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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 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">w</font>, <font color="Olive" title="b6">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#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> &amp; <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b6">w1</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v1</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="b5">w</font>,<font color="Olive" title="b6">w1</font> )</span><br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E13">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">w</font>, <font color="Olive" title="b6">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#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> &amp; <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b6">w1</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v1</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="b5">w</font>,<font color="Olive" title="b6">w1</font></span><br/>

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">u1</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v1</font> implies <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E2:23"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">v2</font>, <font color="Maroon" title="c9">v3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:23"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c8">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v3</font>
 <b>and </b><br/><a NAME="E7:23"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c9">v3</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font>
 <b>and </b><br/><a NAME="E8:23"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c9">v3</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">v1</font>
 <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></i>;<br/>
<a NAME="E9:23"/>
<font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c9">v3</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font>
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E4:23"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E8:23"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="dirort.html#T13" target="_self" title="DIRORT:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font>
 <b>by </b><i><a class="txt" href="#E6:23"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="#E7:23"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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