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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="midsp_1.html#L1">MidStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="midsp_1.html#NM1">MidSp</a>
 ;<br/><b>thus </b><a NAME="E2:44_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#V3">Abelian</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="midsp_2.html#V1" target="_self">midpoint_operator</a>  <a href="rlvect_1.html#L1">LoopStr</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="midsp_1.html#NM1">MidSp</a> <b>by </b><i><a class="txt" href="midsp_2.html#E1:44_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:44_1_1"/><i><font color="Green">E38</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E1:44_1_1_1"/>
(  <a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a>  <a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T24" target="_self">Th24</a>, <a class="ref" href="midsp_2.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E3:44_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>, <a href="midsp_1.html#K19">vectgroup</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:44_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>2</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</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="midsp_2.html#V1" target="_self">midpoint_operator</a>  <a href="rlvect_1.html#L1">LoopStr</a> , <font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:44"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E3:44"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="midsp_1.html#NM1">MidSp</a>
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:44"><i><font color="Green">E37</font></i></a>, <a class="ref" href="midsp_2.html#T23" target="_self">Th23</a></i>;<br/>


</div>
