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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:3"/><i><font color="Green">E20</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E2:3"/><i><font color="Green">E22</font></i>: 
 <a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="analort.html#K5">CORTM</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="analort.html#D8">ANALORT:def 8</a></i>;<br/>
<b>thus </b><a NAME="E3:3"/>
for <font color="Olive">u</font>, <font color="Olive">u1</font>, <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> holds <br/> ( <font color="Olive">u</font>,<font color="Olive">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">w</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w</font> &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">u1</font> &amp; not <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> implies <font color="Olive">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">w</font> &amp; not <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">w</font> implies <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">v1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> implies <font color="Olive">v</font>,<font color="Olive">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">u1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">w</font> implies <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">w</font> ) &amp; (  not <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">v1</font> or <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u</font>,<font color="Olive">u1</font> or <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">u</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font>, <font color="Maroon">u1'</font> = <font color="Maroon">u1</font>, <font color="Maroon">v1'</font> = <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:3_1"/>
<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_1"/>
<font color="Maroon">u'</font>,<font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analort.html#T21">ANALORT:21</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_1"/>
<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E3:3_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_2"/>
<font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">w'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analort.html#T23">ANALORT:23</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_2"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E4:3_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> implies <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_3"/>
( <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">u1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> implies <font color="Maroon">u1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T31">ANALORT:31</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_3"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> implies <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E5:3_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> &amp; not <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_4"/>
( <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> implies <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">w'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T33">ANALORT:33</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_4"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> &amp; not <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> implies <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_5"/>
( <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> implies <font color="Maroon">v'</font>,<font color="Maroon">u'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">u1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="analort.html#T35">ANALORT:35</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_5"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> implies <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E7:3_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_6"/>
( <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> implies <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">w'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T37">ANALORT:37</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_6"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E8:3_1"/>
(  not <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">v1</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_7"/>
(  not <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">u'</font>,<font color="Maroon">u1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T19">ANALORT:19</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_7"/>
(  not <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">v1</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E9:3_1"/>
verum
 ;<br/>


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

<b>thus </b><a NAME="E4:3"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>  ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Olive">w</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u</font>,<font color="Olive">v</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:3_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">w'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1'</font> &amp; <font color="Maroon">w'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u'</font>,<font color="Maroon">v'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="analort.html#T39">ANALORT:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u1'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:3_2"/>
( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>
<b>hence </b><a NAME="E6:3_2"/>
 ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Maroon">w</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> )
 ;<br/>


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

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">w'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1'</font> &amp; <font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">w'</font>,<font color="Maroon">u1'</font> <a href="analort.html#R2">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="analort.html#T41">ANALORT:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u1'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:3"/>
( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">u1</font> )
 
<b>by </b><i>, <a class="ref" href="analort.html#T57">ANALORT:57</a></i>;<br/>
<b>hence </b><a NAME="E10:3"/>
 ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7">CMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Olive">u1</font> )
 ;<br/>


</div>
