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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">a</font>, <font color="Olive" title="b2">b</font>, <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><br/>  for <font color="Olive" title="b4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b5">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">f</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="Olive" title="b5">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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:()"><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#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b2">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</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="Olive" title="b2">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <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> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</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="Olive" title="b1">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</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="c6">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:15"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <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="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</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="c6">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span><br/>

<a NAME="E2:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D10" title="CAT_1:def.10">CAT_1:def 10</a></span>;<br/>
<a NAME="E3:15"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="cat_1.html#T5" title="CAT_1:th.5">CAT_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</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="c6">g</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="#E2:15"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="oppcat_1.html#T16" target="_self" title="OPPCAT_1:th.16">Th17</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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