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

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> ) &amp; ( 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>  st  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">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> ) )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">consider </span><font color="Maroon" title="c3">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/>
<span class="kw">thus </span><a NAME="E2:23"/>
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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>  st  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">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></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">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/>
<span class="kw">let </span><font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">aa</font> = <font color="Maroon" title="c5">a</font> as    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E4:23_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c6">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="c4">p</font>,<font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T55" title="MIDSP_1:th.55">MIDSP_1:55</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">xx</font> =  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">q</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> =  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/>

<a NAME="E6:23_1"/>
<span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">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="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c7">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="midsp_1.html#T62" title="MIDSP_1:th.62">MIDSP_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23_1"/>
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c4">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> implies <font color="Maroon" title="c4">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> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">aa</font> = <font color="Maroon" title="c4">a</font> as    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:23"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T55" title="MIDSP_1:th.55">MIDSP_1:55</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:23"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T55" title="MIDSP_1:th.55">MIDSP_1:55</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:23"/>
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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></span><br/>

<span class="kw">then </span><font color="Maroon" title="c5">aa</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Maroon" title="c5">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>
<br/>.= 
 <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>
;<br/>
<a NAME="E10:23"/><span class="kw">then </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="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="c3">p</font>,<font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="midsp_1.html#T58" title="MIDSP_1:th.58">MIDSP_1:58</a></span>;<br/>
<a NAME="E11:23"/><span class="kw">then </span>
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c3">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="c3">p</font>,<font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T60" title="MIDSP_1:th.60">MIDSP_1:60</a></span>;<br/>
<a NAME="E12:23"/><span class="kw">then </span>
 <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> <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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="midsp_1.html#T59" title="MIDSP_1:th.59">MIDSP_1:59</a></span>;<br/>
<a NAME="E13:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T57" title="MIDSP_1:th.57">MIDSP_1:57</a></span>;<br/>
<font color="Maroon" title="c3">p</font> = 
<font color="Maroon" title="c3">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#D4" title="MIDSP_1:def.4">MIDSP_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="midsp_1.html#D4" title="MIDSP_1:def.4">MIDSP_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><font color="Maroon" title="c4">a</font> = 
 <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="midsp_1.html#T58" title="MIDSP_1:th.58">MIDSP_1:58</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T18" target="_self" title="MIDSP_2:th.18">Th18</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
