<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a>
 ;<br/>

<a NAME="E3:9"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c4">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:9_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c5">m1</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">o1</font> =  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font>;<br/>
<b>set </b><font color="Maroon" title="c8">o2</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font>;<br/>
<a NAME="E4:9_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">o3</font> =  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font>;<br/>
<b>set </b><font color="Maroon" title="c10">o4</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">m1'</font> = <font color="Maroon" title="c5">m1</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font> <b>by </b><i><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m2'</font> = <font color="Maroon" title="c6">m2</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font> <b>by </b><i><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></i>;<br/>
<a NAME="E7:9_1"/><i><font color="Green" title="E12">A6</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></i>;<br/>
<a NAME="E8:9_1"/><i><font color="Green" title="E13">A7</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></i>;<br/>
<a NAME="E9:9_1"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<i><font color="Green" title="E15">A9</font></i>: <font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c11">m1'</font> = 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:9_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="yoneda_1.html#E7:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m2'</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E8:9_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E11:9_1"/><i><font color="Green" title="E16">A10</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<a NAME="E12:9_1"/><i><font color="Green" title="E17">A11</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m2'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E10:9_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="yoneda_1.html#E11:9_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E9:9_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="yoneda_1.html#E12:9_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
;<br/>
<a NAME="E14:9_1"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m2</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>
<a NAME="E15:9_1"/><i><font color="Green" title="E19">A13</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m2'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E10:9_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="yoneda_1.html#E11:9_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E9:9_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="yoneda_1.html#E15:9_1"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
;<br/>
<a NAME="E17:9_1"/><b>then </b>
<font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="yoneda_1.html#E4:9_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="yoneda_1.html#E14:9_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>
<b>hence </b><a NAME="E18:9_1"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E1:9_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="yoneda_1.html#E3:9_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="yoneda_1.html#E7:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="cat_1.html#D24" title="CAT_1:def.24">CAT_1:def 24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon" title="c2">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div>
