<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a>
 ;<br/>

<a NAME="E2:30"/>
( ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><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">u2</font>,<font color="Olive" title="b6">v2</font> ) iff 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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1"/><i><font color="Green" title="E17">A2</font></i>: 
( ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><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">u2</font>,<font color="Olive" title="b6">v2</font> ) implies 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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><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">u2</font>,<font color="Olive" title="b6">v2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">u1</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">u2</font>, <font color="Maroon" title="c7">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>











<b>assume </b><a NAME="E2:30_1_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u2</font>,<font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">u2</font>,<font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c6">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:30_1_1"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c8">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w1</font> &amp; <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> )
 <b>by </b><i><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></i>;<br/>
<div><i><font color="Green" title="E21">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_1"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">u1</font>
 ;<br/><a NAME="E2:30_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>by </b><i><a class="txt" href="dirort.html#E1:30_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="dirort.html#E2:30_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="dirort.html#E5:30_1_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_1"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>by </b><i><a class="txt" href="dirort.html#E5:30_1_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_2"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font>
 ;<br/><b>consider </b><font color="Maroon" title="c10">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>consider </b><font color="Maroon" title="c11">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:30_1_1_2"/><i><font color="Green" title="E23">A8</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">w1</font> &amp; <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c11">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> )
 <b>by </b><i><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></i>;<br/><a NAME="E5:30_1_1_2"/>
( <font color="Maroon" title="c10">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">w1</font> &amp; <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c11">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c11">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c2">u</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E4:30_1_1_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E6:30_1_1_2"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>by </b><i><a class="txt" href="dirort.html#E1:30_1_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:30_1_1"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>by </b><i><a class="txt" href="dirort.html#E6:30_1_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1"/>
( ( 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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> ) implies for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><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">u2</font>,<font color="Olive" title="b6">v2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green" title="E18">A9</font></i>: 
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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">u1</font>, <font color="Maroon" title="c4">u2</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font>, <font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>















<b>assume </b><a NAME="E2:30_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> &amp; <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1"/>
( <font color="Maroon" title="c8">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w1</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> )
 ;<br/><a NAME="E2:30_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font>
 <b>by </b><i><a class="txt" href="dirort.html#E1:30_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="dirort.html#E2:30_1_2"><i><font color="Green" title="E19">A10</font></i></a></i>;<br/><a NAME="E3:30_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">u3</font>, <font color="Olive" title="b2">v3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">u3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v3</font> &amp; <font color="Olive" title="b1">u3</font>,<font color="Olive" title="b2">v3</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> &amp; <font color="Olive" title="b1">u3</font>,<font color="Olive" title="b2">v3</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> )
 <b>by </b><i><a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:30_1_2_1"/>
( <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:30"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="dirort.html#E2:30_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="dirort.html#D4" target="_self" title="DIRORT:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_2"/>
( <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1"/>
( ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><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">u2</font>,<font color="Olive" title="b6">v2</font> ) iff 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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:30_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon" title="c1">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> iff 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">u2</font>, <font color="Olive" title="b6">v2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <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="b5">u2</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v2</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> )
 <b>by </b><i><a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a></i>;<br/>


</div>
