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

<a NAME="E1:31_1"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">F</font> being <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>( ( for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Olive">F</font> <a href="functor0.html#K6">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E2:31_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">G</font> being <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>1</sub></font>() st <br/>( ( for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Olive">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Olive">G</font> <a href="functor0.html#K6">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:31_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>)) holds <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b</font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b</font>) <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b</font>,<font color="Olive">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b</font>) is <a href="funct_1.html#V2">one-to-one</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>))
 ;<br/>

<a NAME="E2:31_1_1"/><i><font color="Green">E54</font></i>: 
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>)),<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">b</font>) &amp; <font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>)), <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font>,<font color="Maroon">f</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>)),<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ] )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, , , </i>;<br/>

<a NAME="E6:31_1_1"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>)) st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Olive">g</font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">b</font>) <a href="funct_1.html#K2">"</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">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, , , </i>;<br/>

<b>thus </b><a NAME="E8:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">b</font>) is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>)) holds <br/>( <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">a</font>) <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b</font>,<font color="Olive">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">a</font>) is <a href="funct_1.html#V2">one-to-one</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>))
 ;<br/>

<a NAME="E2:31_1_2"/><i><font color="Green">E74</font></i>: 
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>))<b> such that </b><br/><a NAME="E4:31_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; <font color="Maroon">P<sub>2</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>)),<font color="Maroon">f</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>)), <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ] )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">a</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, , , </i>;<br/>

<a NAME="E6:31_1_2"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>)),<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Olive">g</font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">a</font>) <a href="funct_1.html#K2">"</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">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, , , </i>;<br/>

<b>thus </b><a NAME="E8:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">a</font>) is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:31_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)) <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">a</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:31_1_3"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E2:31_1_3"/>
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_1_3"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">b</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">a</font>)
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><a NAME="E1:31_1_4"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E2:31_1_4"/>
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_1_4"/>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>)) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">b</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:31_1"/>
<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <a href="yellow18.html#R1">are_equivalent</a> 
 <b>from </b><i><a class="ref" href="yellow18.html#S22">YELLOW18:sch 22</a>(, , , , , );</i><br/>


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