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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds  <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"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Olive" title="b2">g</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Olive" title="b2">g</font></span>)</span>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K21" title="ENS_1:func.21">hom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">Hom</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c3">g</font></span>)</span>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K21" title="ENS_1:func.21">hom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">Hom</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/>





<a NAME="E1:76"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c3">g</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K16" title="ENS_1:func.16">Hom</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ens_1.html#K16" title="ENS_1:func.16">Hom</a> <font color="Maroon" title="c1">C</font> )
 
;<br/>
<a NAME="E2:76"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c3">g</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T51" target="_self" onmouseover="rs('ens_1/T51')" onmouseout="rh()">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:76"/>
<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"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<span class="p4">(<span class="default"><a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c3">g</font></span>)</span>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K21" title="ENS_1:func.21">hom</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">Hom</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="ens_1.html#T5" target="_self" onmouseover="rs('ens_1/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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