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<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="firebrick">::  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="firebrick">::  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="E1:39"/>
( <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="firebrick">::  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="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_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="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" title="CAT_1:def.9">CAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:39_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#D8" target="_self" title="CAT_3:def.8">Def10</a>, <a class="ref" href="cat_3.html#D9" target="_self" title="CAT_3:def.9">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:39"/>
<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="firebrick">::  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="E3:39"/>
(  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#T22" target="_self" title="CAT_3:th.22">Th26</a>, <a class="ref" href="cat_3.html#T29" target="_self" title="CAT_3:th.29">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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