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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_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="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="altcat_4.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E1:67"><i><font color="Green">E41</font></i></a>, <a class="ref" href="altcat_2.html#T34">ALTCAT_2:34</a></i>;<br/>
<a NAME="E4:67"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:67"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E5:67"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E6:67"/><b>then </b><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V3">iso</a> )
 
<b>by </b><i><a class="ref" href="altcat_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:67"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a>  is <a href="altcat_3.html#V3">iso</a>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:67"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a>  <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E6:67"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E9:67"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a>  as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E8:67"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E11:67"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V1">retraction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a><br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:67"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_4.html#E9:67"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E6:67"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>
<a NAME="E12:67"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V2">coretraction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a><br/>

<b>thus </b><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:67"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_4.html#E9:67"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E6:67"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:67"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:67"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_4.html#E9:67"><i><font color="Green">E44</font></i></a>, <a class="txt" href="altcat_4.html#E11:67"><i><font color="Green">E45</font></i></a>, <a class="ref" href="altcat_3.html#T6">ALTCAT_3:6</a></i>;<br/>

<b>thus </b><a NAME="E14:67"/>
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_3.html#K1">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="altcat_4.html#E9:67"><i><font color="Green">E44</font></i></a></i>;<br/>


</div>
