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

<b>let </b><font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">F1</font> be  <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">F2</font> be  <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">t</font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F1</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">a</font> is <a href="altcat_3.html#V3">iso</a>
 ;<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/>
<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> ex <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">a</font></span>)</span> <a href="altcat_3.html#K1">"</a>  );<br/>
<a NAME="E4:48"/><i><font color="Green">E34</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/> 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="48_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="E1:48_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">o</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">o</font></span>)</span> <a href="altcat_3.html#K1">"</a> 
;<br/>

<b>thus </b><a NAME="E3:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">o</font></span>)</span> <a href="altcat_3.html#K1">"</a> ]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</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="E6:48"/><i><font color="Green">E35</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">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</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>
<a NAME="E7:48"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E8:48"/>
<font color="Maroon">f</font> is    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_2"/>
<font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F1</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D2">FUNCTOR2:def 2</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_3.html#K1">"</a>  )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:48_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> is   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="functor3.html#E1:48"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F1</font> ;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:48_3"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:48_3"/><i><font color="Green">E105</font></i>: 
 ex <font color="Olive">aa</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">aa</font></span>)</span> <a href="altcat_3.html#K1">"</a>  )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">fa</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> ;<br/><a NAME="E4:48_3"/><i><font color="Green">E106</font></i>: 
 ex <font color="Olive">bb</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">bb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">bb</font></span>)</span> <a href="altcat_3.html#K1">"</a>  )
 <b>by </b><i/>;<br/><a NAME="E5:48_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i/>;<br/><a NAME="E6:48_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E7:48_3"/><i><font color="Green">E109</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">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</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="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E8:48_3"/><i><font color="Green">E110</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">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</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="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E9:48_3"/><i><font color="Green">E111</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">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E10:48_3"/><i><font color="Green">E112</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">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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="functor3.html#E2:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/><a NAME="E11:48_3"/><b>then </b><i><font color="Green">E113</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">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E12:48_3"/><i><font color="Green">E114</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">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E13:48_3"/><i><font color="Green">E115</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">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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="functor3.html#E2:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">fa</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="functor3.html#E6:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E4"><i><font color="Green">Lemma38</font></i></a>, , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E2:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E4:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E11:48"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">F2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_4"/>
<font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F1</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a><br/>

<b>thus </b><a NAME="E2:48_4"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F1</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1">M3</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being  <a href="subset_1.html#M1">M3</a>(<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="functor3.html#E1:48"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:48"/>
<font color="Maroon">f</font> is    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_5"/>
<font color="Maroon">F2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green">E32</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a><br/>

<b>thus </b><a NAME="E2:48_5"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1">M3</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1">M3</a>(<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="functor3.html#E1:48"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F1</font> ;<br/>
<b>thus </b><a NAME="E14:48"/>
<font color="Maroon">F2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green">E32</font></i></a></i>;<br/>

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

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E16:48"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_3.html#K1">"</a>  )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E17:48"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> 
 <b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green">E33</font></i></a></i>;<br/>

<a NAME="E18:48"/><i><font color="Green">E118</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">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</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="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E19:48"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_3.html#K1">"</a> 
 
<b>by </b><i>, <a class="txt" href="functor3.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>;<br/>
<a NAME="E20:48"/>
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font> is <a href="altcat_3.html#V3">iso</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:48"/>
<font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="functor3.html#E4:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="functor3.html#E6:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="altcat_4.html#T3">ALTCAT_4:3</a></i>;<br/>


</div>
