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

<b>consider </b><font color="Maroon">O</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E3:9_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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>() holds <br/><font color="Maroon">O</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:9_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">O</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">O</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, </i>;<br/>
<a NAME="E5:9_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:9_1_1"/>
<font color="Maroon">y</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>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> = <font color="Maroon">O</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">OM</font> = <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">O</font></span><a href="funct_3.html#K16">:]</a></span> as   <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<a NAME="E7:9_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_2"/><i><font color="Green">E54</font></i>: 
( <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; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:9_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E6:9_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:9_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</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>
 
;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>);<br/>

<a NAME="E9:9_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</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>3</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:9_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">OM</font>
 
<b>by </b><i>, , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">OM</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
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> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E12:9_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>

<a NAME="E13:9_1_2"/>
( <font color="Maroon">i</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">i</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E14:9_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">i</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font><b> such that </b><br/><a NAME="E9:9_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as    <a href="functor0.html#M1">MSUnTrans</a> of <font color="Maroon">OM</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="functor0.html#D5">FUNCTOR0:def 5</a></i>;<br/>
<a NAME="E11:9_1"/>
 <a href="functor0.html#G2">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">M</font> #) is <a href="functor0.html#V10">Covariant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">O</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D2">FUNCTOR0:def 2</a>,<a class="ref" href="functor0.html#D13">FUNCTOR0:def 13</a><br/>

<b>thus </b><a NAME="E1:9_1_3"/>
the <a href="functor0.html#U1">ObjectMap</a> of <a href="functor0.html#G2">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">M</font> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">O</font></span><a href="funct_3.html#K16">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="functor0.html#G2">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">M</font> #) as  <a href="functor0.html#V10">Covariant</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() ;<br/>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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>thus </b><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>)
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">o2</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="E1:9_1_5"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><a NAME="E2:9_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> <font color="Maroon">OM</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span><b> such that </b><br/><a NAME="E4:9_1_5"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:9_1_5"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Olive">x</font>)
 <b>by </b><i/>;<br/><i><font color="Green">E107</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">g</font>)
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<font color="Maroon">g</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">g</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><a NAME="E7:9_1_5"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:9_1_5"/>
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">g</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E15:9_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V8">feasible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a><br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/>
<b>assume </b><a NAME="E2:9_1_6"/><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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/>

<a NAME="E3:9_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">g</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <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">o1</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o2</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:9_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o2</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E6:9_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:9_1_6"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span></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/>


</div><b>end;</b></div>
<div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">o2</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="E1:9_1_7"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><a NAME="E2:9_1_7"/>
( <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">g</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span></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>  )
 <b>by </b><i>, , , <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/><b>hence </b><a NAME="E3:9_1_7"/>
<font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">g</font>)
 <b>by </b><i>, <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:9_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V13">comp-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D22">FUNCTOR0:def 22</a><br/>

<b>assume </b><a NAME="E1:9_1_8"/><i><font color="Green">E113</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">o2</font>,<font color="Maroon">o3</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">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>



<b>set </b><font color="Maroon">a</font> = <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font>;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o3</font>;<br/>
<a NAME="E2:9_1_8"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o1</font>) &amp; <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o2</font>) &amp; <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o3</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:9_1_8"/><b>then </b><i><font color="Green">E115</font></i>: 
( 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">o1</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o2</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; 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">o2</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o3</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o3</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">f</font>) as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o2</font>,<font color="Maroon">o3</font>,<font color="Maroon">g</font>) as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o3</font></span>)</span> <b>by </b><i>, , </i>;<br/>
<a NAME="E6:9_1_8"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">O</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o3</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f'</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">gg</font> = <font color="Maroon">g'</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o3</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>take </b>
<font color="Maroon">gg</font>
;<br/>

<a NAME="E9:9_1_8"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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> 
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E10:9_1_8"/>
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o1</font>,<font color="Maroon">o3</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">gg</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">ff</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E11:9_1_8"/>
( <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <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">gg</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">ff</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9_1"/>
<font color="Maroon">F</font> is    <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1_9"/>
<font color="Maroon">F</font> is <a href="functor0.html#V8">feasible</a>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D26">FUNCTOR0:def 26</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D21">FUNCTOR0:def 21</a>
<div class="add"><b>let </b><font color="Maroon">o</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:9_1_9_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">o</font>)
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o</font>,<font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">o</font>,<font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span>)
<b>by </b><i/>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:9_1_9"/>
( ( <font color="Maroon">F</font> is <a href="functor0.html#V10">Covariant</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V13">comp-preserving</a> ) or ( <font color="Maroon">F</font> is <a href="functor0.html#V11">Contravariant</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V14">comp-reversing</a> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <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>by </b><i>, <a class="ref" href="functor0.html#D27">FUNCTOR0:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E20:9_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>(<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="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>by </b><i>, </i>;<br/>


</div>
