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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:24_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T20">BHSP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E4:24_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:24_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_1_1_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:24_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:24_1_1_2_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:24_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E6:24_1_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  )
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:24_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rusub_5.html#E3:24_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rusub_5.html#E5:24_1_1_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E7:24_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>;<br/>
<a NAME="E8:24_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T7">BHSP_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:24_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:24_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_1_1_3"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  ) )
 ;<br/>
<a NAME="E4:24_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_3_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:24_1_1_3"><i><font color="Green">E18</font></i></a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8">BHSP_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> 0
<b>by </b><i><a class="txt" href="rusub_5.html#E2:24_1_1_3_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E4:24_1_1_3_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:24_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:24_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:24_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_5.html#E3:24_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rusub_1.html#T29">RUSUB_1:29</a></i>;<br/>


</div>
