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<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="firebrick">::  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  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/>

<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E13">A1</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="firebrick">::  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">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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  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#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/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  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#R2" title="DIRORT:pred.2">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">w1</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:23"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:23"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:23"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="dirort.html#D3" target="_self" title="DIRORT:def.3">Def3</a></span>;<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="dirort.html#T11" target="_self" title="DIRORT:th.11">Th13</a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E18">A6</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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