<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:44_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:44_1_1"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E2:44_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></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="coh_sp.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:44_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font> st <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:44_1_2"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:44_1_2"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:44_1_2"/><b>then </b>
(  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:44_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a>, <a class="ref" href="coh_sp.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_2"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></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="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:44_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font> st <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1_3"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E2:44_1_3"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E3:44_1_3"/><i><font color="Green">E30</font></i>: 
(  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></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="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E4:44_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:44_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="coh_sp.html#E2:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="coh_sp.html#T22" target="_self">Th22</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="coh_sp.html#E2:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="coh_sp.html#E3:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="coh_sp.html#E4:44_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="coh_sp.html#E1:44_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="coh_sp.html#E2:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="coh_sp.html#E3:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="coh_sp.html#T23" target="_self">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="coh_sp.html#E2:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="coh_sp.html#E3:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="coh_sp.html#E4:44_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="coh_sp.html#E1:44_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="coh_sp.html#E3:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:44_1"/><i><font color="Green">E28</font></i>: 
( <span class="p1">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K8" target="_self">id$</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="coh_sp.html#D17" target="_self">Def17</a>, <a class="ref" href="coh_sp.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E5:44_1"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="coh_sp.html#D14" target="_self">Def14</a>, <a class="ref" href="coh_sp.html#D15" target="_self">Def15</a></i>;<br/><b>thus </b><a NAME="E6:44_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font> st <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:44_1_4"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:44_1_4"/><i><font color="Green">E30</font></i>: 
 <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></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="coh_sp.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:44_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:44_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="coh_sp.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_4"/>
<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E4:44_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="coh_sp.html#E1:44_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="coh_sp.html#E2:44_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E7:44_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E8:44_1"/><i><font color="Green">E30</font></i>: 
 <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></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="coh_sp.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E9:44_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E7:44_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="coh_sp.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E10:44_1"/>
<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E4:44_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="coh_sp.html#E7:44_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="coh_sp.html#E8:44_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="coh_sp.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:44"/>
 <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="coh_sp.html#K3" target="_self">CSp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K10" target="_self">CDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K11" target="_self">CCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K12" target="_self">CComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K13" target="_self">CId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div>
