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

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





<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="isocat_1.html#NR1" title="ISOCAT_1:NR.1">is_an_isomorphism</a> 
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">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="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:14"/><i><font color="Green" title="E14">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="cat_1.html#D22" title="CAT_1:def.22">CAT_1:def 22</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green" title="E15">A4</font></i>: 
 <a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="isocat_1.html#T12" target="_self" title="ISOCAT_1:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">G</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">B</font>,the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="isocat_1.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="isocat_1.html#E3:14"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="isocat_1.html#T12" target="_self" title="ISOCAT_1:th.12">Th12</a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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/><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T104" title="CAT_1:th.104">CAT_1:104</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E3:14"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="isocat_1.html#E4:14"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
;<br/><b>then </b> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </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#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="isocat_1.html#D2" target="_self" title="ISOCAT_1:def.2">Def2</a></i>
;<br/><b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T103" title="CAT_1:th.103">CAT_1:103</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
