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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">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="Olive" title="b2">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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"> for <font color="Olive" title="b1">m</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="c2">o</font> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  )</span><br/>





<b>consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> , <font color="Maroon" title="c6">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E2:58"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c6">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">f</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="c4">a</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/>
<a NAME="E3:58"/>
( <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<b>hence </b><a NAME="E4:58"/>
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</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="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  )
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</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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