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

<b>set </b><font color="Maroon">G</font> =  <a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font>;<br/>
<b>thus </b><a NAME="E1:20"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> st  <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">xx</font> =  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">q</font></span>)</span>;<br/>
<a NAME="E5:20_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="midsp_1.html#K10">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font>
 
<b>by </b><i>, <a class="ref" href="midsp_1.html#T62">MIDSP_1:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">q</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E7:20_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E8:20_1"/>
 <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:20"/>
 <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Maroon">a</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#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<a NAME="E3:20"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</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#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E7:20"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55">MIDSP_1:55</a></i>;<br/>
<i><font color="Green">E62</font></i>: <font color="Maroon">aa</font> <a href="midsp_1.html#K10">+</a> <font color="Maroon">aa</font> = 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="midsp_1.html#K9">ID</a> <font color="Maroon">M</font>
<b>by </b><i/>
;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E10:20"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">q</font>,<font color="Maroon">r</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#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="midsp_1.html#K10">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K11">vect</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a>, <a class="ref" href="midsp_2.html#T1" target="_self">Th1</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#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">p</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#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="midsp_2.html#T1" target="_self">Th1</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">E64</font></i>: 
<font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T57">MIDSP_1:57</a></i>;<br/>
<font color="Maroon">p</font> = 
<font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>
;<br/>
<b>hence </b><font color="Maroon">a</font> = 
 <a href="midsp_1.html#K9">ID</a> <font color="Maroon">M</font>
<b>by </b><i>, <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#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
;<br/><br/>


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