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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="E1:2"/><i><font color="Green" title="E2">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#K6" title="ANALORT:func.6">CESpace</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="E2:2"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="analort.html#K6" title="ANALORT:func.6">CESpace</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#K4" title="ANALORT:func.4">CORTE</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#D7" title="ANALORT:def.7">ANALORT:def 7</a></i>;<br/>
<b>thus </b><a NAME="E3:2"/>
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#K6" title="ANALORT:func.6">CESpace</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="2_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#K6" title="ANALORT:func.6">CESpace</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#E2:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:2_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="2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#T20" title="ANALORT:th.20">ANALORT:20</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E3:2_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="2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#T22" title="ANALORT:th.22">ANALORT:22</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E4:2_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="2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T30" title="ANALORT:th.30">ANALORT:30</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E5:2_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="2_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T32" title="ANALORT:th.32">ANALORT:32</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E6:2_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="2_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#T34" title="ANALORT:th.34">ANALORT:34</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E7:2_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="2_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T36" title="ANALORT:th.36">ANALORT:36</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E8:2_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="2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:2_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T18" title="ANALORT:th.18">ANALORT:18</a></i>;<br/>
<b>hence </b><a NAME="E2:2_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#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


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

<b>thus </b><a NAME="E9:2_1"/>
verum
 ;<br/>


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

<b>thus </b><a NAME="E4:2"/>
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#K6" title="ANALORT:func.6">CESpace</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#K6" title="ANALORT:func.6">CESpace</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="2_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#K6" title="ANALORT:func.6">CESpace</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#E2:2"><i><font color="Green" title="E3">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="E3:2_2"/><i><font color="Green" title="E4">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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T38" title="ANALORT:th.38">ANALORT:38</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#K6" title="ANALORT:func.6">CESpace</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#E2:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E5:2_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#E3:2_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>
<b>hence </b><a NAME="E6:2_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#K6" title="ANALORT:func.6">CESpace</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#K6" title="ANALORT:func.6">CESpace</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#E2:2"><i><font color="Green" title="E3">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="E7:2"/><i><font color="Green" title="E4">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#R1" title="ANALORT:pred.1">are_COrte_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#E1:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analort.html#T40" title="ANALORT:th.40">ANALORT:40</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#K6" title="ANALORT:func.6">CESpace</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#E2:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E9:2"/>
( <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="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#E7:2"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>
<b>hence </b><a NAME="E10:2"/>
 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#K6" title="ANALORT:func.6">CESpace</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>
