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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:25"/>
for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> holds <br/><font color="Olive">w</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:25_1"/>
<font color="Maroon">w</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">V</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="E3:25_1"/>
<font color="Maroon">w</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</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="E2:25"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> holds <br/><font color="Olive">w</font>,<font color="Olive">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  </span>}</span> 
 
;<br/>
<a NAME="E3:25"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K2" target="_self">Ort_Comp</a> <font color="Maroon">W</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:25"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_5.html#K2" target="_self">Ort_Comp</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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