<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>] <b>means </b>for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,$1 <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:23_1_1"/>
for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>, <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c4">w</font>, <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  )</span><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c4">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">w</font>, <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> </span><br/>

<a NAME="E2:23_1_1_1"/>
<font color="Maroon" title="c4">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T20" title="BHSP_1:th.20">BHSP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon" title="c4">w</font>, <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3" title="BHSP_1:def.3">BHSP_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:23_1_1"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
;<br/>
<a NAME="E4:23_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E2:23_1_1_2"/>
for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  )</span><br/>

<b>assume </b><a NAME="E1:23_1_1_2_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> </span><br/>

<b>consider </b><font color="Maroon" title="c7">v'</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:23_1_1_2_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v'</font> &amp; ( for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,<font color="Maroon" title="c7">v'</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:23_1_1_2"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">u'</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:23_1_1_2_1"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u'</font> &amp; ( for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,<font color="Maroon" title="c8">u'</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:23_1_1_2"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_2_1"/>
( <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  &amp; <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c5">u</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  )
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:23_1_1_2_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="rusub_5.html#E3:23_1_1_2_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="rusub_5.html#E5:23_1_1_2_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<a NAME="E7:23_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D3" title="BHSP_1:def.3">BHSP_1:def 3</a></i>;<br/>
<a NAME="E8:23_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T7" title="BHSP_1:th.7">BHSP_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1_2_1"/>
<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3" title="BHSP_1:def.3">BHSP_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1_2"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:23_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>



<b>assume </b><a NAME="E1:23_1_1_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">v'</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:23_1_1_3"/><i><font color="Green" title="E17">A7</font></i>: 
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v'</font> &amp; ( for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,<font color="Maroon" title="c6">v'</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  ) )
 ;<br/>
<a NAME="E4:23_1_1_3"/>
for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">w</font>,<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  )</span><br/>

<b>assume </b><a NAME="E1:23_1_1_3_1"/>
<font color="Maroon" title="c7">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> </span><br/>

<a NAME="E2:23_1_1_3_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c5">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:23_1_1_3"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/>
<font color="Maroon" title="c7">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> = 
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8" title="BHSP_1:th.8">BHSP_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="rusub_5.html#E2:23_1_1_3_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="bhsp_1.html#D3" title="BHSP_1:def.3">BHSP_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1_1_3_1"/>
<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3" title="BHSP_1:def.3">BHSP_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:23_1_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:23_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:23_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">v</font> where <font color="Olive" title="b2">v</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : for <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b3">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b3">w</font>,<font color="Olive" title="b2">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_5.html#E3:23_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rusub_1.html#T29" title="RUSUB_1:th.29">RUSUB_1:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
