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<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> implies for <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">q</font> );<br/>
<a NAME="E1:84"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E2:84"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> implies for <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">q</font> )
 ;<br/><b>assume </b><a NAME="E2:84_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:84_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><i><font color="Green">E51</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E1:84"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>
;<br/><a NAME="E6:84_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></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="E7:84_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E9:84_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="finseq_1.html#E11:84_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:84_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E10:84_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:84_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E13:84_1_1"><i><font color="Green">E55</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E13:84_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E13:84_1_1"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16:84_1_1"><i><font color="Green">E57</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:84_1_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E16:84_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16:84_1_1"><i><font color="Green">E57</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:84_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <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>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E21:84_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><a NAME="E22:84_1_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Olive">x</font>]
 ;<br/><b>consider </b><font color="Maroon">q2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E24:84_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E25:84_1_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1" target="_self">FINSEQ_1:sch 1</a>(<a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, );<br/></i><a NAME="E26:84_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E27:84_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E28:84_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E29:84_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span>
 ;<br/>

<a NAME="E2:84_1_1_2_1"/><b>then </b><i><font color="Green">E84</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#K2">rng</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:84"><i><font color="Green">E31</font></i></a></i>;<br/>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:84_1_1_2_1_1"/><i><font color="Green">E87</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/><b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:84_1_1_2_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font>
 <b>and </b><br/><a NAME="E5:84_1_1_2_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:84_1_1_2_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E7:84_1_1_2_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a></i>;<br/><a NAME="E9:84_1_1_2_1_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E10:84_1_1_2_1_1"/>
 <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:84_1_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E12:84_1_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E13:84_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="finseq_1.html#E13:84_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="finseq_1.html#E15:84_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E14:84_1_1_2_1_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"><font color="Maroon">x</font></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="E15:84_1_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font>
 ;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_1_1_2_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E4:84_1_1_2_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E6:84_1_1_2_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , </i>;<br/><a NAME="E7:84_1_1_2_1_2"/><i><font color="Green">E131</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:84_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:84_1_1_2_1_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , </i>;<br/><a NAME="E9:84_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:84_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E11:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E12:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:84_1_1_2_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , </i>;<br/><a NAME="E14:84_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font>
 ;<br/><a NAME="E15:84_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="finseq_1.html#E13:84_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="finseq_1.html#E15:84_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E16:84_1_1_2_1_2"/><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"><font color="Maroon">x</font></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="E17:84_1_1_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:84_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:84_1_1_2"/><i><font color="Green">E134</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#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:84_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 
;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:84_1_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E6:84_1_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:84_1_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E9:84_1_1_2"/>
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"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:84_1_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15:84_1_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:84_1_1_2_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:84_1_1_2_2"/><i><font color="Green">E142</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, <a class="ref" href="nat_1.html#T59">NAT_1:59</a></i>;<br/><a NAME="E6:84_1_1_2_2"/>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, </i>;<br/><a NAME="E7:84_1_1_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:84_1_1_2_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E9:84_1_1_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E10:84_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E22:84_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_3"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:84_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:84_1_1_2_3"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E4:84_1_1_2_3"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:84_1_1_2_3"/><b>then </b>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:84_1_1_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:84_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:84_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:84"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:84_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E31:84_1_1"/><i><font color="Green">E146</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_1_1_3_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:84_1_1_3_1"/><i><font color="Green">E148</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:84_1_1_3_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:84_1_1_3_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:84_1_1_3_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a></i>;<br/>
<a NAME="E8:84_1_1_3_1"/><i><font color="Green">E152</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="finseq_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E9:84_1_1_3_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:84_1_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:84_1_1_3_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:84_1_1_3_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E13:84_1_1_3_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E14:84_1_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E5:84_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:84_1_1_3_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &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"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:84_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></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/>


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

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:84_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:84_1_1_3"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &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"><font color="Maroon">x</font></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/>
<a NAME="E4:84_1_1_3"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <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="txt" href="finseq_1.html#E1:84"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:84_1_1_3"/><b>then </b>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> ) &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:84_1_1_3"/><b>then </b>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:84_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:84_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:84_1_1_4"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:84_1_1_4"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:84_1_1_4"/><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">y1</font>, <font color="Maroon">m2</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a></i>;<br/>
<a NAME="E5:84_1_1_4"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">q1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<div><i><font color="Green">E161</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_4_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> )
 ;<br/><a NAME="E2:84_1_1_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:84_1_1_4_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_4_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font>
 ;<br/><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:84_1_1_4_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E3:84_1_1_4_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:84_1_1_4_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E22:84_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E5:84_1_1_4_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E6:84_1_1_4_2"/><i><font color="Green">E168</font></i>: 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><a NAME="E7:84_1_1_4_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E8:84_1_1_4_2"/><b>then </b><i><font color="Green">E169</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:84_1_1_4_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E10:84_1_1_4_2"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E11:84_1_1_4_2"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E29:84_1_1"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E12:84_1_1_4_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E13:84_1_1_4_2"/>
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E22:84_1_1"><i><font color="Green">E78</font></i></a>, </i>;<br/><a NAME="E14:84_1_1_4_2"/><b>then </b>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E15:84_1_1_4_2"/><b>then </b>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E16:84_1_1_4_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="finseq_1.html#E22:84_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="finseq_1.html#E28:84_1_1"><i><font color="Green">E82</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_4_3"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font>
 ;<br/><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:84_1_1_4_3"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E3:84_1_1_4_3"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:84_1_1_4_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E5:84_1_1_4_3"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E6:84_1_1_4_3"/><i><font color="Green">E176</font></i>: 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><a NAME="E7:84_1_1_4_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, </i>;<br/><a NAME="E8:84_1_1_4_3"/><b>then </b><i><font color="Green">E177</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:84_1_1_4_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, </i>;<br/><a NAME="E10:84_1_1_4_3"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E11:84_1_1_4_3"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:84_1_1_4_3"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E13:84_1_1_4_3"/>
<font color="Maroon">m2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E14:84_1_1_4_3"/><b>then </b>
( <font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E15:84_1_1_4_3"/><b>then </b>
( <font color="Maroon">m2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E16:84_1_1_4_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E28:84_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="finseq_1.html#E29:84_1_1"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_4_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">j1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:84_1_1_4_4"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E2:84_1_1_4_4"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 ;<br/><b>given </b><font color="Maroon">j2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E3:84_1_1_4_4"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E4:84_1_1_4_4"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 ;<br/><a NAME="E5:84_1_1_4_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="finseq_1.html#E28:84_1_1"><i><font color="Green">E82</font></i></a>, </i>;<br/><a NAME="E6:84_1_1_4_4"/><b>then </b><i><font color="Green">E185</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E7:84_1_1_4_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a>, </i>;<br/><a NAME="E8:84_1_1_4_4"/><b>then </b><i><font color="Green">E186</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:84_1_1_4_4"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="txt" href="finseq_1.html#E24:84_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="finseq_1.html#E27:84_1_1"><i><font color="Green">E81</font></i></a>, </i>;<br/><a NAME="E10:84_1_1_4_4"/><b>then </b>
( <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E11:84_1_1_4_4"/><b>then </b>
( <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E12:84_1_1_4_4"/><b>then </b>
( <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/><a NAME="E13:84_1_1_4_4"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:84_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E29:84_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E14:84_1_1_4_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E25:84_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="finseq_1.html#E28:84_1_1"><i><font color="Green">E82</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:84_1_1_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a>, <a class="txt" href="finseq_1.html#E22:84_1_1"><i><font color="Green">E78</font></i></a>, </i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, </i>
;<br/><a NAME="E34:84_1_1"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><div><i><font color="Green">E188</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:84_1_1_6"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 ;<br/><div><i><font color="Green">E190</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_6_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:84_1_1_6_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E3:84_1_1_6_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_6_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:84_1_1_6_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 <b>and </b><br/><a NAME="E2:84_1_1_6_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E3:84_1_1_6_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T29" target="_self">Th29</a>, , <a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:84_1_1_6_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_6_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:84_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:84_1_1_6"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:84_1_1_7"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q2</font>
 ;<br/><b>hence </b><font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E21:84_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, , </i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<font color="Maroon">q</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#K3" target="_self">len</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div><a NAME="E37:84_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/><b>then </b><i><font color="Green">E193</font></i>:  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q2</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/> <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T9" target="_self">Th9</a></i>
;<br/><b>hence </b><a NAME="E40:84_1_1"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E5:84_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="finseq_1.html#E9:84_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T21" target="_self">Th21</a>, , <a class="ref" href="finseq_1.html#T23" target="_self">Th23</a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:84_1"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:84"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self">FINSEQ_1:sch 3</a>(, );<br/></i>
<b>hence </b><a NAME="E4:84"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> implies  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 ;<br/>


</div>
