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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:21"/>
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> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:21_1"/>
<font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:21"/>
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="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">N</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">v1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:21_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E6:21_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:21"/>
<font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
