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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="topreal1.html#K5">L~</a> <font color="Olive">f</font> is <a href="compts_1.html#V6">compact</a>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T28">TOPREAL1:28</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> holds <br/> <a href="topreal1.html#K5">L~</a> <font color="Olive">f</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:16_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:16_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E6:16_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">q</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_2_2"/>
( <font color="Maroon">q</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">q</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:16_1_1_2_2_1"/>
<font color="Maroon">q</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b> <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> = 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
 <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T12">SPPOL_2:12</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_2_2_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:16_1_1_2_2_2"/>
not <font color="Maroon">q</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b> <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="sppol_2.html#T23">SPPOL_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="sppol_2.html#T12">SPPOL_2:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E3:16_1_1_2_2_2"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, <a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:16_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>thus </b><a NAME="E4:16_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i/>;<br/>


</div>
