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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">g</font>, <font color="Olive" title="b2">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  holds <br/><font color="Olive" title="b2">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">g</font>, <font color="Maroon" title="c3">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"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">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="c3">f</font> &amp; <font color="Maroon" title="c2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  implies <font color="Maroon" title="c3">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">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="c3">f</font>
 ; <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">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  or <font color="Maroon" title="c3">f</font> is  <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">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="E3:36"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">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">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:36"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c4">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">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</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#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></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="c3">f</font> 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="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font>
; <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="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></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> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">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="c3">f</font></span>)</span> )</span><br/>

<a NAME="E5:36"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="cat_1.html#T17" title="CAT_1:th.17">CAT_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></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> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="cat_1.html#T17" title="CAT_1:th.17">CAT_1:17</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="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">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="c3">f</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> = 
 <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">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</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> <font color="Maroon" title="c3">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="cat_1.html#T17" title="CAT_1:th.17">CAT_1:17</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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