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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="oppcat_1.html#M1">Contravariant_Functor</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E9</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="yoneda_1.html#V1" target="_self">faithful</a>
 ;<br/>

<a NAME="E3:11"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:11_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E4:11_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<a NAME="E6:11_1"/><i><font color="Green">E14</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</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="cat_1.html#T19">CAT_1:19</a></i>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="oppcat_1.html#T33">OPPCAT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T33">OPPCAT_1:33</a></i>
;<br/>
<a NAME="E8:11_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="oppcat_1.html#T33">OPPCAT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T33">OPPCAT_1:33</a></i>
;<br/>
<a NAME="E10:11_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="yoneda_1.html#E4:11_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yoneda_1.html#E8:11_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<b>hence </b><a NAME="E11:11_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yoneda_1.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yoneda_1.html#E3:11_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E6:11_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yoneda_1.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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