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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f1</font>, <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f1</font> holds <br/>( <font color="Olive">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) holds <br/> <a href="rvsum_1.html#K16">Product</a> <font color="Olive">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Olive">f2</font>;<br/>
<a NAME="E1:64"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:64_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) or  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64"/><i><font color="Green">E39</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_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/>

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:64_2"/>
for <font color="Olive">f1</font>, <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">f1</font> holds <br/>( <font color="Olive">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) holds <br/> <a href="rvsum_1.html#K16">Product</a> <font color="Olive">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Olive">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:64_2_1"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:64_2_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/>( <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon">g1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:64_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E15:64_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:64_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E15:64_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E7:64_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:64_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E15:64_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E9:64_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E19:64_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E10:64_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E15:64_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:64_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 ;<br/><a NAME="E2:64_2_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:64_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E22:64_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_4.html#E21:64_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E3:64"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:64_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:64_2_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E19:64_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:64_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><b>hence </b><a NAME="E7:64_2_1_1"/>
( <font color="Maroon">g1</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g2</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">g1</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:64"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E4:64"><i><font color="Green">E74</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E12:64_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E22:64_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_4.html#E21:64_2_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E13:64_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a>, <a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E14:64_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a>, <a class="txt" href="nat_4.html#E19:64_2_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">k1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<a NAME="E15:64_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:64_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E21:64_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E17:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E15:64_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:64_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E19:64_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a>, , <a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E20:64_2_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a>, , <a class="txt" href="nat_4.html#E18:64_2_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E21:64_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a>, <a class="txt" href="nat_4.html#E22:64_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_4.html#E19:64_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nat_4.html#E21:64_2_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E22:64_2_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:64"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E23:64_2_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> holds <br/><font color="Maroon">g1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E24:64_2_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E24:64_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:64"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E2:64_2_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E25:64_2_1"/>
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E26:64_2_1"/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:64"><i><font color="Green">E74</font></i></a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:64"/><i><font color="Green">E73</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E4:64"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 ;<br/>

<b>assume </b><a NAME="E5:64"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/>( <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">f1</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>hence </b><a NAME="E6:64"/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/>


</div>
