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

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

<a NAME="E1:28"/>
( ( for <font color="Olive">u</font>, <font color="Olive">u1</font>, <font color="Olive">u2</font>, <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">v2</font>, <font color="Olive">w</font>, <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">v1</font> &amp; <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font> &amp; <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u2</font>,<font color="Olive">v2</font> &amp; not <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> &amp; not <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> holds <br/><font color="Olive">u2</font>,<font color="Olive">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font> ) implies for <font color="Olive">u</font>, <font color="Olive">u1</font>, <font color="Olive">u2</font>, <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">v2</font>, <font color="Olive">w</font>, <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">v1</font> &amp; <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font> &amp; not <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w1</font> &amp; not <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> holds <br/><font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u2</font>,<font color="Olive">v2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">u</font>, <font color="Olive">u1</font>, <font color="Olive">u2</font>, <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">v2</font>, <font color="Olive">w</font>, <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">v1</font> &amp; <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font> &amp; <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u2</font>,<font color="Olive">v2</font> &amp; not <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> &amp; not <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> holds <br/><font color="Olive">u2</font>,<font color="Olive">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w1</font>
 ;<br/>

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>















<b>assume </b><a NAME="E2:28_1"/><i><font color="Green">E22</font></i>: 
( <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> )
 ;<br/>

<a NAME="E3:28_1"/><i><font color="Green">E28</font></i>: 
( ( 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> &amp; not <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>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> ) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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="28_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:28_1_1_1"/><i><font color="Green">E29</font></i>: 
(  not <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> or ( 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> &amp; not <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>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> ) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_1_1"/>
<font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w</font>,<font color="Maroon">w1</font>
 ;<br/><a NAME="E2:28_1_1_1_1"/><b>then </b>
<font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">w1</font>,<font color="Maroon">w</font>
 <b>by </b><i/>;<br/><a NAME="E3:28_1_1_1_1"/><b>then </b>
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:28_1_1_1_1"/>
( ( 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> &amp; not <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>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> ) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_1"/>
( ( 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> &amp; not <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>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> ) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1_1"/>
( ( 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> &amp; not <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>,<font color="Maroon">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> ) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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> )
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_2"/>
( <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">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</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> )
 ;<br/><a NAME="E2:28_1_2"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_2"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_3"/>
( <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">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 ;<br/><a NAME="E2:28_1_3"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_3"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_4"/>
( <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">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> &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> )
 ;<br/><a NAME="E2:28_1_4"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_4"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> or <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_1_4"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_5"/>
( <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">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> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 ;<br/><a NAME="E2:28_1_5"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_5"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font> or <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_1_5"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_6"/>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</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> )
 ;<br/><a NAME="E2:28_1_6"/><b>then </b>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</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> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_6"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E4:28_1_6"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:28_1_6"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_7"/>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 ;<br/><a NAME="E2:28_1_7"/><b>then </b>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_7"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E4:28_1_7"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:28_1_7"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_8"/>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <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> &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> )
 ;<br/><a NAME="E2:28_1_8"/><b>then </b>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</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> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_8"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_1_8"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_9"/>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <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> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 ;<br/><a NAME="E2:28_1_9"/><b>then </b>
( <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v1</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">w</font>,<font color="Maroon">w1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v2</font>,<font color="Maroon">u2</font> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_9"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v2</font>,<font color="Maroon">u2</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_1_9"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:28_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <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="txt" href="dirort.html#E1:28_1"><i><font color="Green">E20</font></i></a>, , <a class="txt" href="dirort.html#E2:28_1"><i><font color="Green">E22</font></i></a>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:28"/>
( <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> implies <font color="Maroon">AS</font> is <a href="dirort.html#V3" target="_self">right_transitive</a> )
 <b>by </b><i>, </i>;<br/>


</div>
