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<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">G</font> =  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>thus </b><a NAME="E1:20"/>
for <font color="Olive" title="b1">a</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="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> st  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</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="c4">aa</font> = <font color="Maroon" title="c3">a</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="c5">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/>
<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="E4:20_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c4">aa</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="c5">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>set </b><font color="Maroon" title="c7">xx</font> =  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/>
<a NAME="E5:20_1"/><i><font color="Green" title="E20">A2</font></i>: 
<span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">q</font></span>)</span></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="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">aa</font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E4:20_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="midsp_1.html#T62" title="MIDSP_1:th.62">MIDSP_1:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> =  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">q</font></span>)</span> as   <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> <b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">x</font>
;<br/>

<b>thus </b><a NAME="E7:20_1"/>
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E5:20_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">a</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>assume </b><a NAME="E2:20"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c4">aa</font> = <font color="Maroon" title="c3">a</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="c5">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/>
<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="E6:20"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c4">aa</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="c5">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/>
<i><font color="Green" title="E21">A5</font></i>: <font color="Maroon" title="c4">aa</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Maroon" title="c4">aa</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
<b>by </b><i><a class="txt" href="midsp_2.html#E2:20"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>
<br/>.= 
 <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <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="c7">r</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="E9:20"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c4">aa</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="c6">q</font>,<font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T55" title="MIDSP_1:th.55">MIDSP_1:55</a></i>;<br/>
<a NAME="E10:20"/>
<span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">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="c6">q</font>,<font color="Maroon" title="c7">r</font></span>)</span> <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="c5">p</font>,<font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E6:20"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="midsp_2.html#E7:20"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="midsp_2.html#E9:20"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="midsp_1.html#T58" title="MIDSP_1:th.58">MIDSP_1:58</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">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="c5">p</font>,<font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T60" title="MIDSP_1:th.60">MIDSP_1:60</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">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="c6">q</font>,<font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E6:20"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="midsp_2.html#E9:20"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="midsp_1.html#T59" title="MIDSP_1:th.59">MIDSP_1:59</a></i>;<br/>
<a NAME="E13:20"/><b>then </b><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">q</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T57" title="MIDSP_1:th.57">MIDSP_1:57</a></i>;<br/>
<font color="Maroon" title="c5">p</font> = 
<font color="Maroon" title="c5">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D4" title="MIDSP_1:def.4">MIDSP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="txt" href="midsp_2.html#E13:20"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="midsp_1.html#D4" title="MIDSP_1:def.4">MIDSP_1:def 4</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c3">a</font> = 
 <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="txt" href="midsp_2.html#E6:20"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="midsp_1.html#T58" title="MIDSP_1:th.58">MIDSP_1:58</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
<b>by </b><i><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></i>
;<br/><br/>


</div>
