<?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>let </b><font color="Maroon" title="c2">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</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="c7">x</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>;<br/>













<b>set </b><font color="Maroon" title="c8">w</font> = the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font>;<br/>
<b>set </b><font color="Maroon" title="c9">G</font> = the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font>;<br/>
<a NAME="E1:58"/><i><font color="Green" title="E41">A1</font></i>: 
( the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></i>;<br/>
<a NAME="E2:58"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c1">M</font>,the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font> &amp; the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</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>,the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></i>;<br/>
<b>thus </b><a NAME="E3:58"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T4" target="_self" title="MIDSP_2:th.4">Th4</a></i>;<br/>

<b>thus </b><a NAME="E4:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T5" target="_self" title="MIDSP_2:th.5">Th5</a></i>;<br/>

<b>thus </b><a NAME="E5:58"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T6" target="_self" title="MIDSP_2:th.6">Th6</a></i>;<br/>

<b>thus </b><a NAME="E6:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T7" target="_self" title="MIDSP_2:th.7">Th7</a></i>;<br/>

<b>thus </b><a NAME="E7:58"/>
for <font color="Olive" title="b1">b</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="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</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="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">b</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="c11">x</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>;<br/>



<b>consider </b><font color="Maroon" title="c12">a</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="E2:58_1"/><i><font color="Green" title="E43">A3</font></i>: 
the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T8" target="_self" title="MIDSP_2:th.8">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">a</font>
;<br/>

<b>thus </b><a NAME="E3:58_1"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:58_1"><i><font color="Green" title="E43">A3</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T9" target="_self" title="MIDSP_2:th.9">Th9</a></i>;<br/>

<b>thus </b><a NAME="E9:58"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></i>;<br/>

<b>thus </b><a NAME="E10:58"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c6">d</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#T16" target="_self" title="MIDSP_2:th.16">Th16</a></i>;<br/>

<b>thus </b><a NAME="E11:58"/>
( ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="midsp_2.html#D4" target="_self" title="MIDSP_2:def.4">Def4</a></i>;<br/>


</div>
