarrow ((p\rightarrow q)\rightarrow(p\rightarrow r))$ - (3), (2) and axiom 2, 2 MPs
$((p\rightarrow q)\rightarrow (p\rightarrow r))\rightarrow(p\rightarrow ((p\rightarrow q)\rightarrow r))$ - Axiom 3
$p\rightarrow (p\rightarrow ((p\rightarrow q)\rightarrow r))$ - (4), (5) and axiom 2, 2 MPs
$(p\rightarrow q)\rightarrow r$ - (6) and axiom 4, MP

