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<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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>] <b>means </b> <a href="topreal1.html#K5">L~</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:22"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>]
 
;<br/>
<a NAME="E2:22"/><i><font color="Green">E43</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><br/> for <font color="Olive">p</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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>let </b><font color="Maroon">p</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1"/>
( <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">f</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="22_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b> <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T63">FINSEQ_5:63</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>;<br/><a NAME="E3:22_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>set </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1;<br/><a NAME="E4:22_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>;<br/><a NAME="E5:22_1_1_2"/><i><font color="Green">E50</font></i>: 
not  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>thus </b> <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</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">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T66">FINSEQ_5:66</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:22"/>
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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:22"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


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