<?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="E1:22"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V2" target="_self">bach_transitive</a> 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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V2" target="_self">bach_transitive</a>
 ;<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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><a NAME="E2:22_1"/><i><font color="Green">E15</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#R2" 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>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E3:22_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</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="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="dirort.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1"/>
<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#E1:22_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dirort.html#E2:22_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="dirort.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:22"/>
( ( 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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> ) implies <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V2" target="_self">bach_transitive</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_2"/>
( ( 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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> ) 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="22_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_1"/><i><font color="Green">E14</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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>
 ;<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="E2:22_2_1"/><i><font color="Green">E15</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2_1_1"/>
( <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> &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> )
 ;<br/><a NAME="E2:22_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="dirort.html#R2" 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#E2:22_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="dirort.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E3:22_2_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#E1:22_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dirort.html#E2:22_2_1"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_2"/>
( ( 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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> ) 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></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="dirort.html#V2" target="_self">bach_transitive</a> iff 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> 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#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> )
 <b>by </b><i><a class="txt" href="dirort.html#E1:22"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
