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

<a NAME="E1:23_1"/><i><font color="Green">E48</font></i>: 
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="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>) is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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>();<br/>

<a NAME="E1:23_1_1"/>
( <font color="Maroon">a</font> is   <a href="waybel_0.html#NM8">LATTICE</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:23_1_1"/>
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>) is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:23_1"/><i><font color="Green">E49</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>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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>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="E1:23_1_2"/><i><font color="Green">E50</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="E2:23_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:23_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:23_1_2"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) is   <a href="struct_0.html#NM6">Function</a> of <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>) &amp; <font color="Maroon">P<sub>2</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>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>)] )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:23_1_2"/>
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <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>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>now </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>(), <font color="Maroon">c</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="E1:23_1_3"/><i><font color="Green">E53</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><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>(), <font color="Maroon">c'</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="E2:23_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c</font>) )
 ;<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/><b>let </b><font color="Maroon">g'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">c'</font>;<br/><b>assume </b><a NAME="E3:23_1_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>) )
 ;<br/><a NAME="E4:23_1_3"/><i><font color="Green">E70</font></i>: 
(  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:23_1_3"/>
(  <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> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <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>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c</font>) )
 <b>by </b><i>, </i>;<br/><a NAME="E6:23_1_3"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</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> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</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">b'</font>,<font color="Maroon">c'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i/>;<br/><a NAME="E7:23_1_3"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> )
 <b>by </b><i/>;<br/><a NAME="E8:23_1_3"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>] )
 <b>by </b><i>, , </i>;<br/><b>then </b><font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font></span>)</span></span>)</span>) = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>)
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><a NAME="E10:23_1_3"/>
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4"><b>now </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>();<br/><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>();<br/><b>assume </b><a NAME="E1:23_1_4"/>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>)
 ;<br/><a NAME="E2:23_1_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>) &amp;  <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="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span>) = 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a'</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being  <a href="functor0.html#V9">strict</a> <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>()<b> such that </b><br/><a NAME="E6:23_1"/><i><font color="Green">E76</font></i>: 
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="Maroon">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>)
 <b>and </b><br/><a NAME="E7:23_1"/><i><font color="Green">E77</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</font> <a href="functor0.html#K6">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S8">YELLOW18:sch 8</a>(, , , );<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E8:23_1"/>
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="Maroon">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>(<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>)
 <b>by </b><i/>;<br/>

<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="E9:23_1"/><i><font color="Green">E78</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="E10:23_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
<font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font></span>)</span>)
 <b>by </b><i>, </i>;<br/>


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