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<b>let </b><font color="Maroon">C</font> be   <a href="cat_4.html#NM1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">a</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">b</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E34</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:38"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> is <a href="cat_1.html#V3">monic</a> or <font color="Maroon">g</font> is <a href="cat_1.html#V3">monic</a> )
 ;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V3">monic</a>
 ;<br/><b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">c</font>, <font color="Maroon">f2</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">c</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:38_1"/><i><font color="Green">E39</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:38_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E4:38_1"/>
( <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> &amp; <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="cat_4.html#T1" target="_self">Th1</a>, , <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:38_1"/><b>then </b>
<font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:38_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="cat_4.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:38"/><i><font color="Green">E41</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">g</font> is <a href="cat_1.html#V3">monic</a>
 ;<br/><b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">c</font>, <font color="Maroon">f2</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">c</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:38_2"/><i><font color="Green">E44</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:38_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E4:38_2"/>
( <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> &amp; <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , , , <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:38_2"/><b>then </b>
<font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:38_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="cat_4.html#T1" target="_self">Th1</a>, , <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38"/>
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> is <a href="cat_1.html#V3">monic</a>
 <b>by </b><i>, , , <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/>


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