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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="dirort.html#NM1">Oriented_Orthogonality_Space</a>;<br/>

<a NAME="E1:29"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> ) implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; not <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>8</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V4" target="_self">left_transitive</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V3" target="_self">right_transitive</a> )
 
<b>by </b><i><a class="ref" href="dirort.html#T17" target="_self">Th17</a></i>;<br/>
<b>assume </b><a NAME="E2:29_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>















<b>assume </b><a NAME="E3:29_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E4:29_1"/><i><font color="Green">E19</font></i>: 
( ( not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E2:29_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:29_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:29_1_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E4:29_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:29_1_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:29_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:29_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:29_1_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E5:29_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:29_1_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:29_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:29_1_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:29_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E2:29_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:29_1_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E4:29_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:29_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E2:29_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="dirort.html#E3:29_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:29_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/>
<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_2"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:29_1_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E4:29_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:29_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_3"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E2:29_1_3"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:29_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dirort.html#E2:29_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dirort.html#E4:29_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="dirort.html#D5" target="_self">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:29_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dirort.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="dirort.html#E5:29_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="dirort.html#E6:29_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="dirort.html#E7:29_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="dirort.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V4" target="_self">left_transitive</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V2" target="_self">bach_transitive</a> )
 <b>by </b><i><a class="ref" href="dirort.html#D4" target="_self">Def4</a>, <a class="ref" href="dirort.html#D6" target="_self">Def6</a></i>;<br/>


</div>
