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<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>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>let </b><font color="Maroon">d</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/>
<b>set </b><font color="Maroon">G</font> = the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E1:58"/><i><font color="Green">E25</font></i>: 
( the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> is <a href="midsp_2.html#V1" target="_self">midpoint_operator</a> &amp; the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> is <a href="rlvect_1.html#V4">add-associative</a> &amp; the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp; the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> is <a href="rlvect_1.html#V6">right_complementable</a> &amp; the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> is <a href="rlvect_1.html#V3">Abelian</a> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E2:58"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">M</font>,the <a href="midsp_2.html#U1" target="_self">algebra</a> of <font color="Maroon">W</font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> the <a href="midsp_2.html#U2" target="_self">function</a> of <font color="Maroon">W</font> &amp; 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>thus </b><a NAME="E3:58"/>
<font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11" target="_self">0.</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E4:58"/>
( <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>  <a href="midsp_2.html#K11" target="_self">0.</a> <font color="Maroon">W</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E5:58"/>
<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>  <a href="rlvect_1.html#K5">-</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E6:58"/>
( <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">c</font>,<font color="Maroon">d</font> implies <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E7:58"/>
for <font color="Olive">b</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">a</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="58_1"><b>proof </b></a><div class="add">

<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">x</font> be   <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">W</font>;<br/>



<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:58_1"/><i><font color="Green">E42</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><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, <a class="txt" href="midsp_2.html#E2:58_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E3:58_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#T9" target="_self">Th9</a></i>;<br/>


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

<b>thus </b><a NAME="E8:58"/>
( <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> implies <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#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E9:58"/>
( <font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> iff <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E10:58"/>
( <font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font> iff <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9" target="_self">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, , </i>;<br/>

<b>thus </b><a NAME="E11:58"/>
( ( <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> implies <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_2.html#K10" target="_self">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; ( <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_2.html#K10" target="_self">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <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>, </i>;<br/>


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