<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:11"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E2:11"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T48">FUNCT_3:48</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E2:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E5:11"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:11_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:11_1_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E4:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i><a class="txt" href="hilbert3.html#E3:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="hilbert3.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:11_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E4:11_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E5:11_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:11_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E7:11_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="hilbert3.html#E3:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="hilbert3.html#E1:11_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:11_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:11"/>
<span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>


</div>
