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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a>  implies ( <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> ) )</span><br/>

<a NAME="E1:32"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( ( 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="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c2">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/>
<span class="kw">assume </span><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">u1</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c8">v2</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="c7">u2</font>,<font color="Maroon" title="c8">v2</font> &amp; <font color="Maroon" title="c7">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v2</font> implies <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c8">v2</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<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="c7">u2</font>,<font color="Maroon" title="c8">v2</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c7">u2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:32_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c2">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w1</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><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/><span class="kw">assume </span><a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font></span><br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:32_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<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="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></span>;<br/><a NAME="E6:32_1_1"/>
<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="c3">u</font>,<font color="Maroon" title="c3">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:32_1_1"/>
<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1_2"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font></span><br/><a NAME="E2:32_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c9">w1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_2"/>
<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:32_1"/>
<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E3:32"/>
( ( 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="32_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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></span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; not <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w1</font> &amp; not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> implies <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> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:32_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:32_2"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:32_2"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_2_1"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:32_2_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/><a NAME="E3:32_2_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E4:32_2"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E1:32_2_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>;<br/><a NAME="E4:32_2_1"/><span class="kw">then </span>
 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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="dirort.html#D4" target="_self" title="DIRORT:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:32_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:32"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
