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

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

<b>set </b><font color="Maroon">w</font> =  <a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font>;<br/>
<a NAME="E1:33"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><br/> for <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>  ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">x</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">xx</font> = <font color="Maroon">x</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">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">xx</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>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E4:33_1"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">r</font> holds <br/><font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><a NAME="E1:33_2"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then </b> <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
 <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:33_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T59">MIDSP_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33"/>
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>





<a NAME="E1:33_3"/>
( <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</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">p</font>,<font color="Maroon">q</font> &amp; <span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</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/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> = 
<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>
<b>by </b><i/>
<br/>.= 
 <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T60">MIDSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
 <a href="midsp_2.html#K5" target="_self">vect</a> <font color="Maroon">M</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">M</font>, <a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font>
 <b>by </b><i>, , </i>;<br/>


</div>
