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

<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>and </b><br/><a NAME="E3:80"/><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green" title="E67">A4</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E5:80"/><i><font color="Green" title="E68">A5</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E6:80"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E7:80"/><i><font color="Green" title="E70">A7</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>and </b><br/><a NAME="E8:80"/><i><font color="Green" title="E71">A8</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font>
 ;<br/>

<a NAME="E9:80"/><i><font color="Green" title="E72">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green" title="E73">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 ;<br/><a NAME="E2:80_1"/><b>then </b>
( <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/><b>hence </b><a NAME="E3:80_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sppol_2.html#E7:80"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c4">q</font> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T50" target="_self" title="SPPOL_2:th.50">Th50</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">f1'</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>;<br/>
<a NAME="E12:80"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:80"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">l</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">f2'</font> = <font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font>;<br/>
<b>set </b><font color="Maroon" title="c11">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">Z</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E15:80"/><i><font color="Green" title="E74">A11</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/>
<a NAME="E16:80"/><b>then </b><i><font color="Green" title="E75">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">l</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f1q</font> = <font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> as  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T59" title="FINSEQ_5:th.59">FINSEQ_5:59</a></i>;<br/>
<a NAME="E18:80"/><i><font color="Green" title="E76">A13</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_2"/><i><font color="Green" title="E77">A14</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon" title="c14">l</font> = <font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>;<br/><a NAME="E2:80_2"/><i><font color="Green" title="E78">A15</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_2"><i><font color="Green" title="E77">A14</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/><a NAME="E3:80_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_2"><i><font color="Green" title="E77">A14</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>;<br/><a NAME="E4:80_2"/><b>then </b><i><font color="Green" title="E79">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:80_2"/>
<font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_2"><i><font color="Green" title="E77">A14</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:80_2"/><i><font color="Green" title="E80">A17</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:80_2"/><i><font color="Green" title="E81">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_2"><i><font color="Green" title="E80">A17</font></i></a>, <a class="ref" href="finseq_5.html#T55" title="FINSEQ_5:th.55">FINSEQ_5:55</a></i>;<br/><a NAME="E10:80_2"/>
<font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_2"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_2"><i><font color="Green" title="E80">A17</font></i></a>, <a class="ref" href="finseq_5.html#T54" title="FINSEQ_5:th.54">FINSEQ_5:54</a></i>;<br/><a NAME="E11:80_2"/><b>then </b>
<font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sppol_2.html#E18:80"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="txt" href="sppol_2.html#E9:80_2"><i><font color="Green" title="E81">A18</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/><a NAME="E12:80_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E13:80_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:80"/><b>then </b>
not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60" title="FINSEQ_5:th.60">FINSEQ_5:60</a></i>;<br/>
<a NAME="E21:80"/><b>then </b><i><font color="Green" title="E77">A19</font></i>: 
not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:80"/><i><font color="Green" title="E78">A20</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E9:80"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="finseq_5.html#T44" title="FINSEQ_5:th.44">FINSEQ_5:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_3"/><i><font color="Green" title="E79">A21</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 ;<br/><a NAME="E2:80_3"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T43" title="FINSEQ_5:th.43">FINSEQ_5:43</a></i>;<br/><b>hence </b><a NAME="E3:80_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:80"><i><font color="Green" title="E74">A11</font></i></a>, <a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E22:80"><i><font color="Green" title="E78">A20</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_3"><i><font color="Green" title="E79">A21</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:80"/><b>then </b>
not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:80"/><b>then </b><i><font color="Green" title="E79">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E21:80"><i><font color="Green" title="E77">A19</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E26:80"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60" title="FINSEQ_5:th.60">FINSEQ_5:60</a></i>;<br/>
<a NAME="E27:80"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T21" title="FINSEQ_5:th.21">FINSEQ_5:21</a>, <a class="ref" href="finseq_5.html#T58" title="FINSEQ_5:th.58">FINSEQ_5:58</a></i>;<br/>
<a NAME="E28:80"/><b>then </b><i><font color="Green" title="E80">A23</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E29:80"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E30:80"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font> )
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self" title="SPPOL_2:th.18">Th18</a></i>;<br/>
<a NAME="E31:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E32:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="sppol_2.html#E28:80"><i><font color="Green" title="E80">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E33:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="sppol_2.html#E25:80"><i><font color="Green" title="E79">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T42" title="ZFMISC_1:th.42">ZFMISC_1:42</a></i>;<br/>
<a NAME="E34:80"/><b>then </b><i><font color="Green" title="E81">A24</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E35:80"/>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c13">f1q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
;<br/>
<a NAME="E36:80"/><b>then </b><i><font color="Green" title="E82">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E34:80"><i><font color="Green" title="E81">A24</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E37:80"/><i><font color="Green" title="E83">A26</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E38:80"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E39:80"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E40:80"/><b>then </b><i><font color="Green" title="E84">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E41:80"/><b>then </b><i><font color="Green" title="E85">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E42:80"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E43:80"/><b>then </b><i><font color="Green" title="E86">A29</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E44:80"/><b>then </b><i><font color="Green" title="E87">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="sppol_2.html#E15:80"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E45:80"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E40:80"><i><font color="Green" title="E84">A27</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E46:80"/><b>then </b><i><font color="Green" title="E88">A31</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E47:80"/><i><font color="Green" title="E89">A32</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E48:80"/><i><font color="Green" title="E90">A33</font></i>: 
not <font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E44:80"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E49:80"/>
<font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T28" target="_self" title="SPPOL_2:th.28">Th28</a></i>;<br/>
<a NAME="E50:80"/><b>then </b><i><font color="Green" title="E91">A34</font></i>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T29" target="_self" title="SPPOL_2:th.29">Th29</a></i>;<br/>
<a NAME="E51:80"/><i><font color="Green" title="E92">A35</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 
;<br/>
<a NAME="E52:80"/><i><font color="Green" title="E93">A36</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E53:80"/><b>then </b><i><font color="Green" title="E94">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>
<i><font color="Green" title="E95">A38</font></i>: <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T57" title="FINSEQ_5:th.57">FINSEQ_5:57</a></i>
;<br/>
<a NAME="E55:80"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E56:80"/><b>then </b><i><font color="Green" title="E96">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E51:80"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="sppol_2.html#E52:80"><i><font color="Green" title="E93">A36</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<div><i><font color="Green" title="E97">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_5"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:80_5"/><b>then </b><i><font color="Green" title="E98">A41</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:80_5"/><i><font color="Green" title="E99">A42</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>and </b><br/><a NAME="E5:80_5"/>
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>and </b><br/><a NAME="E6:80_5"/><i><font color="Green" title="E100">A43</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self" title="SPPOL_2:th.13">Th13</a></i>;<br/><a NAME="E7:80_5"/><i><font color="Green" title="E101">A44</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E8:80_5"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E9:80_5"/><b>then </b><i><font color="Green" title="E102">A45</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E7:80_5"><i><font color="Green" title="E101">A44</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:80_5"/><i><font color="Green" title="E103">A46</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_5"><i><font color="Green" title="E99">A42</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E13:80_5"/><i><font color="Green" title="E104">A47</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E11:80_5"><i><font color="Green" title="E103">A46</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self" title="SPPOL_2:th.10">Th10</a></i>;<br/><a NAME="E14:80_5"/><b>then </b>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:80_5"><i><font color="Green" title="E100">A43</font></i></a>, <a class="txt" href="sppol_2.html#E9:80_5"><i><font color="Green" title="E102">A45</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E15:80_5"/><b>then </b><i><font color="Green" title="E105">A48</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,1 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/><a NAME="E16:80_5"/><i><font color="Green" title="E106">A49</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80"><i><font color="Green" title="E73">A10</font></i></a>, <a class="txt" href="sppol_2.html#E37:80"><i><font color="Green" title="E83">A26</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_5_1"/>
( <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1"><b>suppose </b></a><a NAME="E1:80_5_1_1"/><i><font color="Green" title="E107">A50</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:80_5_1_1"/><i><font color="Green" title="E108">A51</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80_5"><i><font color="Green" title="E106">A49</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_5_1_1_1_1"/>
( <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E2:80_5_1_1"><i><font color="Green" title="E108">A51</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:80_5_1_1_1_1_1"/><i><font color="Green" title="E109">A52</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:80_5_1_1_1_1_1"/><i><font color="Green" title="E110">A53</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_5_1_1_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E1:80_5_1_1_1_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_5_1_1_1_1_1"><i><font color="Green" title="E110">A53</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:80_5_1_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:80_5_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_5_1_1_1_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="finseq_5.html#T53" title="FINSEQ_5:th.53">FINSEQ_5:53</a></i>;<br/><b>hence </b><a NAME="E3:80_5_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_5"><i><font color="Green" title="E98">A41</font></i></a>, <a class="ref" href="topreal1.html#T28" title="TOPREAL1:th.28">TOPREAL1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:80_5_1_1_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:80_5_1_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:80_5_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:80_5_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:80_5"><i><font color="Green" title="E100">A43</font></i></a>, <a class="txt" href="sppol_2.html#E9:80_5"><i><font color="Green" title="E102">A45</font></i></a>, <a class="txt" href="sppol_2.html#E13:80_5"><i><font color="Green" title="E104">A47</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_5_1_1"><i><font color="Green" title="E107">A50</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_5_1_1_1_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:80_5_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E111">A54</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:80_5_1_1_1_1_1_1_1_2"/>
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_5_1_1_1_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/><b>hence </b><a NAME="E7:80_5_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sppol_2.html#E18:80"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_5_1_1_1_1_1_1_1_2"><i><font color="Green" title="E111">A54</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:80_5_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:80_5_1_1_1_1_2"/>
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:80_5_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_5_1_1"><i><font color="Green" title="E107">A50</font></i></a></i>;<br/><b>hence </b><a NAME="E3:80_5_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:80_5"><i><font color="Green" title="E105">A48</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:80_5_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_5_1_2"><b>suppose </b></a><a NAME="E1:80_5_1_2"/>
<font color="Maroon" title="c16">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:80_5_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:80_5_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80_5"><i><font color="Green" title="E106">A49</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:80_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:80_5"><i><font color="Green" title="E105">A48</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E58:80"/><i><font color="Green" title="E98">A55</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c9">l</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="sppol_2.html#E51:80"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="sppol_2.html#E53:80"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="sppol_2.html#E54:80"><i><font color="Green" title="E95">A38</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<div><i><font color="Green" title="E99">A56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:80_6"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E40:80"><i><font color="Green" title="E84">A27</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:80_6"/><b>then </b><i><font color="Green" title="E100">A57</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_6_1_1"/><i><font color="Green" title="E101">A58</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:80_6_1_1"/><i><font color="Green" title="E102">A59</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E58:80"><i><font color="Green" title="E98">A55</font></i></a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_6"><i><font color="Green" title="E100">A57</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:80_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1 = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T2" target="_self" title="SPPOL_2:th.2">Th2</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self" title="SPPOL_2:th.10">Th10</a></i>
;<br/><a NAME="E6:80_6_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E7:80_6_1_1"/><b>then </b><i><font color="Green" title="E103">A60</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_6_1_1"><i><font color="Green" title="E102">A59</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:80_6_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E9:80_6_1_1"/><b>then </b>
not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E57:80"><i><font color="Green" title="E97">A40</font></i></a></i>;<br/><a NAME="E10:80_6_1_1"/><b>then </b>
not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E11:80_6_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E54:80"><i><font color="Green" title="E95">A38</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_6_1_1"><i><font color="Green" title="E101">A58</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_6_1_1"><i><font color="Green" title="E103">A60</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:80_6_1"/><i><font color="Green" title="E101">A61</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 ;<br/><a NAME="E3:80_6_1"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E4:80_6_1"/><b>then </b><i><font color="Green" title="E102">A62</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_6"><i><font color="Green" title="E100">A57</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E5:80_6_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_6_1"><i><font color="Green" title="E101">A61</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><b>hence </b><a NAME="E6:80_6_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_6_1"><i><font color="Green" title="E102">A62</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:80_6"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E100">A63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_7_1"/>
( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_7_1_1"><b>suppose </b></a><a NAME="E1:80_7_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:80_7_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E3:80_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T27" target="_self" title="SPPOL_2:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_7_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:80_7_1_2"/><i><font color="Green" title="E101">A64</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:80_7_1_2"/><i><font color="Green" title="E102">A65</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E3:80_7_1_2"/><i><font color="Green" title="E103">A66</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_7_1_2"><i><font color="Green" title="E101">A64</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:80_7_1_2"/><i><font color="Green" title="E104">A67</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E44:80"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:80_7_1_2"/><i><font color="Green" title="E105">A68</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_7_1_2"><i><font color="Green" title="E103">A66</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_2"><i><font color="Green" title="E104">A67</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:80_7_1_2"/><i><font color="Green" title="E106">A69</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_2"><i><font color="Green" title="E104">A67</font></i></a></i>;<br/><a NAME="E9:80_7_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80_7_1_2"><i><font color="Green" title="E104">A67</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>;<br/><a NAME="E10:80_7_1_2"/><b>then </b><i><font color="Green" title="E107">A70</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E53:80"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="sppol_2.html#E58:80"><i><font color="Green" title="E98">A55</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_2"><i><font color="Green" title="E104">A67</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_7_1_2"><i><font color="Green" title="E105">A68</font></i></a>, <a class="txt" href="sppol_2.html#E8:80_7_1_2"><i><font color="Green" title="E106">A69</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E11:80_7_1_2"/>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_7_1_2"><i><font color="Green" title="E102">A65</font></i></a>, <a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E12:80_7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80_7_1_2"><i><font color="Green" title="E104">A67</font></i></a>, <a class="txt" href="sppol_2.html#E10:80_7_1_2"><i><font color="Green" title="E107">A70</font></i></a>, <a class="ref" href="sppol_2.html#T31" target="_self" title="SPPOL_2:th.31">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_7_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:80_7_1_3"/><i><font color="Green" title="E101">A71</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E2:80_7_1_3"/><i><font color="Green" title="E102">A72</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:80_7_1_3"/>
<font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_7_1_3"><i><font color="Green" title="E101">A71</font></i></a>, <a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E4:80_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E50:80"><i><font color="Green" title="E91">A34</font></i></a>, <a class="txt" href="sppol_2.html#E59:80"><i><font color="Green" title="E99">A56</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_7_1_3"><i><font color="Green" title="E102">A72</font></i></a>, <a class="ref" href="sppol_2.html#T30" target="_self" title="SPPOL_2:th.30">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_7_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:80_7_1_4"/><i><font color="Green" title="E101">A73</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:80_7_1_4"/><i><font color="Green" title="E102">A74</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:80_7_1_4"/><i><font color="Green" title="E103">A75</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_7_1_4"><i><font color="Green" title="E101">A73</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:80_7_1_4"/><i><font color="Green" title="E104">A76</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E44:80"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:80_7_1_4"/><i><font color="Green" title="E105">A77</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_7_1_4"><i><font color="Green" title="E103">A75</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_4"><i><font color="Green" title="E104">A76</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:80_7_1_4"/><i><font color="Green" title="E106">A78</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_4"><i><font color="Green" title="E104">A76</font></i></a></i>;<br/><a NAME="E9:80_7_1_4"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80_7_1_4"><i><font color="Green" title="E104">A76</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>;<br/><a NAME="E10:80_7_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E53:80"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="sppol_2.html#E58:80"><i><font color="Green" title="E98">A55</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_4"><i><font color="Green" title="E104">A76</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_7_1_4"><i><font color="Green" title="E105">A77</font></i></a>, <a class="txt" href="sppol_2.html#E8:80_7_1_4"><i><font color="Green" title="E106">A78</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><b>hence </b><a NAME="E11:80_7_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="sppol_2.html#E50:80"><i><font color="Green" title="E91">A34</font></i></a>, <a class="txt" href="sppol_2.html#E59:80"><i><font color="Green" title="E99">A56</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_7_1_4"><i><font color="Green" title="E102">A74</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_7_1_4"><i><font color="Green" title="E104">A76</font></i></a>, <a class="ref" href="sppol_2.html#T32" target="_self" title="SPPOL_2:th.32">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E61:80"/>
<font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T35" target="_self" title="SPPOL_2:th.35">Th35</a></i>;<br/>
<a NAME="E62:80"/><b>then </b><i><font color="Green" title="E101">A79</font></i>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T36" target="_self" title="SPPOL_2:th.36">Th36</a></i>;<br/>
<a NAME="E63:80"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E65"><i><font color="Green" title="E58">Lm16</font></i></a></i>;<br/>
<a NAME="E64:80"/><b>then </b><i><font color="Green" title="E102">A80</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>;<br/>
<a NAME="E65:80"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E24"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/>
<a NAME="E66:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="sppol_2.html#E64:80"><i><font color="Green" title="E102">A80</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E67:80"/><b>then </b><i><font color="Green" title="E103">A81</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T42" title="ZFMISC_1:th.42">ZFMISC_1:42</a></i>;<br/>
<div><i><font color="Green" title="E104">A82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:80_8"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>and </b><br/><a NAME="E2:80_8"/><i><font color="Green" title="E105">A83</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 ;<br/><a NAME="E3:80_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_8"><i><font color="Green" title="E105">A83</font></i></a></i>;<br/><a NAME="E4:80_8"/><b>then </b><i><font color="Green" title="E106">A84</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:80_8"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E6:80_8"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E58:80"><i><font color="Green" title="E98">A55</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_8"><i><font color="Green" title="E106">A84</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E105">A85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:80_9"/><i><font color="Green" title="E106">A86</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>and </b><br/><a NAME="E2:80_9"/><i><font color="Green" title="E107">A87</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:80_9"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E4:80_9"/><b>then </b><i><font color="Green" title="E108">A88</font></i>: 
not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E57:80"><i><font color="Green" title="E97">A40</font></i></a></i>;<br/><a NAME="E5:80_9"/><i><font color="Green" title="E109">A89</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E58:80"><i><font color="Green" title="E98">A55</font></i></a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_9"><i><font color="Green" title="E107">A87</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E7:80_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E110">A90</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T2" target="_self" title="SPPOL_2:th.2">Th2</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T10" target="_self" title="SPPOL_2:th.10">Th10</a></i>
;<br/><div><i><font color="Green" title="E111">A91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_9_2"><b>now </b></a><div class="add"><a NAME="E1:80_9_2"/><i><font color="Green" title="E112">A92</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E2:80_9_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:80_9_2"/><i><font color="Green" title="E113">A93</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 ;<br/><a NAME="E5:80_9_2"/><b>then </b><i><font color="Green" title="E114">A94</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_9_2"><i><font color="Green" title="E112">A92</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:80_9_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E7:80_9_2"/><b>then </b><i><font color="Green" title="E115">A95</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_9_2"><i><font color="Green" title="E113">A93</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_9_2"><i><font color="Green" title="E114">A94</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E8:80_9_2"/><i><font color="Green" title="E116">A96</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><i><font color="Green" title="E117">A97</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E8:80_9_2"><i><font color="Green" title="E116">A96</font></i></a>, <a class="ref" href="finseq_5.html#T53" title="FINSEQ_5:th.53">FINSEQ_5:53</a></i>
<br/>.= 
<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_9_2_2"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_2_2_1"><b>suppose </b></a><a NAME="E1:80_9_2_2_1"/><i><font color="Green" title="E118">A98</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:80_9_2_2_1"/><b>then </b><i><font color="Green" title="E119">A99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:80_9_2"><i><font color="Green" title="E117">A97</font></i></a></i>;<br/><a NAME="E3:80_9_2_2_1"/><i><font color="Green" title="E120">A100</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E4:80_9_2_2_1"/>
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:80_9_2_2_1"/><b>then </b><i><font color="Green" title="E121">A101</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_9_2_2_1"><i><font color="Green" title="E120">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:80_9_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:80_9_2"><i><font color="Green" title="E117">A97</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_9_2_2_1"><i><font color="Green" title="E118">A98</font></i></a></i>;<br/><a NAME="E7:80_9_2_2_1"/><b>then </b><i><font color="Green" title="E122">A102</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_9_2_2_1"><i><font color="Green" title="E119">A99</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_9_2_2_1"><i><font color="Green" title="E121">A101</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E8:80_9_2_2_1"/><i><font color="Green" title="E123">A103</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_9_2"><i><font color="Green" title="E112">A92</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E9:80_9_2_2_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_9_2"><i><font color="Green" title="E113">A93</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E10:80_9_2_2_1"/><b>then </b><i><font color="Green" title="E124">A104</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80_9_2_2_1"><i><font color="Green" title="E123">A103</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:80_9_2_2_1"/><i><font color="Green" title="E125">A105</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E12:80_9_2_2_1"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E13:80_9_2_2_1"/><b>then </b><i><font color="Green" title="E126">A106</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80_9_2_2_1"><i><font color="Green" title="E123">A103</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>assume </b><a NAME="E14:80_9_2_2_1"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font>
 ;<br/><a NAME="E15:80_9_2_2_1"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80_9"><i><font color="Green" title="E110">A90</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_9_2_2_1"><i><font color="Green" title="E122">A102</font></i></a>, <a class="txt" href="sppol_2.html#E13:80_9_2_2_1"><i><font color="Green" title="E126">A106</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:80_9_2_2_1"/><b>then </b>
<font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">k</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E10:80_9_2_2_1"><i><font color="Green" title="E124">A104</font></i></a>, <a class="txt" href="sppol_2.html#E11:80_9_2_2_1"><i><font color="Green" title="E125">A105</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><b>hence </b><a NAME="E18:80_9_2_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_2_2_2"><b>suppose </b></a><a NAME="E1:80_9_2_2_2"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:80_9_2_2_2"/><b>then </b>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80_9"><i><font color="Green" title="E106">A86</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:80_9_2_2_2"/><b>then </b>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:80_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E5:80_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 ;<br/><a NAME="E6:80_9_2_2_2"/><b>then </b><i><font color="Green" title="E118">A107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:80_9_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:80_9_2"><i><font color="Green" title="E117">A97</font></i></a></i>;<br/><a NAME="E8:80_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:80_9_2_2_2"><i><font color="Green" title="E118">A107</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:80_9_2_2_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E10:80_9_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:80_9_2_2_2"/>
not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80_9"><i><font color="Green" title="E110">A90</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_9_2"><i><font color="Green" title="E115">A95</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:80_9"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80_9"><i><font color="Green" title="E110">A90</font></i></a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E11:80_9"/><b>then </b><i><font color="Green" title="E112">A108</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_9"><i><font color="Green" title="E109">A89</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_9_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:80_9_3"/><i><font color="Green" title="E113">A109</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font></span>)</span>
 ;<br/><a NAME="E3:80_9_3"/>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E11:80_9"><i><font color="Green" title="E112">A108</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_9_3"><i><font color="Green" title="E113">A109</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:80_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_9"><i><font color="Green" title="E108">A88</font></i></a>, <a class="txt" href="sppol_2.html#E9:80_9"><i><font color="Green" title="E111">A91</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_9_3"><i><font color="Green" title="E113">A109</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E13:80_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>hence </b><a NAME="E14:80_9"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_10"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:80_10"/><i><font color="Green" title="E106">A110</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>and </b><br/><a NAME="E4:80_10"/><i><font color="Green" title="E107">A111</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>and </b><br/><a NAME="E5:80_10"/><i><font color="Green" title="E108">A112</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self" title="SPPOL_2:th.13">Th13</a></i>;<br/><a NAME="E6:80_10"/><i><font color="Green" title="E109">A113</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_10"><i><font color="Green" title="E107">A111</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>;<br/><a NAME="E7:80_10"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E8:80_10"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="sppol_2.html#E51:80"><i><font color="Green" title="E92">A35</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E9:80_10"/><b>then </b><i><font color="Green" title="E110">A114</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80_10"><i><font color="Green" title="E108">A112</font></i></a>, <a class="txt" href="sppol_2.html#E6:80_10"><i><font color="Green" title="E109">A113</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:80_10"/><b>then </b><i><font color="Green" title="E111">A115</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c14">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/><a NAME="E11:80_10"/><i><font color="Green" title="E112">A116</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:80"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_10_1"/>
( <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> or <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1"><b>suppose </b></a><a NAME="E1:80_10_1_1"/><i><font color="Green" title="E113">A117</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_10_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:80"><i><font color="Green" title="E75">A12</font></i></a></i>;<br/><a NAME="E3:80_10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f2</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_10"><i><font color="Green" title="E106">A110</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_10_1_1"><i><font color="Green" title="E113">A117</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:80_10_1_1"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:80_10"><i><font color="Green" title="E110">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:80_10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E9:80"><i><font color="Green" title="E72">A9</font></i></a>, <a class="txt" href="sppol_2.html#E56:80"><i><font color="Green" title="E96">A39</font></i></a>, <a class="txt" href="sppol_2.html#E11:80_10"><i><font color="Green" title="E112">A116</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2"><b>suppose </b></a><a NAME="E1:80_10_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_10_1_2"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_10"><i><font color="Green" title="E107">A111</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:80_10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80_10"><i><font color="Green" title="E111">A115</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E71:80"/><b>then </b>
( not <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E57:80"><i><font color="Green" title="E97">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E72:80"/><b>then </b>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E67:80"><i><font color="Green" title="E103">A81</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E73:80"/><b>then </b><i><font color="Green" title="E106">A118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E62:80"><i><font color="Green" title="E101">A79</font></i></a>, <a class="txt" href="sppol_2.html#E68:80"><i><font color="Green" title="E104">A82</font></i></a>, <a class="txt" href="sppol_2.html#E69:80"><i><font color="Green" title="E105">A85</font></i></a>, <a class="ref" href="sppol_2.html#T37" target="_self" title="SPPOL_2:th.37">Th37</a></i>;<br/>
<a NAME="E74:80"/>
<font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T41" target="_self" title="SPPOL_2:th.41">Th41</a></i>;<br/>
<a NAME="E75:80"/><b>then </b><i><font color="Green" title="E107">A119</font></i>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T42" target="_self" title="SPPOL_2:th.42">Th42</a></i>;<br/>
<a NAME="E76:80"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="sppol_2.html#E43:80"><i><font color="Green" title="E86">A29</font></i></a>, <a class="txt" href="sppol_2.html#E51:80"><i><font color="Green" title="E92">A35</font></i></a>, <a class="txt" href="sppol_2.html#E53:80"><i><font color="Green" title="E94">A37</font></i></a>, <a class="txt" href="sppol_2.html#E54:80"><i><font color="Green" title="E95">A38</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/>
<a NAME="E77:80"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E75:80"><i><font color="Green" title="E107">A119</font></i></a>, <a class="txt" href="sppol_2.html#E56"><i><font color="Green" title="E52">Lm14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sppol_2.html#E36:80"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="sppol_2.html#E46:80"><i><font color="Green" title="E88">A31</font></i></a>, <a class="txt" href="sppol_2.html#E60:80"><i><font color="Green" title="E100">A63</font></i></a>, <a class="txt" href="sppol_2.html#E73:80"><i><font color="Green" title="E106">A118</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">g1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">g2</font>
;<br/>

<b>thus </b><a NAME="E79:80"/><i><font color="Green" title="E108">A120</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T47" title="FINSEQ_5:th.47">FINSEQ_5:47</a></i>;<br/>

<a NAME="E80:80"/><i><font color="Green" title="E109">A121</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><i><font color="Green" title="E110">A122</font></i>: <font color="Maroon" title="c14">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c14">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E47:80"><i><font color="Green" title="E89">A32</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E47:80"><i><font color="Green" title="E89">A32</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c2">p1</font>
<b>by </b><i><a class="txt" href="sppol_2.html#E2:80"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="sppol_2.html#E47:80"><i><font color="Green" title="E89">A32</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green" title="E111">A123</font></i>: <font color="Maroon" title="c4">q</font> = 
<font color="Maroon" title="c7">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T48" title="FINSEQ_5:th.48">FINSEQ_5:48</a></i>
<br/>.= 
<font color="Maroon" title="c7">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g1</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T45" title="FINSEQ_5:th.45">FINSEQ_5:45</a></i>
;<br/>

<a NAME="E83:80"/><i><font color="Green" title="E112">A124</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E84:80"/><i><font color="Green" title="E113">A125</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><i><font color="Green" title="E114">A126</font></i>: <font color="Maroon" title="c14">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g2</font></span>)</span> = 
<font color="Maroon" title="c14">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c14">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E41:80"><i><font color="Green" title="E85">A28</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E83:80"><i><font color="Green" title="E112">A124</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T56" title="FINSEQ_5:th.56">FINSEQ_5:56</a></i>
;<br/><br/>

<a NAME="E86:80"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E87:80"/><b>then </b><i><font color="Green" title="E115">A127</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font> )
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self" title="SPPOL_2:th.18">Th18</a></i>;<br/>
<a NAME="E88:80"/><i><font color="Green" title="E116">A128</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E89:80"/><i><font color="Green" title="E117">A129</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green" title="E118">A130</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E51:80"><i><font color="Green" title="E92">A35</font></i></a>, <a class="ref" href="finseq_5.html#T24" title="FINSEQ_5:th.24">FINSEQ_5:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E47:80"><i><font color="Green" title="E89">A32</font></i></a>, <a class="txt" href="sppol_2.html#E54:80"><i><font color="Green" title="E95">A38</font></i></a>, <a class="ref" href="sppol_2.html#T19" target="_self" title="SPPOL_2:th.19">Th19</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>
<br/>.= 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font>
<b>by </b><i><a class="txt" href="sppol_2.html#E41:80"><i><font color="Green" title="E85">A28</font></i></a>, <a class="txt" href="sppol_2.html#E48:80"><i><font color="Green" title="E90">A33</font></i></a>, <a class="ref" href="sppol_2.html#T23" target="_self" title="SPPOL_2:th.23">Th23</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>
;<br/>
<b>thus </b><a NAME="E91:80"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_15"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:80_15_1"/><i><font color="Green" title="E119">A131</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_15_1_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E1:80_15_1"><i><font color="Green" title="E119">A131</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_1_1_1"><b>suppose </b></a><a NAME="E1:80_15_1_1_1"/><i><font color="Green" title="E120">A132</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 ;<br/><div class="add"><a NAME="E2:80_15_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g2</font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E79:80"><i><font color="Green" title="E108">A120</font></i></a>, <a class="txt" href="sppol_2.html#E80:80"><i><font color="Green" title="E109">A121</font></i></a>, <a class="txt" href="sppol_2.html#E81:80"><i><font color="Green" title="E110">A122</font></i></a>, <a class="txt" href="sppol_2.html#E89:80"><i><font color="Green" title="E117">A129</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E3:80_15_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E87:80"><i><font color="Green" title="E115">A127</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_1_1_1"><i><font color="Green" title="E120">A132</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_1_1_2"><b>suppose </b></a><a NAME="E1:80_15_1_1_2"/><i><font color="Green" title="E120">A133</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 ;<br/><div class="add"><a NAME="E2:80_15_1_1_2"/>
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g2</font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E82:80"><i><font color="Green" title="E111">A123</font></i></a>, <a class="txt" href="sppol_2.html#E84:80"><i><font color="Green" title="E113">A125</font></i></a>, <a class="txt" href="sppol_2.html#E85:80"><i><font color="Green" title="E114">A126</font></i></a>, <a class="txt" href="sppol_2.html#E88:80"><i><font color="Green" title="E116">A128</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E3:80_15_1_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E87:80"><i><font color="Green" title="E115">A127</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_1_1_2"><i><font color="Green" title="E120">A133</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<a NAME="E2:80_15"/><i><font color="Green" title="E119">A134</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E90:80"><i><font color="Green" title="E118">A130</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E3:80_15"/><i><font color="Green" title="E120">A135</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_5.html#T45" title="FINSEQ_5:th.45">FINSEQ_5:45</a></i>;<br/>
<a NAME="E4:80_15"/><i><font color="Green" title="E121">A136</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E5:80_15"/><i><font color="Green" title="E122">A137</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<i><font color="Green" title="E123">A138</font></i>: <font color="Maroon" title="c4">q</font> = 
<font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
<br/>.= 
<font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E4:80_15"><i><font color="Green" title="E121">A136</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_15_3"/>
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3_1"><b>suppose </b></a><a NAME="E1:80_15_3_1"/><i><font color="Green" title="E124">A139</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_15_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_15_3_1_1"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:80_15_3_1_1"/><i><font color="Green" title="E125">A140</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <b>and </b><br/><a NAME="E4:80_15_3_1_1"/><i><font color="Green" title="E126">A141</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g1</font>
 <b>and </b><br/><a NAME="E5:80_15_3_1_1"/><i><font color="Green" title="E127">A142</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self" title="SPPOL_2:th.13">Th13</a></i>;<br/><a NAME="E6:80_15_3_1_1"/><i><font color="Green" title="E128">A143</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80_15_3_1_1"><i><font color="Green" title="E126">A141</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_15_3_1_1"><i><font color="Green" title="E127">A142</font></i></a>, <a class="ref" href="sppol_2.html#T9" target="_self" title="SPPOL_2:th.9">Th9</a></i>;<br/><a NAME="E7:80_15_3_1_1"/><i><font color="Green" title="E129">A144</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E8:80_15_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E37:80"><i><font color="Green" title="E83">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E10:80_15_3_1_1"/><i><font color="Green" title="E130">A145</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 ;<br/><a NAME="E11:80_15_3_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80"><i><font color="Green" title="E73">A10</font></i></a>, <a class="txt" href="sppol_2.html#E37:80"><i><font color="Green" title="E83">A26</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E12:80_15_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E7:80_15_3_1_1"><i><font color="Green" title="E129">A144</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:80_15_3_1_1"/><b>then </b><i><font color="Green" title="E131">A146</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80_15_3_1_1"><i><font color="Green" title="E130">A145</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:80_15_3_1_1"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E15:80_15_3_1_1"/><b>then </b><i><font color="Green" title="E132">A147</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E13:80_15_3_1_1"><i><font color="Green" title="E131">A146</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:80_15_3_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sppol_2.html#E6:80_15"><i><font color="Green" title="E123">A138</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_1"><i><font color="Green" title="E124">A139</font></i></a>, <a class="txt" href="sppol_2.html#E7:80_15_3_1_1"><i><font color="Green" title="E129">A144</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E17:80_15_3_1_1"/><b>then </b><i><font color="Green" title="E133">A148</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:80_15_3_1_1"><i><font color="Green" title="E130">A145</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:80_15_3_1_1"/><b>then </b><i><font color="Green" title="E134">A149</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_15"><i><font color="Green" title="E120">A135</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_15_3_1_1"><i><font color="Green" title="E126">A141</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_15_3_1_1_1"/>
( <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font> or <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:80_15_3_1_1_1_1"/><i><font color="Green" title="E135">A150</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font>
 ;<br/><div class="add"><a NAME="E2:80_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:80_15_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_15_3_1_1"><i><font color="Green" title="E125">A140</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:80_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:80_15_3_1_1"><i><font color="Green" title="E128">A143</font></i></a>, <a class="txt" href="sppol_2.html#E15:80_15_3_1_1"><i><font color="Green" title="E132">A147</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_1_1_1_1"><i><font color="Green" title="E135">A150</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:80_15_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:80_15_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_15"><i><font color="Green" title="E120">A135</font></i></a>, <a class="txt" href="sppol_2.html#E6:80_15"><i><font color="Green" title="E123">A138</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_1"><i><font color="Green" title="E124">A139</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_15_3_1_1"><i><font color="Green" title="E126">A141</font></i></a>, <a class="txt" href="sppol_2.html#E17:80_15_3_1_1"><i><font color="Green" title="E133">A148</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_1_1_1_1"><i><font color="Green" title="E135">A150</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:80_15_3_1_1_1_2"/>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">j</font>
 ;<br/><div class="add"><a NAME="E2:80_15_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E18:80_15_3_1_1"><i><font color="Green" title="E134">A149</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:80_15_3_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c15">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E4:80_15_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c16">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E5:80_15_3_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:80_15_3_1_1"><i><font color="Green" title="E128">A143</font></i></a>, <a class="txt" href="sppol_2.html#E15:80_15_3_1_1"><i><font color="Green" title="E132">A147</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:80_15_3_1"/><b>then </b><i><font color="Green" title="E125">A151</font></i>: 
not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><i><font color="Green" title="E126">A152</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E10:80"><i><font color="Green" title="E73">A10</font></i></a>, <a class="txt" href="sppol_2.html#E6:80_15"><i><font color="Green" title="E123">A138</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_1"><i><font color="Green" title="E124">A139</font></i></a>, <a class="ref" href="sppol_2.html#T45" target="_self" title="SPPOL_2:th.45">Th45</a></i>
;<br/><a NAME="E5:80_15_3_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E64"><i><font color="Green" title="E57">Lm15</font></i></a></i>;<br/><a NAME="E6:80_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E7:80_15_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42" title="ZFMISC_1:th.42">ZFMISC_1:42</a></i>;<br/><a NAME="E8:80_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80_15_3_1"><i><font color="Green" title="E125">A151</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><a NAME="E9:80_15_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80_15"><i><font color="Green" title="E119">A134</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_15_3_1"><i><font color="Green" title="E126">A152</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E10:80_15_3_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_15_3_2"><b>suppose </b></a><a NAME="E1:80_15_3_2"/><i><font color="Green" title="E124">A153</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><a NAME="E2:80_15_3_2"/>
<font color="Maroon" title="c3">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sppol_2.html#E5:80_15"><i><font color="Green" title="E122">A137</font></i></a>, <a class="ref" href="finseq_5.html#T44" title="FINSEQ_5:th.44">FINSEQ_5:44</a></i>;<br/><b>then </b><i><font color="Green" title="E125">A154</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_2"><i><font color="Green" title="E124">A153</font></i></a>, <a class="ref" href="finseq_5.html#T53" title="FINSEQ_5:th.53">FINSEQ_5:53</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><i><font color="Green" title="E126">A155</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="sppol_2.html#E3:80_15_3_2"><i><font color="Green" title="E125">A154</font></i></a>, <a class="ref" href="topreal1.html#T28" title="TOPREAL1:th.28">TOPREAL1:28</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/><a NAME="E5:80_15_3_2"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="sppol_2.html#E64"><i><font color="Green" title="E57">Lm15</font></i></a></i>;<br/><b>hence </b><a NAME="E6:80_15_3_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="sppol_2.html#E2:80_15"><i><font color="Green" title="E119">A134</font></i></a>, <a class="txt" href="sppol_2.html#E1:80_15_3_2"><i><font color="Green" title="E124">A153</font></i></a>, <a class="txt" href="sppol_2.html#E4:80_15_3_2"><i><font color="Green" title="E126">A155</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><font color="Maroon" title="c1">P</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E6:80"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="sppol_2.html#E8:80"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="sppol_2.html#T25" target="_self" title="SPPOL_2:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="sppol_2.html#E90:80"><i><font color="Green" title="E118">A130</font></i></a>, <a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>
;<br/>


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