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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>





<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  implies (  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">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>  &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" target="_self" title="CAT_1:def.17">CAT_1:def 17</a><br/>

<b>given </b><font color="Maroon" title="c4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><b> such that </b><a NAME="E2:67_1"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E3:67_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">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="cat_1.html#E1:67_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:67_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<b>thus </b><a NAME="E4:67_1"/>
 ex <font color="Olive" title="b1">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:67_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:67_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:67"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>and </b><br/><a NAME="E3:67"/><i><font color="Green" title="E43">A4</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">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> 
 ;<br/>

<b>given </b><font color="Maroon" title="c4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><b> such that </b><a NAME="E4:67"/><i><font color="Green" title="E44">A5</font></i>: 
 ex <font color="Olive" title="b1">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>

<b>thus </b><a NAME="E5:67"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="cat_1.html#E2:67"><i><font color="Green" title="E42">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" target="_self" title="CAT_1:def.17">CAT_1:def 17</a><br/>

<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<b>thus </b><a NAME="E6:67"/>
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:67"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="cat_1.html#E3:67"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="cat_1.html#E4:67"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/>


</div>
