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<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green">E9</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> ) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K6" target="_self">commaComp</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green">E10</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> ) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K6" target="_self">commaComp</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>);<br/><b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K17">`11</a> 
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:9_1_2"/><b>then </b><i><font color="Green">E11</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>);<br/><b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K18">`12</a> 
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:9_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>9</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:9_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="commacat.html#E1:9_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="commacat.html#E6:9_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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