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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>;<br/>

<b>set </b><font color="Maroon" title="c2">w</font> =  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:33"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>  ex <font color="Olive" title="b3">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c4">x</font> as    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55" title="MIDSP_1:th.55">MIDSP_1:55</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">q</font>
;<br/>

<b>thus </b><a NAME="E4:33_1"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E3:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>





<b>assume </b><a NAME="E1:33_2"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
 ;<br/>

<b>then </b> <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> = 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></i>
<br/>.= 
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E3:33_2"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T59" title="MIDSP_1:th.59">MIDSP_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">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" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>





<a NAME="E1:33_3"/>
( <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>
<br/>.= 
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T60" title="MIDSP_1:th.60">MIDSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
 <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:33"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>;<br/>


</div>
