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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F</font> <a href="isocat_1.html#NR1" target="_self">is_an_isomorphism</a> 
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:14"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D22">CAT_1:def 22</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T31">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">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T104">CAT_1:104</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="isocat_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><b>then </b> <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="isocat_1.html#K1" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="isocat_1.html#K1" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T103">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">Obj</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K12">Obj</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="isocat_1.html#K1" target="_self">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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