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

<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">O</font>,<span class="p3">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> : verum </span>}</span> ;<br/>
<a NAME="E1:13_1_1"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">O</font>,<span class="p3">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> : verum </span>}</span>  holds <br/> ex <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">w</font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">O</font>,<span class="p3">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<b>thus </b><a NAME="E4:13_1_1_1"/>
 ex <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">w</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">O</font>,<span class="p3">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> : verum </span>}</span>  as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E3:13_1_1"/>
for <font color="Olive">u</font>, <font color="Olive">v1</font>, <font color="Olive">v2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> holds <br/><font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:13_1_1_2"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">v1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:13_1_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O'</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E37</font></i>: <font color="Maroon">v2</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O'</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<font color="Maroon">O</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O'</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">O'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:13_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/>
<a NAME="E5:13_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_1_3"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_3_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:13_1_1_3_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">O</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:13_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_1_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E3:13_1_1_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E4:13_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:13_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_4"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:13_1_1_4"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">y</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<a NAME="E7:13_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_4_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:13_1_1_4_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">u'</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:13_1_1_4_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:13_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>, <a href="zfmisc_1.html#K1">bool</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:13_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">O</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:13_1_1_5"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:13_1_1_5"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O'</font>,<span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">O</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">O'</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<font color="Maroon">y</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">O</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E7:13_1_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:13_1_1"/>
for <font color="Olive">O</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div>
