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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) );<br/>
<a NAME="E1:92_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> 0
 ;<br/>

<b>take </b>
 <a href="finseq_1.html#K6" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> 
;<br/>

<b>thus </b><a NAME="E2:92_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>

<b>thus </b><a NAME="E3:92_1_1_1"/>
for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:92_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:92_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1_2_1"/>
not <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1_1_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</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="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:92_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E7:92_1_1_2_1"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:92_1_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_1_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E2:92_1_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_1_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_1_2_1"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<a NAME="E3:92_1_1_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_1_2_1"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E7:92_1_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:92_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:92_1_1_2_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:92_1_1_2_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:92_1_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, <a class="txt" href="finseq_1.html#E9:92_1_1_2_1"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<a NAME="E6:92_1_1_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:92_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:92_1_1_2_1"/><i><font color="Green">E57</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</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; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_1_2_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E12:92_1_1_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E14:92_1_1_2_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_1_2_1"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><i><font color="Green">E78</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> = 
<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#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="finseq_1.html#E1:92"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="finseq_1.html#E9:92_1_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><a NAME="E16:92_1_1_2_1"/>
for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</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="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_1_2_1_4"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:92_1_1_2_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_1_2_1_4_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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#K8" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:92_1_1_2_1_4_1"/><b>then </b>
<font color="Maroon">l</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:92_1_1_2_1_4_1"/><b>then </b>
<font color="Maroon">l</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="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:92_1_1_2_1_4_1"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:92_1_1_2_1_4_1"/><b>then </b>
<font color="Maroon">l</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#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:92_1_1_2_1_4_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92_1_1_2_1_4"/><i><font color="Green">E81</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1_2_1_4_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:92_1_1_2_1_4_2"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</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#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E3:92_1_1_2_1_4_2"/><b>then </b>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E9:92_1_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:92_1_1_2_1_4_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" 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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" 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>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>
;<br/><b>hence </b><a NAME="E6:92_1_1_2_1_4_2"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</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="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1_2_1_4_3"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:92_1_1_2_1_4_3"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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#K8" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:92_1_1_2_1_4_3"/><b>then </b>
<font color="Maroon">m</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K12" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" target="_self">*&gt;</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:92_1_1_2_1_4_3"/><b>then </b>
<font color="Maroon">m</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="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:92_1_1_2_1_4_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:92_1_1_2_1_4_3"/><b>then </b>
<font color="Maroon">m</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>, </i>;<br/><a NAME="E7:92_1_1_2_1_4_3"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E8:92_1_1_2_1_4_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/><a NAME="E9:92_1_1_2_1_4_3"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_1_2_1"><i><font color="Green">E58</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#D3" target="_self">Def3</a>, </i>;<br/><b>hence </b><a NAME="E10:92_1_1_2_1_4_3"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E9:92_1_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:92_1_1_2_1_4"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E17:92_1_1_2_1"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_1_2"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92_1_1"/>
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:92_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 <b>by </b><i/>;<br/>


</div>
