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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<a NAME="E2:28"/>
( <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> implies  not <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_5.html#K2" title="RUSUB_5:func.2">Ort_Comp</a> <font color="Maroon" title="c2">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>assume </b><a NAME="E2:28_1"/>
<font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_5.html#K2" title="RUSUB_5:func.2">Ort_Comp</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<a NAME="E3:28_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K2" title="RUSUB_5:func.2">Ort_Comp</a> <font color="Maroon" title="c2">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v1</font> where <font color="Olive" title="b1">v1</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="b2">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="b2">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="b2">w</font>,<font color="Olive" title="b1">v1</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_5.html#D3" target="_self" title="RUSUB_5:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">v1</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="E6:28_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</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="c4">v1</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  ) )
 ;<br/>
<a NAME="E7:28_1"/>
<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">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#E1:28_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="rusub_5.html#E6:28_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><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="E9:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
(  not <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W</font> or  not <font color="Maroon" title="c3">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_5.html#K2" title="RUSUB_5:func.2">Ort_Comp</a> <font color="Maroon" title="c2">W</font> )
 ;<br/>


</div>
