<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#R1" target="_self">are_isomorphic</a>  implies (  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E42</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" target="_self">CAT_1:def 17</a><br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><a NAME="E2:67_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E3:67_1"/>
(  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:67_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_1.html#E2:67_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self">Th70</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E4:67_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:67_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_1.html#E2:67_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self">Th70</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:67"/><i><font color="Green">E42</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:67"/><i><font color="Green">E43</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><a NAME="E4:67"/><i><font color="Green">E44</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E5:67"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_1.html#E2:67"><i><font color="Green">E42</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" target="_self">CAT_1:def 17</a><br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E6:67"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:67"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_1.html#E3:67"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cat_1.html#E4:67"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self">Th70</a></i>;<br/>


</div>
