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<div class="add">

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

<a NAME="E1:21"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> implies for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a>
 ;<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">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>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">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>











<b>assume </b><a NAME="E2:21_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> )
 ;<br/>

<a NAME="E3:21_1"/><b>then </b>
 ex <font color="Olive">u</font>, <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Maroon">v</font>,<font color="Maroon">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">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/>
<b>hence </b><a NAME="E4:21_1"/>
<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>
 <b>by </b><i>, <a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
( ( for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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 <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<a NAME="E1:21_2"/>
( ( for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E2:21_2_1"/><i><font color="Green">E50</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">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><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> )
 ;<br/><a NAME="E2:21_2_1_1"/><b>then </b>
<font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="dirort.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E3:21_2_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</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">w</font>,<font color="Maroon">w1</font> )
 <b>by </b><i>, <a class="ref" href="dirort.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_2_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</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">w</font>,<font color="Maroon">w1</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_2"/>
( ( for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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 <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">AS</font> is <a href="dirort.html#V4" target="_self">left_transitive</a> iff for <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#R2" target="_self">//</a> <font color="Olive">u2</font>,<font color="Olive">v2</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">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> )
 <b>by </b><i/>;<br/>


</div>
