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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM1" title="CAT_4:NM.1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">e</font>, <font color="Maroon" title="c7">s</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c8">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>;<br/><b>let </b><font color="Maroon" title="c9">h</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>;<br/><b>let </b><font color="Maroon" title="c10">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c2">a</font>;<br/><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">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> 
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</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> 
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c2">a</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> 
 <b>and </b><br/><a NAME="E4:61"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c7">s</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> 
 ;<br/>

<a NAME="E5:61"/><i><font color="Green" title="E32">A5</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c6">e</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#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c7">s</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>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E6:61"/><b>then </b>
( <font color="Maroon" title="c8">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span> &amp; <font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:61"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="cat_4.html#E3:61"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="cat_4.html#E4:61"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="cat_1.html#T54" title="CAT_1:th.54">CAT_1:54</a></i>;<br/>
<a NAME="E7:61"/><b>then </b>
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c2">a</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#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c7">s</font></span>)</span>,<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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="cat_4.html#K18" title="CAT_4:func.18">[x]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p4">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p4">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="cat_4.html#K18" title="CAT_4:func.18">[x]</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">s</font></span>)</span></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E3:61"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="cat_4.html#E4:61"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="cat_4.html#E5:61"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E8:61"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_4.html#K18" title="CAT_4:func.18">[x]</a> <font color="Maroon" title="c9">h</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_4.html#K18" title="CAT_4:func.18">[x]</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="cat_4.html#K18" title="CAT_4:func.18">[x]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c11">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:61"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:61"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="cat_4.html#T43" target="_self" title="CAT_4:th.43">Th43</a></i>;<br/>


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