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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/><b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:31"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Olive">x</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">x</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#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">x</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="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">w</font>,<font color="Olive">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:31_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( 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="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">w</font>,<font color="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  ) )
 ;<br/>
<a NAME="E5:31_1"/>
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="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_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="E1:31_1_1"/>
<font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:31_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#D1" target="_self">Def1</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_1"/>
<font color="Maroon">w</font>,<font color="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</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">v1</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon">x</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/>


</div><b>end;</b></div>
<a NAME="E3:31"/><b>then </b><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</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><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:31"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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> holds <br/><font color="Olive">x</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#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">x</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>
 ;<br/>

<a NAME="E2:31_2"/><b>then </b>
<font color="Maroon">x</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> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:31_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( 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="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  ) )
 ;<br/>
<a NAME="E5:31_2"/>
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="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">w</font>,<font color="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_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="E1:31_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:31_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#D1" target="_self">Def1</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:31_2_1"/>
<font color="Maroon">w</font>,<font color="Maroon">v</font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</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="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">w</font>,<font color="Olive">v1</font> <a href="bhsp_1.html#R1">are_orthogonal</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:31_2"/>
<font color="Maroon">x</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#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:31"/><b>then </b>
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> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:31"/><b>then </b>
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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:31"/>
 <a href="rusub_5.html#K3" target="_self">Ort_Comp</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</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="rusub_1.html#T24">RUSUB_1:24</a></i>;<br/>


</div>
