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

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:5"/><i><font color="Green">E38</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M1">OrthogonalFamily</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">I</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E4:5"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">y</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:5"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">I</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">I</font>,<font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="bhsp_5.html#D5">BHSP_5:def 5</a></i>;<br/>
<a NAME="E7:5"/><i><font color="Green">E57</font></i>: 
 <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a>  <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">H</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="bhsp_5.html#D5">BHSP_5:def 5</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y2</font> holds <br/>the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:5_1"/>
<font color="Maroon">y1</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="bhsp_5.html#D8">BHSP_5:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1"/>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:5"/>
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> = 
<font color="Maroon">y</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:5_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:5"/><b>then </b>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">I</font>,<font color="Maroon">p</font></span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">I</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">H</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="bhsp_5.html#T9">BHSP_5:9</a></i>;<br/>
<b>hence </b><a NAME="E11:5"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">I</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">I</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>


</div>
