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<div class="add">

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







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E35">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> 
 ;<br/>

<b>thus </b><a NAME="E2:58"/>
( <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> implies (  <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">g</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">g</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">g</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> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c5">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><a NAME="E1:58_1"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font> )
 <b>and </b><br/><a NAME="E2:58_1"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</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> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12" target="_self" title="CAT_1:def.12">CAT_1:def 12</a><br/>

<a NAME="E3:58_1"/><i><font color="Green" title="E38">A4</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:58"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="cat_1.html#T23" target="_self" title="CAT_1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:58_1"/><i><font color="Green" title="E39">A5</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> 
 <b>by </b><i><a class="txt" href="cat_1.html#E1:58_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="cat_1.html#T18" target="_self" title="CAT_1:th.18">Th18</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font> as    <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> <b>by </b><i><a class="txt" href="cat_1.html#E1:58_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="cat_1.html#E3:58_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="cat_1.html#T22" target="_self" title="CAT_1:th.22">Th22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g</font>
;<br/>

<b>thus </b><a NAME="E6:58_1"/>
( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</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="Maroon" title="c6">g</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#E1:58"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:58_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="cat_1.html#E3:58_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="cat_1.html#E4:58_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self" title="CAT_1:def.13">Def13</a></i>;<br/>


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

<b>assume </b><a NAME="E3:58"/><i><font color="Green" title="E36">A6</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="c5">g</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><b> such that </b><a NAME="E4:58"/><i><font color="Green" title="E37">A7</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">g</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="Maroon" title="c5">g</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>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12" target="_self" title="CAT_1:def.12">CAT_1:def 12</a><br/>

<a NAME="E6:58"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:58"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:58"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="cat_1.html#T23" target="_self" title="CAT_1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E7:58"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:58"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:58"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="cat_1.html#E4:58"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self" title="CAT_1:def.13">Def13</a></i>;<br/>


</div>
