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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>let </b><font color="Maroon">AS</font> be   <a href="dirort.html#NM1">Oriented_Orthogonality_Space</a>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E20</font></i>: 
(  <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">AS</font> <a href="hidden.html#R1">=</a>  <a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:24"/><i><font color="Green">E22</font></i>: 
 <a href="analort.html#K6">CESpace</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#K4">CORTE</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#D7">ANALORT:def 7</a></i>;<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>thus </b><a NAME="E1:24_1"/>
( <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> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">w1'</font> = <font color="Maroon">w1</font>, <font color="Maroon">u2'</font> = <font color="Maroon">u2</font>, <font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E2:24_1_1"/>
( <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">w'</font>,<font color="Maroon">w1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">w'</font>,<font color="Maroon">w1'</font>,<font color="Maroon">u2'</font>,<font color="Maroon">v2'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">w'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1'</font> &amp; not <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> implies <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u2'</font>,<font color="Maroon">v2'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T42">ANALORT:42</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1"/>
( <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> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> )
 <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>thus </b><a NAME="E1:24_2"/>
( <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> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">w1'</font> = <font color="Maroon">w1</font>, <font color="Maroon">u2'</font> = <font color="Maroon">u2</font>, <font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E2:24_2_1"/>
( <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">w'</font>,<font color="Maroon">w1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u2'</font>,<font color="Maroon">v2'</font>,<font color="Maroon">w'</font>,<font color="Maroon">w1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">w'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1'</font> &amp; not <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> implies <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u2'</font>,<font color="Maroon">v2'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T46">ANALORT:46</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2_1"/>
( <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> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> )
 <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>thus </b><a NAME="E1:24_3"/>
( <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> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">w1'</font> = <font color="Maroon">w1</font>, <font color="Maroon">u2'</font> = <font color="Maroon">u2</font>, <font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E2:24_3_1"/>
( <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">w'</font>,<font color="Maroon">w1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u2'</font>,<font color="Maroon">v2'</font> <a href="analort.html#R1">are_COrte_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">u1'</font> &amp; not <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> implies <font color="Maroon">u2'</font>,<font color="Maroon">v2'</font>,<font color="Maroon">w'</font>,<font color="Maroon">w1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T48">ANALORT:48</a></i>;<br/>
<b>hence </b><a NAME="E3:24_3_1"/>
( <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> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font> )
 <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>thus </b><a NAME="E1:24_4"/>
( <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> implies <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="24_4_1"><b>proof </b></a><div class="add">

<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">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><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E2:24_4_1"/>
( <font color="Maroon">u'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v'</font>,<font color="Maroon">v1'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> implies <font color="Maroon">v'</font>,<font color="Maroon">v1'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">u'</font> <a href="analort.html#R1">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T18">ANALORT:18</a></i>;<br/>
<b>hence </b><a NAME="E3:24_4_1"/>
( <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> implies <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="dirort.html#T1" target="_self">Th1</a>, <a class="ref" href="analort.html#T56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
( <font color="Maroon">AS</font> is <a href="dirort.html#V5" target="_self">Euclidean_like</a> &amp; <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> &amp; <font color="Maroon">AS</font> is <a href="dirort.html#V3" target="_self">right_transitive</a> &amp; <font color="Maroon">AS</font> is <a href="dirort.html#V2" target="_self">bach_transitive</a> )
 <b>by </b><i>, , , , , , </i>;<br/>


</div>
