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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">M</font> be  <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">N</font> be  <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:3"/>
<font color="Maroon">M</font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">w2</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font>;<br/>
<a NAME="E4:3"/>
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> <font color="Maroon">N</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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="E1:3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font>;<br/>
<a NAME="E4:3_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w1</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:3_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E6:3_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E7:3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>
<a NAME="E8:3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E9:3_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:3"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></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/>
<a NAME="E6:3"/>
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> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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="E1:3_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:3_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v'</font> &amp; <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E6:3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E8:3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>
<a NAME="E9:3_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E10:3_2"/>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w1</font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E12:3_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w1</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>
<a NAME="E13:3_2"/>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_5.html#D1" target="_self">RUSUB_5:def 1</a><br/>

<b>thus </b><a NAME="E8:3"/>
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
