<?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>let </b><font color="Maroon">W</font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">M</font>;<br/>



<b>thus </b><a NAME="E1:57"/>
for <font color="Olive">a</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="midsp_2.html#NM2">Vector</a> of <font color="Maroon">W</font>  ex <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> st <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="midsp_2.html#NM2">Vector</a> of <font color="Maroon">W</font>;<br/>



<b>set </b><font color="Maroon">w</font> = the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E1:57_1"/>
the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</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>,the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green">E25</font></i>: 
the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E4:57_1"/>
<font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T7" target="_self">Th7</a></i>;<br/>


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

<b>thus </b><a NAME="E2:57"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/>

<b>set </b><font color="Maroon">w</font> = the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E2:57_2"/>
the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</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>,the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:57_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T7" target="_self">Th7</a>, </i>;<br/>


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

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





<b>set </b><font color="Maroon">w</font> = the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E3:57"/>
the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</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>,the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:57"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/>


</div>
