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<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>;<br/>
<a NAME="E2:3"/><i><font color="Green" title="E4">A2</font></i>: 
 <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="analort.html#K5" title="ANALORT:func.5">CORTM</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="analort.html#D8" title="ANALORT:def.8">ANALORT:def 8</a></i>;<br/>
<b>thus </b><a NAME="E3:3"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b1">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b5">w</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b5">w</font> &amp; ( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b4">v1</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b2">u1</font> &amp; not <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> implies <font color="Olive" title="b2">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v1</font> ) &amp; ( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b4">v1</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b5">w</font> &amp; not <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">w</font> implies <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b4">v1</font> ) &amp; ( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b4">v1</font> implies <font color="Olive" title="b3">v</font>,<font color="Olive" title="b1">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b2">u1</font> ) &amp; ( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b4">v1</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">w</font> implies <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b5">w</font> ) &amp; (  not <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> or <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> or <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</font>,<font color="Olive" title="b1">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" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c11">v'</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c12">w'</font> = <font color="Maroon" title="c9">w</font>, <font color="Maroon" title="c13">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c14">v1'</font> = <font color="Maroon" title="c8">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="dirort.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:3_1"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c9">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" title="c10">u'</font>,<font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="analort.html#T21" title="ANALORT:th.21">ANALORT:21</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_1"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E3:3_1"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c9">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" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c12">w'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="analort.html#T23" title="ANALORT:th.23">ANALORT:23</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_2"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E4:3_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> implies <font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c13">u1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v'</font> implies <font color="Maroon" title="c13">u1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v1'</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T31" title="ANALORT:th.31">ANALORT:31</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_3"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> implies <font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E5:3_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w</font> &amp; not <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">w</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c8">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" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c12">w'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T33" title="ANALORT:th.33">ANALORT:33</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_4"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w</font> &amp; not <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">w</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c8">v1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> implies <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c6">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" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c13">u1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="analort.html#T35" title="ANALORT:th.35">ANALORT:35</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_5"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> implies <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">u</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c6">u1</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E7:3_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">w</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">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" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c12">w'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T37" title="ANALORT:th.37">ANALORT:37</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_6"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">w</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>


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

<b>thus </b><a NAME="E8:3_1"/>
(  not <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> or <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> or <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">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" title="c10">u'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c10">u'</font>,<font color="Maroon" title="c13">u1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c11">v'</font>,<font color="Maroon" title="c14">v1'</font>,<font color="Maroon" title="c13">u1'</font>,<font color="Maroon" title="c10">u'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T19" title="ANALORT:th.19">ANALORT:19</a></i>;<br/>
<b>hence </b><a NAME="E2:3_1_7"/>
(  not <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> or <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> or <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">u</font> )
 <b>by </b><i><a class="ref" href="analort.html#T57" title="ANALORT:th.57">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" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>  ex <font color="Olive" title="b4">u1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> st <br/>( <font color="Olive" title="b3">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">u1</font> &amp; <font color="Olive" title="b3">w</font>,<font color="Olive" title="b4">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">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" title="c5">u</font>, <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c9">v'</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c10">w'</font> = <font color="Maroon" title="c7">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="dirort.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">u1'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:3_2"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c10">w'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">u1'</font> &amp; <font color="Maroon" title="c10">w'</font>,<font color="Maroon" title="c11">u1'</font>,<font color="Maroon" title="c8">u'</font>,<font color="Maroon" title="c9">v'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T39" title="ANALORT:th.39">ANALORT:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">u1</font> = <font color="Maroon" title="c11">u1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <b>by </b><i><a class="txt" href="dirort.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<a NAME="E5:3_2"/>
( <font color="Maroon" title="c7">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">u1</font> &amp; <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c12">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E3:3_2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>
<b>hence </b><a NAME="E6:3_2"/>
 ex <font color="Olive" title="b1">u1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> st <br/>( <font color="Maroon" title="c7">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u1</font> &amp; <font color="Maroon" title="c7">w</font>,<font color="Olive" title="b1">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> )
 ;<br/>


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

<b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c9">v'</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c10">w'</font> = <font color="Maroon" title="c7">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="dirort.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">u1'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:3"/><i><font color="Green" title="E5">A4</font></i>: 
( <font color="Maroon" title="c10">w'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">u1'</font> &amp; <font color="Maroon" title="c8">u'</font>,<font color="Maroon" title="c9">v'</font>,<font color="Maroon" title="c10">w'</font>,<font color="Maroon" title="c11">u1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="analort.html#T41" title="ANALORT:th.41">ANALORT:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">u1</font> = <font color="Maroon" title="c11">u1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <b>by </b><i><a class="txt" href="dirort.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<a NAME="E9:3"/>
( <font color="Maroon" title="c7">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">u1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c12">u1</font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E7:3"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>;<br/>
<b>hence </b><a NAME="E10:3"/>
 ex <font color="Olive" title="b1">u1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> st <br/>( <font color="Maroon" title="c7">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">w</font>,<font color="Olive" title="b1">u1</font> )
 ;<br/>


</div>
