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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>thus </b><a NAME="E1:20"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st  <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55">MIDSP_1:55</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<a NAME="E5:20_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_1.html#K11">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E4:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="midsp_1.html#T62">MIDSP_1:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E7:20_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E5:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E8:20_1"/>
 <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E7:20_1"><i><font color="Green">E23</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:20"/>
 <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:20"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:20"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55">MIDSP_1:55</a></i>;<br/>
<i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K11">+</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E3:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>
<br/>.= 
 <a href="midsp_1.html#K10">ID</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:20"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55">MIDSP_1:55</a></i>;<br/>
<a NAME="E11:20"/>
<span class="p1">(<span class="default"><a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_1.html#K11">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E7:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="midsp_2.html#E8:20"><i><font color="Green">E23</font></i></a>, <a class="txt" href="midsp_2.html#E10:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="midsp_1.html#T58">MIDSP_1:58</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
 <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T60">MIDSP_1:60</a></i>;<br/>
<a NAME="E13:20"/><b>then </b>
 <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K12">vect</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E7:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="midsp_2.html#E10:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="midsp_1.html#T59">MIDSP_1:59</a></i>;<br/>
<a NAME="E14:20"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T57">MIDSP_1:57</a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E14:20"><i><font color="Green">E25</font></i></a>, <a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>
;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
 <a href="midsp_1.html#K10">ID</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E7:20"><i><font color="Green">E22</font></i></a>, <a class="ref" href="midsp_1.html#T58">MIDSP_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self">Th18</a></i>
;<br/><br/>


</div>
