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<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> ] <b>means </b>for <font color="Olive">f</font>, <font color="Olive">g</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">A</font>,<font color="Maroon">B</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> iff ( <font color="Olive">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">g</font> &amp; <font color="Olive">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">f</font>,<font color="Olive">g</font> ) );<br/>
<a NAME="E1:27_1_1"/><i><font color="Green">E20</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><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="E1:27_1_1_1"/>
<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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> )
 <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">f</font> = <font color="Maroon">f</font>, <font color="Maroon">g</font> = <font color="Maroon">g</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">A</font>,<font color="Maroon">B</font> <b>by </b><i>, <a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">o</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">o</font> &amp; <font color="Maroon">a<sub>2</sub></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">f</font> <a href="functor0.html#K3">.</a> <font color="Olive">o</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="functor0.html#K3">.</a> <font color="Olive">o</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> );<br/>
<a NAME="E5:27_1_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">j</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">f</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>
<b>thus </b><a NAME="E4:27_1_1_1_1"/>
 ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a</font>,<font color="Olive">j</font>]
 <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">N</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:27_1_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1">MSUALG_9:sch 1</a>(<a class="ref" href="functor2.html#D1" target="_self">Def1</a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> );<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:27_1_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">N</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<b>let </b><font color="Maroon">f'</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">g'</font> be  <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">A</font>,<font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E10:27_1_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f'</font>,<font color="Maroon">g'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>

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

<b>thus </b><a NAME="E11:27_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> implies ( <font color="Maroon">f'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f'</font>,<font color="Maroon">g'</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:27_1_1_1_2"/>
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_1_2"/>
( <font color="Maroon">f'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f'</font>,<font color="Maroon">g'</font> )
 <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, </i>;<br/>


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

<b>thus </b><a NAME="E12:27_1_1_1"/>
( <font color="Maroon">f'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f'</font>,<font color="Maroon">g'</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f'</font>,<font color="Maroon">g'</font> )
 ;<br/>

<a NAME="E2:27_1_1_1_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f'</font> <a href="functor2.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">g'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27_1_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E5:27_1_1_1_3"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>
<a NAME="E6:27_1_1_1_3"/><i><font color="Green">E71</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1"><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="E1:27_1_1_1_3_1"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">i''</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:27_1_1_1_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font>
 <b>and </b><br/><a NAME="E5:27_1_1_1_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</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">f</font> <a href="functor0.html#K3">.</a> <font color="Maroon">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="functor0.html#K3">.</a> <font color="Maroon">i''</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:27_1_1_1_3_1"/><i><font color="Green">E74</font></i>: 
<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">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="functor0.html#K3">.</a> <font color="Maroon">i''</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="txt" href="functor2.html#E3:27_1_1_1"><i><font color="Green">E28</font></i></a>, , </i>;<br/>
<a NAME="E7:27_1_1_1_3_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i''</font> is    <a href="subset_1.html#M1">Element</a> of <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">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="functor0.html#K3">.</a> <font color="Maroon">i''</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i>, <a class="txt" href="functor2.html#E3:27_1_1_1"><i><font color="Green">E28</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E8:27_1_1_1_3_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#D3" target="_self">Def3</a>, <a class="txt" href="functor2.html#E7:27_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:27_1_1_1_3"/><i><font color="Green">E75</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_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="E1:27_1_1_1_3_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:27_1_1_1_3_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_1_3_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:27_1_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="functor2.html#E5:27_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:27_1_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, , </i>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E91</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</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>for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</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">A</font>,<font color="Maroon">B</font>  st <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Olive">h</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/>for <font color="Olive">t1</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">f</font>,<font color="Olive">g</font><br/> for <font color="Olive">t2</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">g</font>,<font color="Olive">h</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">t2</font>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp;  ex <font color="Olive">v</font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Olive">t1</font>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</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> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">o</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"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">o</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<a NAME="E2:27_1_1_2"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p3">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">ff</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">h</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:27_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:27_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ff</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:27_1_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:27_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:27_1_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:27_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">f</font>, <font color="Maroon">j</font> = <font color="Maroon">g</font>, <font color="Maroon">k</font> = <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, , </i>;<br/>
<i><font color="Green">E114</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<i><font color="Green">E115</font></i>: <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<a NAME="E16:27_1_1_2"/>
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> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Maroon">o</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_2_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:27_1_1_2_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E5:27_1_1_2_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:27_1_1_2_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:27_1_1_2_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</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">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E9:27_1_1_2_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">i'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">x1</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font> <b>by </b><i>, </i>;<br/>
<a NAME="E11:27_1_1_2_3"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">j'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">x2</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">j'</font>,<font color="Maroon">k'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">j'</font>,<font color="Maroon">k'</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">x2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">x1</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">i'</font>,<font color="Maroon">k'</font> ;<br/>
<a NAME="E14:27_1_1_2_3"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">i'</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">vv</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">i'</font>,<font color="Maroon">k'</font> )
 
<b>by </b><i>, <a class="ref" href="functor2.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E15:27_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">k'</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></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/>
<a NAME="E16:27_1_1_2_3"/><b>then </b>
<font color="Maroon">vv</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">k'</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:27_1_1_2_3"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">vv</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:27_1_1_2_3"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</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">A</font>,<font color="Maroon">B</font>  st <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Olive">h</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> holds <br/>for <font color="Olive">t1</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">f</font>,<font color="Olive">g</font><br/> for <font color="Olive">t2</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">g</font>,<font color="Olive">h</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">t2</font>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">v</font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> holds <br/><font color="Maroon">vv</font> <a href="pboole.html#R6">=</a> <font color="Olive">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Olive">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">g</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">h</font> be  <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">A</font>,<font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E1:27_1_1_2_3_1"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/>

<b>let </b><font color="Maroon">t1</font> be    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/><b>let </b><font color="Maroon">t2</font> be    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">g</font>,<font color="Maroon">h</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:27_1_1_2_3_1"/><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:27_1_1_2_3_1"/>
 ex <font color="Olive">v</font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font>
 ;<br/>

<a NAME="E4:27_1_1_2_3_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, <a class="ref" href="functor2.html#D5" target="_self">Def5</a>, <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> ;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">g</font>,<font color="Maroon">h</font> <b>by </b><i/>;<br/>
<a NAME="E7:27_1_1_2_3_1"/>
( <font color="Maroon">x2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">x1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1_2_3_1"/>
<font color="Maroon">vv</font> <a href="pboole.html#R6">=</a> <font color="Maroon">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">t1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:27_1_1_2_3"/>
 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"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>,<font color="Maroon">o</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:27_1_1_2"/>
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> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Maroon">o</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span><b> such that </b><br/><a NAME="E6:27_1_1"/><i><font color="Green">E134</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#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/> ex <font color="Olive">v</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</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> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">v</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>(<a class="ref" href="functor2.html#D1" target="_self">Def1</a>);<br/></i>
<b>set </b><font color="Maroon">F</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #);<br/>
<a NAME="E7:27_1_1"/><i><font color="Green">E135</font></i>: 
not  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><b>proof </b></a><div class="add">

<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">A</font>,<font color="Maroon">B</font>;<br/>
<a NAME="E2:27_1_1_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_3"/>
not  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:27_1_1"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #) is <a href="altcat_1.html#V2">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><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 <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #), <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #), <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:27_1_1_4"/>
( <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/>

<a NAME="E2:27_1_1_4"/><b>then </b>
( the <a href="altcat_1.html#U1">Arrows</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #) <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="altcat_1.html#U1">Arrows</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #) <a href="binop_1.html#K1">.</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E3:27_1_1_4"/><b>then </b><i><font color="Green">E137</font></i>: 
( <font color="Maroon">a</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="tarski.html#K4">]</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">a1</font> = <font color="Maroon">o1</font>, <font color="Maroon">a2</font> = <font color="Maroon">o2</font>, <font color="Maroon">a3</font> = <font color="Maroon">o3</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">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:27_1_1_4"/><i><font color="Green">E139</font></i>: 
<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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:27_1_1_4"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</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>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:27_1_1_4"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">a1</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">y</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E10:27_1_1_4"/><i><font color="Green">E142</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:27_1_1_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E13:27_1_1_4"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">a2</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a2</font>,<font color="Maroon">a3</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a2</font>,<font color="Maroon">a3</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E15:27_1_1_4"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">y</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a1</font>,<font color="Maroon">a3</font>
 
;<br/>
<a NAME="E16:27_1_1_4"/><i><font color="Green">E146</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E17:27_1_1_4"/>
( <font color="Maroon">a1</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">a3</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">a1</font>,<font color="Maroon">a3</font> )
 
<b>by </b><i>, , <a class="ref" href="functor2.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E18:27_1_1_4"/><b>then </b>
<font color="Maroon">a</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">o3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:27_1_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>hence </b><a NAME="E20:27_1_1_4"/>
not <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#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">c</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a>  <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E10:27_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>

<b>thus </b><a NAME="E11:27_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</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">A</font>,<font color="Maroon">B</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <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</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> iff ( <font color="Olive">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">g</font> &amp; <font color="Olive">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">f</font>,<font color="Olive">g</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">g</font> be  <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">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:27_1_1_5"/>
( <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</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> implies ( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_5_1"/>
<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</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:27_1_1_5_1"/><b>then </b><i><font color="Green">E147</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</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 
;<br/>
<a NAME="E3:27_1_1_5_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:27_1_1_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></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>hence </b><a NAME="E5:27_1_1_5_1"/>
( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:27_1_1_5"/>
( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> implies <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</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_5_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:27_1_1_5_2"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:27_1_1_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:27_1_1_5_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:27_1_1_5_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</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>


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


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

<b>let </b><font color="Maroon">f</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">g</font> be  <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">A</font>,<font color="Maroon">B</font>, <font color="Maroon">h</font> be  <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">A</font>,<font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E12:27_1_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">h</font> )
 ;<br/>

<b>let </b><font color="Maroon">t1</font> be    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/><b>let </b><font color="Maroon">t2</font> be    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">g</font>,<font color="Maroon">h</font>;<br/>



<a NAME="E13:27_1_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E14:27_1_1"/><i><font color="Green">E151</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p3">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>
<a NAME="E15:27_1_1"/><i><font color="Green">E152</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E16:27_1_1"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:27_1_1"/><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</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"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:27_1_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">a</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span><b> such that </b><br/><a NAME="E20:27_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>and </b><br/><a NAME="E21:27_1_1"/><i><font color="Green">E156</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> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span>]
 <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E22:27_1_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">f</font>, <font color="Maroon">j</font> = <font color="Maroon">g</font>, <font color="Maroon">k</font> = <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<i><font color="Green">E158</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<a NAME="E26:27_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E27:27_1_1"/>
<font color="Maroon">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E28:27_1_1"/><b>then </b>
( <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">h</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E29:27_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E30:27_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E31:27_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">t2</font>,<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">t1</font>
 
;<br/>
<b>hence </b><a NAME="E32:27_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font> &amp; <font color="Olive">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">t2</font>,<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/>


</div>
