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<a NAME="E1:11_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>;<br/><a NAME="E1:11_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/><b>hence </b><a NAME="E2:11_1_1_1"/>
for <font color="Olive">c1</font>, <font color="Olive">c2</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c2</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">c1</font>,<font color="Olive">c2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">m</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">O</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">i</font> = <font color="Maroon">c</font> <a href="cat_1.html#K7">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K1">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>;<br/>
<a NAME="E3:11_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K1">.--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K1">.--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T7">CAT_1:7</a>, <a class="ref" href="cat_1.html#T8">CAT_1:8</a></i>;<br/>
<a NAME="E4:11_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<a NAME="E6:11_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>now </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="E1:11_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:11_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font>
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:11_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span>;<br/><a NAME="E1:11_1_1_3"/>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:11_1_1_3"/>
for <font color="Olive">c1</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c1</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">c1</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:11_1_1"/>
 <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> is  <a href="cat_1.html#V1">strict</a>  <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>


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