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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> ) );<br/>
<a NAME="E1:22_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="E1:22_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i2</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="E3:22_1_1_1"/><i><font color="Green">E28</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:22_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:22_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">y0</font> = <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>;<br/><a NAME="E2:22_1_1_1_1_1_1"/>
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:22_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:22_1_1_1_1_1_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E6:22_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E8:22_1_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:22_1_1_1_1_1_1"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E10:22_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E12:22_1_1_1_1_1_1"/>
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</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="E1:22_1_1_1_1_1_1_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1_1_1_1_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:22_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E14:22_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E15:22_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">y0</font> = <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>;<br/><a NAME="E2:22_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:22_1_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:22_1_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_1_1_1_1_1_2"/>
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</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="E1:22_1_1_1_1_1_2_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1_1_1_2_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:22_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E7:22_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:22_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_3"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">y0</font> = <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span>;<br/><a NAME="E2:22_1_1_1_1_1_3"/>
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:22_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:22_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:22_1_1_1_1_1_3"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E6:22_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E7:22_1_1_1_1_1_3"/><i><font color="Green">E76</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E8:22_1_1_1_1_1_3"/>
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</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="E1:22_1_1_1_1_1_3_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1_1_1_3_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:22_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E10:22_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:22_1_1_1_1_1_3"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:22_1_1_1_1_1_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">y0</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i2</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:22_1_1_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</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="E1:22_1_1_1_1_1_4_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i2</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1_1_1_4_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</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="E3:22_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i2</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E4:22_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i2</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:22_1_1_1_1_1_4"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:22_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">x3</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x3</font>,<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x3</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E4:22_1_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="relset_1.html#NM3">Relation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:22_1_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T170">FUNCT_2:170</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:22_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E8:22_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:22_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i>, </i>;<br/>


</div>
