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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">fs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">fss</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Olive">fs</font>  st ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">fs</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">fs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Olive">fss</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Olive">fss</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">fs</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">fss</font> be    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">fs</font>;<br/><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:4_1"/><b>then </b>
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E3:4_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:4"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>let </b><font color="Maroon">fs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">fss</font> be    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">fs</font>;<br/><b>assume </b><a NAME="E2:4_2"/><i><font color="Green">E52</font></i>: 
( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>;<br/><b>consider </b><font color="Maroon">L</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:4_2"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E5:4_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:4_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">fss</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><a NAME="E7:4_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>;<br/><a NAME="E8:4_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:4_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E11:4_2"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T27">GRAPH_2:27</a></i>;<br/><a NAME="E12:4_2"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E14:4_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i/>;<br/><a NAME="E15:4_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">d</font>]
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:4_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><b>set </b><font color="Maroon">y</font> = <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>;<br/><a NAME="E18:4_2"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">xx1</font> = <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon">fssq</font> = <font color="Maroon">fss</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">fssq</font> = <font color="Maroon">fss</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">fs</font> <b>by </b><i><a class="ref" href="graph_2.html#T29">GRAPH_2:29</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E22:4_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E23:4_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">fsD</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">fsC</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E25:4_2"/><i><font color="Green">E70</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fsD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fsC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">fs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsD</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fsC</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E26:4_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">xx1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">fsA</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">fsB</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E28:4_2"/><i><font color="Green">E74</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fsA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fsB</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">fsD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsA</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fsB</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E29:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsD</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:4_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsD</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E31:4_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsB</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E32:4_2"/><b>then </b>
<font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsB</font> <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E33:4_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">fsB</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><i><font color="Green">E77</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fs</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsD</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsC</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsC</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsC</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsC</font>
 ;<br/><a NAME="E2:4_2_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fsC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:4_2_2"/>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a></i>;<br/><b>hence </b><a NAME="E4:4_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fsC</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E36:4_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsC</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/><a NAME="E37:4_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">z</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E39:4_2"/><i><font color="Green">E83</font></i>: 
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/><a NAME="E40:4_2"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">z</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E41:4_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E42:4_2"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsA</font>
 ;<br/><a NAME="E2:4_2_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">fsA</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsD</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:4_2_3"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsA</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsD</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E4:4_2_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsD</font>
 <b>by </b><i/>;<br/><a NAME="E5:4_2_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">fsA</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, , </i>;<br/><a NAME="E6:4_2_3"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsD</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><b>hence </b><a NAME="E7:4_2_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fsA</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:4_2_4"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:4_2_4"/><i><font color="Green">E98</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xx1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:4_2_4"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsA</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:4_2_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E9:4_2_4"/>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:4_2_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fsD</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:4_2_4"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsD</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E12:4_2_4"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fsA</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E13:4_2_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fsA</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E45:4_2"/><b>then </b>
<font color="Maroon">fssq</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fsA</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">fssq</font> = <font color="Maroon">fssq</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">fsA</font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5"><b>now </b></a><div class="add"><a NAME="E1:4_2_5"/><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fssq</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>;<br/><a NAME="E2:4_2_5"/><i><font color="Green">E102</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>;<br/><div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:4_2_5_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_2_5_1"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>thus </b><a NAME="E4:4_2_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a>, , </i>;<br/></div><b>end;</b></div><a NAME="E4:4_2_5"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">fss</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:4_2_5_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_5_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:4_2_5_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:4_2_5_2_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xx1</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:4_2_5_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:4_2_5_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:4_2_5_2_1"/><b>then </b>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E10:4_2_5_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:4_2_5_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:4_2_5_2"/><b>then </b><i><font color="Green">E111</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_2_5_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E6:4_2_5_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E7:4_2_5_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E9:4_2_5_2"/><i><font color="Green">E115</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:4_2_5_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:4_2_5_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:4_2_5_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:4_2_5_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i/>;<br/><a NAME="E14:4_2_5_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, </i>;<br/><a NAME="E15:4_2_5_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E16:4_2_5_2"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:4_2_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E18:4_2_5_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E19:4_2_5_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/></div><b>end;</b></div><a NAME="E6:4_2_5"/><b>then </b>
<font color="Maroon">fssq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fss</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>then </b><i><font color="Green">E118</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">fssq</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">fss</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/><b>hence </b><a NAME="E8:4_2_5"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">x1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">x2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E9:4_2_5"/><i><font color="Green">E120</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E10:4_2_5"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:4_2_5"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E12:4_2_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E13:4_2_5"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:4_2_5"/>
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fss</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E15:4_2_5"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E16:4_2_5"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:4_2_5"/>
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fssq</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E18:4_2_5"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>consider </b><font color="Maroon">lk</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:4_2_5"/><i><font color="Green">E127</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">lk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
;<br/><a NAME="E22:4_2_5"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E23:4_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E24:4_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E25:4_2_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E26:4_2_5"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><a NAME="E27:4_2_5"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:4_2_5_5_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 ;<br/><a NAME="E2:4_2_5_5_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E3:4_2_5_5_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E5:4_2_5_5_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">z'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:4_2_5_5_1"/><i><font color="Green">E135</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2_5_5_1_1"/>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:4_2_5_5_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_2_5_5_1_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xx1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:4_2_5_5_1_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xx1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:4_2_5_5_1_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:4_2_5_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E7:4_2_5_5_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:4_2_5_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:4_2_5_5"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_5_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_5_5_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_5_5_2_1_1"><b>suppose </b></a><a NAME="E1:4_2_5_5_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font>
 ;<br/><div class="add"><a NAME="E2:4_2_5_5_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E3:4_2_5_5_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:4_2_5_5_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_5_5_2_1_2"><b>suppose </b></a><a NAME="E1:4_2_5_5_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_5_5_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_2_5_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 ;<br/></div><b>end;</b></div><a NAME="E29:4_2_5"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_5_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:4_2_5_6"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:4_2_5_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><a NAME="E31:4_2_5"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2_5_7"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><a NAME="E2:4_2_5_7"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:4_2_5_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">fssq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fssq</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E4:4_2_5_7"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">xx1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><a NAME="E5:4_2_5_7"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xx1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E6:4_2_5_7"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E33:4_2_5"/><b>then </b>
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fssq</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><b>hence </b><a NAME="E34:4_2_5"/>
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E48:4_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fssq</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><a NAME="E49:4_2"/><b>then </b><i><font color="Green">E142</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6"><b>now </b></a><div class="add"><a NAME="E1:4_2_6"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><a NAME="E2:4_2_6"/><i><font color="Green">E144</font></i>: 
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fss</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E3:4_2_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:4_2_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:4_2_6"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">fss</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span>;<br/><i><font color="Green">E147</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <font color="Maroon">fss</font>
<b>by </b><i><a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2_6_2"/><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:4_2_6_2"/><i><font color="Green">E149</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:4_2_6_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E6:4_2_6_2"/><i><font color="Green">E152</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:4_2_6_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, , , </i>;<br/><b>set </b><font color="Maroon">k2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E8:4_2_6_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:4_2_6_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:4_2_6_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">k2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fss</font></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E12:4_2_6_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k2</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E13:4_2_6_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="glib_003.html#E2:4"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:4_2_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fss</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E9:4_2_6"/>
<font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a>, </i>;<br/></div><b>end;</b></div><a NAME="E51:4_2"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/><a NAME="E52:4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E53:4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><a NAME="E54:4_2"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fsA</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E55:4_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:4"/><b>then </b><i><font color="Green">E156</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:4"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">fs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">fss</font> be    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">fs</font>;<br/>



<b>assume </b><a NAME="E6:4"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 ;<br/>

<a NAME="E7:4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:4"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">fss</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fs</font>
 <b>by </b><i>, </i>;<br/>


</div>
