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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <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/>
<a NAME="E2:23_1_1"/>
( ( 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> ) &amp; ( 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> st  <a href="midsp_2.html#K1" target="_self">Double</a> <font color="Olive">b<sub>1</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> holds <br/><font color="Olive">b<sub>1</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> ) )
 
<b>by </b><i><a class="txt" href="midsp_2.html#E23"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<a NAME="E3:23_1_1"/><b>then </b>
 <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>1</sub></font> is <a href="midsp_2.html#V1" target="_self">midpoint_operator</a>
 
<b>by </b><i><a class="ref" href="midsp_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V1">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="midsp_2.html#V1" target="_self">midpoint_operator</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V3">Abelian</a> )
 ;<br/>


</div>
