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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>thus </b><a NAME="E1:58"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">c<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D4" target="_self">YELLOW20:def 4</a><br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E2:58"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V21">bijective</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E4:58"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>



<b>assume </b><a NAME="E5:58"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E6:58"/><i><font color="Green">E34</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E4:58"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E7:58"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E4:58"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E5:58"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E6:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="yellow20.html#E6:58"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow20.html#E7:58"><i><font color="Green">E35</font></i></a>, <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
;<br/>


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