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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#V6" title="CAT_1:attr.6">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">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/>



<b>thus </b><a NAME="E1:40"/>
( <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">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:()"><i><font color="Red">::  thesis: </font></i></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#V6" title="CAT_1:attr.6">invertible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_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="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">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#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>hence </b><a NAME="E2:40_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> )
 <b>by </b><i><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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:40"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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#V6" title="CAT_1:attr.6">invertible</a> )</span><br/>

<b>hence </b><a NAME="E3:40"/>
(  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#V6" title="CAT_1:attr.6">invertible</a> )
 <b>by </b><i><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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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