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

<b>let </b><font color="Maroon">n</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="membered.html#T2">MEMBERED:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X'</font> as   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="numbers.html#K1">REAL</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a>  implies  ex <font color="Olive">m</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) ) );<br/>
<a NAME="E3:7_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a> ]
 
;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/><b>thus </b><a NAME="E2:7_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:7_1_1_1_1_1_1"/>
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>take </b><font color="Maroon">bb</font> = <font color="Maroon">b</font>;<br/><a NAME="E3:7_1_1_1_1_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.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="E4:7_1_1_1_1_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E5:7_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/><b>hence </b><a NAME="E6:7_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">bb</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:7_1_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:7_1_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">bb</font> = <font color="Maroon">b</font>;<br/><a NAME="E2:7_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.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="E3:7_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E4:7_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/><a NAME="E5:7_1_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:7_1_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:7_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">bb</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E2:7_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E3:7_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/><a NAME="E4:7_1_1_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:7_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:7_1_1"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">m</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:7_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>and </b><br/><a NAME="E8:7_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>
<a NAME="E9:7_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">m</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a><br/>

<b>thus </b><a NAME="E1:7_1_1_2"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X'</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E11:7_1_1"/>
 <a href="seq_4.html#K2">upper_bound</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_3"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:7_1_1_3"/>
for <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p</font> ) holds <br/><font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:7_1_1_3"/>
 <a href="seq_4.html#K2">upper_bound</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T63">SEQ_4:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:7_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K2">upper_bound</a> <font color="Maroon">A</font> implies ( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">k</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E13:7_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E14:7_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">k</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E15:7_1_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_4"/><i><font color="Green">E61</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E2:7_1_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>

<a NAME="E3:7_1_1_4"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_4"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:7_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K2">upper_bound</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div>
