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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">E</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">E</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font><br/> for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>  st  ex <font color="Olive" title="b4">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b5">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ex <font color="Olive" title="b6">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> st <font color="Olive" title="b6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b5">d</font></span>)</span> holds <br/>( <font color="Olive" title="b3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b3">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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="b3">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font><br/> for <font color="Olive" title="b2">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Olive" title="b1">G</font>  st  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ex <font color="Olive" title="b5">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> st <font color="Olive" title="b5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">d</font></span>)</span> holds <br/>( <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>  st  ex <font color="Olive" title="b2">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ex <font color="Olive" title="b4">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">d</font></span>)</span> holds <br/>( <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">o</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ex <font color="Olive" title="b3">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">d</font></span>)</span> implies ( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:4"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ex <font color="Olive" title="b3">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">d</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )</span><br/>

<a NAME="E2:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">c</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Olive" title="b2">d</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>, <font color="Olive" title="b3">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> : <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">d</font></span>)</span> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="commacat.html#D5" target="_self" title="COMMACAT:def.5">Def5</a></span>;<br/>
<a NAME="E3:4"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c8">d</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c9">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E5:4"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:4"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c7">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E7:4"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E8:4"/>
( <font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:4"/>
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</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="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E7:4"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
