<?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="E1:12"/><i><font color="Green">E29</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="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">L</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E3:12"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<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="E5:12"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">S</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> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> =  <a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">L</font>,<font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E7:12"/><i><font color="Green">E40</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">L</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xd</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:12_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">L</font>,<font color="Maroon">p</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="bhsp_5.html#D4">BHSP_5:def 4</a></i>;<br/><a NAME="E2:12_1_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">L</font>,<font color="Maroon">p</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1"/>
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">L</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:12"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 ;<br/>

<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="bhsp_5.html#D4">BHSP_5:def 4</a></i>
<br/>.= 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E3:12_2"/>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:12"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<a NAME="E1:12_3"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E2:12_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 
;<br/>
<a NAME="E3:12_3"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E4:12_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<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> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:12"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">S</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">L</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , <a class="ref" href="bhsp_5.html#D5">BHSP_5:def 5</a></i>;<br/>


</div>
