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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>let </b><font color="Maroon" title="c4">AS</font> be   <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E14">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> &amp; <font color="Maroon" title="c4">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="E2:24"/><i><font color="Green" title="E15">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/>
<div><i><font color="Green" title="E16">A3</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" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>;<br/><b>thus </b><a NAME="E1:24_1"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <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">u2</font>,<font color="Maroon" title="c10">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" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="dirort.html#E2:24"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E2:24_1_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">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="c15">w'</font>,<font color="Maroon" title="c18">w1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">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="c15">w'</font>,<font color="Maroon" title="c18">w1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</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="c15">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T42" title="ANALORT:th.42">ANALORT:42</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <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">u2</font>,<font color="Maroon" title="c10">v2</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E17">A4</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" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>;<br/><b>thus </b><a NAME="E1:24_2"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <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">u2</font>,<font color="Maroon" title="c10">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" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="dirort.html#E2:24"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E2:24_2_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">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="c14">v'</font>,<font color="Maroon" title="c17">v1'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</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="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</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="c15">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T46" title="ANALORT:th.46">ANALORT:46</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2_1"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <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">u2</font>,<font color="Maroon" title="c10">v2</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">ANALORT:56</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E18">A5</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" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>;<br/><b>thus </b><a NAME="E1:24_3"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <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">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">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" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="dirort.html#E2:24"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E2:24_3_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">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="c14">v'</font>,<font color="Maroon" title="c17">v1'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</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="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</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="c13">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T48" title="ANALORT:th.48">ANALORT:48</a></i>;<br/>
<b>hence </b><a NAME="E3:24_3_1"/>
( <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="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <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">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">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" 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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>;<br/><b>thus </b><a NAME="E1:24_4"/>
( <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> implies <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="24_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c10">v'</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c11">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c12">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#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="dirort.html#E2:24"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E2:24_4_1"/>
( <font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c11">u1'</font>,<font color="Maroon" title="c10">v'</font>,<font color="Maroon" title="c12">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="c10">v'</font>,<font color="Maroon" title="c12">v1'</font>,<font color="Maroon" title="c11">u1'</font>,<font color="Maroon" title="c9">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#E1:24"><i><font color="Green" title="E14">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="E3:24_4_1"/>
( <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> implies <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="txt" href="dirort.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="analort.html#T56" title="ANALORT:th.56">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" title="c4">AS</font> is <a href="dirort.html#V5" title="DIRORT:attr.5">Euclidean_like</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:24"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="dirort.html#E4:24"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="dirort.html#E5:24"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="dirort.html#D4" target="_self" title="DIRORT:def.4">Def4</a>, <a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self" title="DIRORT:def.6">Def6</a>, <a class="ref" href="dirort.html#D7" target="_self" title="DIRORT:def.7">Def7</a></i>;<br/>


</div>
