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

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="rsspace.html#K13">l2_Space</a> , <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="rsspace.html#K13">l2_Space</a> , <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:2_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:2_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i/>;<br/><a NAME="E5:2_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

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

0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , , <a class="ref" href="rsspace.html#T21">RSSPACE:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E2:2_1_2"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:2_1"/>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace.html#D11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/><b>hence </b><a NAME="E7:2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a> 
 <b>by </b><i>, , <a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a> 
 ;<br/><a NAME="E2:2_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> 0
<b>by </b><i>, , <a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>
<br/>.= 
0

;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> = 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="rsspace.html#T20">RSSPACE:20</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

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

<a NAME="E1:2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:2_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E47</font></i>: <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> = 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<a NAME="E7:2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font> = 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T24">SEQ_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E10:2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> = 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T26">SEQ_1:26</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E12:2"/>
( ( <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a>  ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a>  implies <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, , , , , , , <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>


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