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

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E5:19_1"/>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:19_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> ) holds <br/><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:19_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:19_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E4:19_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:19_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2"/>
for <font color="Olive">z</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:19_2_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<a NAME="E3:19_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:19_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:19_2_1_1_2_1"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:19_2_1_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/><a NAME="E4:19_2_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/><b>hence </b><a NAME="E5:19_2_1_1_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:19"/>
<font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:19"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T63">SEQ_4:63</a></i>;<br/>


</div>
