<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c3">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c3">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:69"/>
<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a> </span><br/>

<a NAME="E2:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T31" title="NATTRA_1:th.31">NATTRA_1:31</a></span>;<br/>
<a NAME="E3:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font><span class="kw"> such that </span><br/><a NAME="E5:69"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c5">s</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:69"/>
<font color="Maroon" title="c3">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_1.html#D24" title="CAT_1:def.24">CAT_1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a9</font>  holds <br/>(  not <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a9</font>  holds <br/>(  not <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E2:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c7">a9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c9">f2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font> or <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font></span><br/>

<span class="lab"><font color="Green" title="E48">A7</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c9">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
;<br/>
<span class="lab"><font color="Green" title="E49">A8</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a9</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a9</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></span>
;<br/>
<a NAME="E6:69_1"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c7">a9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c7">a9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></span>;<br/>
<a NAME="E7:69_1"/>
<font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a9</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="nattra_1.html#D10" title="NATTRA_1:def.10">NATTRA_1:def 10</a></span>;<br/>
<a NAME="E8:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a9</font> is  <a href="cat_1.html#V3" title="CAT_1:attr.3">monic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="cat_1.html#T73" title="CAT_1:th.73">CAT_1:73</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c8">f1</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="nattra_1.html#T16" title="NATTRA_1:th.16">NATTRA_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c9">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E6:69_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E8:69_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E5:69_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="cat_1.html#T60" title="CAT_1:th.60">CAT_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c9">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="nattra_1.html#T16" title="NATTRA_1:th.16">NATTRA_1:16</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
