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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<i><font color="Green">E23</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K8" target="_self">dom</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="modcat_1.html#D11" target="_self">Def11</a></i>
<br/>.= 
 <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font>

;<br/>
<i><font color="Green">E24</font></i>:  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K9" target="_self">cod</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="modcat_1.html#D12" target="_self">Def12</a></i>
<br/>.= 
 <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>

;<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then </b> <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="modcat_1.html#D14" target="_self">Def14</a></i>
<br/>.= 
 <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>

;<br/><b>hence </b><a NAME="E3:31_3"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="modcat_1.html#E4:31"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_4"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then </b> <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="modcat_1.html#E3:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="modcat_1.html#E4:31"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
 <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">c<sub>7</sub></font>

;<br/><b>hence </b><a NAME="E3:31_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="modcat_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> iff  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:31"><i><font color="Green">E25</font></i></a></i>;<br/>


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