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<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">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E1:29"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b1">x</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> <span class="p2">(<span class="default"><a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:29_1"/>
for <font color="Olive" title="b1">y</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">y</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</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="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</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="E1:29_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b>
<font color="Maroon" title="c5">y</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#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> : 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">v</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#D4" target="_self" title="RUSUB_5:def.4">Def4</a></i>;<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="E5:29_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c5">y</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</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/>
<b>thus </b><a NAME="E6:29_1_1"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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#E1:29_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="rusub_5.html#E5:29_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> : 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="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">v</font> <a href="bhsp_1.html#R1" title="BHSP_1:pred.1">are_orthogonal</a>  </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:29_1"/>
<font color="Maroon" title="c3">x</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> <span class="p2">(<span class="default"><a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font></span>)</span></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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29"/>
<font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><a href="rusub_5.html#K3" title="RUSUB_5:func.3">Ort_Comp</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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