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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E1:94"/>
 <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<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E2:94"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>3</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cat_3.html#T69" target="_self">Th69</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D18" target="_self">CAT_3:def 18</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_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="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>5</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>



<b>assume </b><a NAME="E3:94"/><i><font color="Green">E55</font></i>: 
 <a href="cat_3.html#K3" target="_self">doms</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E4:94"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:94_1"/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self">Th67</a></i>;<br/><b>thus </b> <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:94"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <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<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:94"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E7:94"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E8:94"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<b>thus </b><a NAME="E9:94"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <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<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E8:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_3.html#E1:94_2"><i><font color="Green">E58</font></i></a></i>
;<br/>


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

<b>assume </b><a NAME="E10:94"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>7</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="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E11:94"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E10:94"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <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<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E8:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_3.html#K2" target="_self">.--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E11:94"><i><font color="Green">E59</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>
;<br/><br/>


</div>
