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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="cat_1.html#M2" target="_self">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>





<b>thus </b><a NAME="E1:111"/>
( <font color="Maroon">T</font> is <a href="cat_1.html#V10" target="_self">faithful</a> implies for <font color="Olive">c</font>, <font color="Olive">c'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds   <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Olive">c</font>,<font color="Olive">c'</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">T</font> is <a href="cat_1.html#V10" target="_self">faithful</a>
 ;<br/>

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">c'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>

<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_1_1"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:111_1_1"/><b>then </b>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E3:111_1_1"/>
 <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T59">PARTFUN1:59</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">f2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:111_1_2_1"/>
( <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 ;<br/><a NAME="E2:111_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f1</font> is    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">f2</font> is    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:111_1_2_1"/><b>then </b>
( <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T42" target="_self">Th42</a></i>;<br/><b>hence </b><a NAME="E4:111_1_2_1"/>
( <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> implies <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T43" target="_self">Th43</a>, <a class="ref" href="cat_1.html#T46" target="_self">Th46</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:111_1_2"/>
 <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T45" target="_self">Th45</a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:111_1"/>
 <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T44" target="_self">Th44</a></i>;<br/>


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

<b>assume </b><a NAME="E2:111"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">c</font>, <font color="Olive">c'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds   <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Olive">c</font>,<font color="Olive">c'</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">c'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D24" target="_self">CAT_1:def 24</a><br/>

<b>assume </b><a NAME="E3:111"/><i><font color="Green">E51</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">f1</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font>, <font color="Maroon">f2</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font>;<br/>

<a NAME="E4:111"/>
(  <a href="cat_1.html#K6" target="_self">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="cat_1.html#K13" target="_self">.</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp;  <a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K16" target="_self">hom</a> <font color="Maroon">T</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T43" target="_self">Th43</a>, <a class="ref" href="cat_1.html#T44" target="_self">Th44</a>, , <a class="ref" href="cat_1.html#T40" target="_self">Th40</a>, <a class="ref" href="cat_1.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E5:111"/>
( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font> implies <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>


</div>
