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

<b>thus </b><a NAME="E1:18_1_2"/>
for <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="relset_1.html#NM3">Relation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>  st ( 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/><font color="Olive">f1</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> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 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/><font color="Olive">f2</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> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> ) holds <br/><font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f1</font> be   <a href="relset_1.html#NM3">Relation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <font color="Maroon">f2</font> be   <a href="relset_1.html#NM3">Relation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>;<br/>

<b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green">E46</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/><font color="Maroon">f1</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> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 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/><font color="Maroon">f2</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> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Olive">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> ) )
 ;<br/>

<a NAME="E2:18_1_2_1"/>
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/><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_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="E1:18_1_2_1_1"/><i><font color="Green">E47</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="E3:18_1_2_1_1"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_1_2_1_1"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</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> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relset_1.html#K10">.:</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>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1_2_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T54">RELSET_1:54</a></i>;<br/>


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


</div>
