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<b>let </b><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>;<br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="isocat_1.html#NR5" title="ISOCAT_1:NR.5">are_equivalent</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="isocat_1.html#M1" title="ISOCAT_1:mode.1">Equivalence</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">G</font> being    <a href="isocat_1.html#M1" title="ISOCAT_1:mode.1">Equivalence</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</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#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:64"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="isocat_1.html#T56" target="_self" title="ISOCAT_1:th.56">Th56</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E45">A4</font></i>: 
 <a href="cat_1.html#K15" title="CAT_1:func.15">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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E3:64"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="nattra_1.html#T31" title="NATTRA_1:th.31">NATTRA_1:31</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green" title="E46">A5</font></i>: 
 <a href="cat_1.html#K15" title="CAT_1:func.15">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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></i>;<br/>
<b>consider </b><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#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E8:64"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c5">s</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a>
 <b>by </b><i><a class="txt" href="isocat_1.html#E5:64"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E9:64"/><i><font color="Green" title="E48">A7</font></i>: 
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E4:64"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="nattra_1.html#T31" title="NATTRA_1:th.31">NATTRA_1:31</a></i>;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E11:64"/><i><font color="Green" title="E50">A9</font></i>: 
 ex <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font> st <font color="Olive" title="b1">t</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E9:64"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E12:64"/><i><font color="Green" title="E51">A10</font></i>: 
<font color="Maroon" title="c4">G</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E4:64"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="isocat_1.html#T57" target="_self" title="ISOCAT_1:th.57">Th57</a></i>;<br/>
<b>thus </b><a NAME="E13:64"/>
<font color="Maroon" title="c3">F</font> is <a href="cat_1.html#V10" title="CAT_1:attr.10">full</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D23" title="CAT_1:def.23">CAT_1:def 23</a><br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E52">A11</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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> 
 ;<br/>

<b>let </b><font color="Maroon" title="c8">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font>;<br/>

<a NAME="E2:64_1"/><i><font color="Green" title="E53">A12</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T118" title="CAT_1:th.118">CAT_1:118</a></i>;<br/>
<a NAME="E3:64_1"/><i><font color="Green" title="E54">A13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c4">G</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">g</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font> ;<br/>
<a NAME="E5:64_1"/><i><font color="Green" title="E55">A14</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</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="txt" href="isocat_1.html#E6:64"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></i>;<br/>
<a NAME="E6:64_1"/><i><font color="Green" title="E56">A15</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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="txt" href="isocat_1.html#E1:64_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="isocat_1.html#E3:64_1"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<a NAME="E7:64_1"/>
<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E3:64"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="nattra_1.html#D11" title="NATTRA_1:def.11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E8:64_1"/><b>then </b><i><font color="Green" title="E57">A16</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></i>;<br/>
<a NAME="E9:64_1"/><i><font color="Green" title="E58">A17</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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="txt" href="isocat_1.html#E5:64_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="isocat_1.html#E6:64_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E10:64_1"/><i><font color="Green" title="E59">A18</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</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> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E2:64_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="isocat_1.html#E8:64_1"><i><font color="Green" title="E57">A16</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">f1</font> = <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> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font> <b>by </b><i><a class="ref" href="cat_1.html#T118" title="CAT_1:th.118">CAT_1:118</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f2</font> = <font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a'</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c7">a'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font> <b>by </b><i><a class="ref" href="cat_1.html#T118" title="CAT_1:th.118">CAT_1:118</a></i>;<br/>
<b>take </b><font color="Maroon" title="c12">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c10">f1</font></span>)</span>;<br/>

<a NAME="E13:64_1"/><i><font color="Green" title="E60">A19</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</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="txt" href="isocat_1.html#E6:64"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></i>;<br/>
<a NAME="E14:64_1"/>
<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> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E8:64"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="nattra_1.html#D10" title="NATTRA_1:def.10">NATTRA_1:def 10</a></i>;<br/>
<a NAME="E15:64_1"/><b>then </b><i><font color="Green" title="E61">A20</font></i>: 
<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> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E5:64_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="cat_1.html#T73" title="CAT_1:th.73">CAT_1:73</a></i>;<br/>
<a NAME="E16:64_1"/><i><font color="Green" title="E62">A21</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c7">a'</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E8:64"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="nattra_1.html#D10" title="NATTRA_1:def.10">NATTRA_1:def 10</a></i>;<br/>
<a NAME="E17:64_1"/><i><font color="Green" title="E63">A22</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">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="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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">a'</font></span>)</span> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<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>)</span>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:64_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="txt" href="isocat_1.html#E10:64_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="nattra_1.html#T16" title="NATTRA_1:th.16">NATTRA_1:16</a></i>;<br/>
<font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</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="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a'</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<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>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E9:64_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="cat_1.html#T57" title="CAT_1:th.57">CAT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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">a'</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<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">a'</font></span>)</span> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<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>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E13:64_1"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="isocat_1.html#E16:64_1"><i><font color="Green" title="E62">A21</font></i></a>, <a class="ref" href="cat_1.html#D14" title="CAT_1:def.14">CAT_1:def 14</a></i>
<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">a'</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">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="c12">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E8:64_1"><i><font color="Green" title="E57">A16</font></i></a>, <a class="txt" href="isocat_1.html#E9:64_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="isocat_1.html#E13:64_1"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="isocat_1.html#E17:64_1"><i><font color="Green" title="E63">A22</font></i></a>, <a class="ref" href="cat_1.html#T54" title="CAT_1:th.54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</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="c12">f</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</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>
<b>by </b><i><a class="txt" href="isocat_1.html#E6:64"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="isocat_1.html#E10:64_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>
;<br/>
<a NAME="E19:64_1"/><b>then </b><i><font color="Green" title="E64">A23</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</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="c12">f</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E5:64_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="isocat_1.html#E6:64_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="isocat_1.html#E15:64_1"><i><font color="Green" title="E61">A20</font></i></a>, <a class="ref" href="cat_1.html#T65" title="CAT_1:th.65">CAT_1:65</a></i>;<br/>
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> = 
<font color="Maroon" title="c4">G</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c8">g</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</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="c12">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="isocat_1.html#E19:64_1"><i><font color="Green" title="E64">A23</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="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="c12">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<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="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c8">g</font> = 
<font color="Maroon" title="c3">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E12:64"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="cat_1.html#D24" title="CAT_1:def.24">CAT_1:def 24</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E14:64"/>
<font color="Maroon" title="c3">F</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a>
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:64"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="isocat_1.html#T57" target="_self" title="ISOCAT_1:th.57">Th57</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>;<br/>

<b>take </b>
<font color="Maroon" title="c4">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font>
;<br/>

<a NAME="E15:64"/><i><font color="Green" title="E52">A24</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a>, <a class="ref" href="cat_1.html#T118" title="CAT_1:th.118">CAT_1:118</a></i>;<br/>
<a NAME="E16:64"/>
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E17:64"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E11:64"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="isocat_1.html#E15:64"><i><font color="Green" title="E52">A24</font></i></a>, <a class="ref" href="isocat_1.html#T9" target="_self" title="ISOCAT_1:th.9">Th9</a></i>;<br/>


</div>
