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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  iff ( <font color="Olive" title="b1">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  iff ( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:43"/>
( <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  implies ( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_1.html#D12" title="CAT_1:def.12">CAT_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:43_1"/>
( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D10" target="_self" title="CAT_3:def.10">Def10</a>, <a class="ref" href="cat_3.html#D11" target="_self" title="CAT_3:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:43"/>
<font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  or <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E3:43"/>
(  not <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  or <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T26" target="_self" title="CAT_3:th.26">Th26</a>, <a class="ref" href="cat_3.html#T33" target="_self" title="CAT_3:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
