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<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="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_3.html#V1" title="CAT_3:attr.1">retraction</a>  iff <font color="Olive" title="b1">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  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_3.html#V1" title="CAT_3:attr.1">retraction</a>  iff <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:42"/>
( <font color="Maroon" title="c2">f</font> is  <a href="cat_3.html#V1" title="CAT_3:attr.1">retraction</a>  implies <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  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> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  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_3.html#V1" title="CAT_3:attr.1">retraction</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c3">i</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">i</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>
 <span class="kw">and </span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">i</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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D8" target="_self" title="CAT_3:def.8">CAT_3:def 8</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> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D9" target="_self" title="CAT_3:def.9">CAT_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<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></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="oppcat_1.html#T16" title="OPPCAT_1:th.16">OPPCAT_1:16</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T20" title="OPPCAT_1:th.20">OPPCAT_1:20</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T10" title="OPPCAT_1:th.10">OPPCAT_1:10</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">given </span><font color="Maroon" title="c3">i</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span><span class="kw"> such that </span><a NAME="E3:42"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D9" target="_self" title="CAT_3:def.9">CAT_3: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> </span><br/>

<span class="kw">take </span>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D8" target="_self" title="CAT_3:def.8">CAT_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> <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> )</span><br/>

<span class="kw">thus </span><span class="lab"><font color="Green" title="E26">A5</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>
; <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> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> <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></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="oppcat_1.html#T16" title="OPPCAT_1:th.16">OPPCAT_1:16</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T10" title="OPPCAT_1:th.10">OPPCAT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<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></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T20" title="OPPCAT_1:th.20">OPPCAT_1:20</a></span>
<br/>.= 
 <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</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>
