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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">c<sub>1</sub></font> is  <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E1:7_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E2:7_1_1"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
;<br/>
<a NAME="E3:7_1_1"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<div><i><font color="Green" title="E8">A3</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" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">v</font>] )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">u</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">v</font> as  <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="topgen_5.html#E1:7_1_1_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:7_1_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">z</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E7:7_1_1_1"/><i><font color="Green" title="E11">A6</font></i>: 
for <font color="Olive" title="b1">z</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></i>;<br/><a NAME="E8:7_1_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a><br/>

<b>let </b><font color="Maroon" title="c8">z</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:7_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/>
<a NAME="E3:7_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E7:7_1_1_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_1"/>
not  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T2" title="RCOMP_2:th.2">RCOMP_2:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:7_1_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a><br/>

<b>let </b><font color="Maroon" title="c8">z</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:7_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/>
<a NAME="E3:7_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:7_1_1_1"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_2"/>
not <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="square_1.html#T113" title="SQUARE_1:th.113">SQUARE_1:113</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:7_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">v</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:7_1_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:7_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">X</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="topgen_5.html#E1:7_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="topgen_5.html#E3:7_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="topgen_5.html#E4:7_1_1"><i><font color="Green" title="E8">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 ;<br/>


</div>
