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

<b>assume </b><a NAME="E1:2"/><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">Y</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>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="E2:2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; ( 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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</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="E4:2"/><i><font color="Green">E34</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">Y</font> &amp;  <a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y</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/>
<a NAME="E5:2"/><i><font color="Green">E38</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">I</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="2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_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="E1:2_1_1"/><i><font color="Green">E40</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">I</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">I</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="E2:2_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="E3:2_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">I</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="E2:2_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">I</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</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">I</font>,<font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E2:2_2"/>
 <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">I</font>,<font color="Maroon">p</font></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E4:2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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/><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> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">I</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">I</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/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E6:2_2"/>
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E7:2"/><b>then </b>
 <a href="bhsp_5.html#K4">Func_Seq</a> <font color="Maroon">I</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:2"/>
 <a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y</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>
 <b>by </b><i>, , , <a class="ref" href="bhsp_5.html#D5">BHSP_5:def 5</a></i>;<br/>


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