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

<b>let </b><font color="Maroon">A</font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:10"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>case </b></a><a NAME="E1:10_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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="E1:10_1_1_1_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i>, <a class="ref" href="measure5.html#T14">MEASURE5:14</a></i>;<br/><a NAME="E3:10_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:10_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>set </b><font color="Maroon">c</font> =  <a href="supinf_1.html#K1">+infty</a> ;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E4:10_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_1_1_1_2_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E7:10_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_1_1_1_2_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:10_1_1_1_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>
<a NAME="E11:10_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E12:10_1_1_1_1_1_2_1_1"/>
<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_1_1_1_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_1_1_1_2_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1_2_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_1_2_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E6:10_1_1_1_1_1_2_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E2:10_1_1_1_1_1_2_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:10_1_1_1_1_1_2_1_2_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:10_1_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_1_1_1_2_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_2_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:10_1_1_1_1_1_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:10_1_1_1_1_1_2"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><b>hence </b><a NAME="E6:10_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, , , , <a class="ref" href="measure6.html#T35">MEASURE6:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:10_1_1_1_1_1_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E6:10_1_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_1_1_1_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E9:10_1_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E11:10_1_1_1_1_1_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E12:10_1_1_1_1_1_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_4_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">1o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">1r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">1o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">1r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">1o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1r</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">2o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">2r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">2o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">2r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">2o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">2r</font> )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E11:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>
<a NAME="E12:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E1:10_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>
<a NAME="E13:10_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font> is   <a href="supinf_1.html#NM3">R_eal</a> &amp; <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font> is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">1o1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">1r1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E15:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">1o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font> &amp; <font color="Maroon">1r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font> )
 ;<br/>
<a NAME="E16:10_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font> is   <a href="supinf_1.html#NM3">R_eal</a> &amp; <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font> is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">2o1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">2r1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E18:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">2o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font> &amp; <font color="Maroon">2r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font> )
 ;<br/>
<a NAME="E19:10_1_1_1_1_1_4_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">1r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1o1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:10_1_1_1_1_1_4_1_1"/>
<font color="Maroon">2r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">2o1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E21:10_1_1_1_1_1_4_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, , , <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, , <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn2.html#E1:10_1_1_1"><i><font color="Green">E25</font></i></a>, , , , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_1_1_1_4_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_4_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1_4_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_1_4_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E6:10_1_1_1_1_1_4_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E2:10_1_1_1_1_1_4_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_4_1_2_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:10_1_1_1_1_1_4_1_2_1_1_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E3:10_1_1_1_1_1_4_1_2_1_1_1"/>
<font color="Maroon">o</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E1:10_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, <a class="txt" href="urysohn2.html#E1:10_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_1_1_1_4_1_2_1_1_1"/>
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_1_1_1_4_1_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_4_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_4_1_2_1_1_2"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1_4_1_2_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:10_1_1_1_1_1_4_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_1_4_1_2_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="txt" href="urysohn2.html#E1:10_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_1_1_1_4_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_4_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_4_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:10_1_1_1_1_1_4_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_4_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:10_1_1_1_1_1_4"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E14:10_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/><a NAME="E15:10_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E16:10_1_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_5"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1_5"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_1_1_1_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><a NAME="E6:10_1_1_1_1_1_5"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_1_1_1_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E9:10_1_1_1_1_1_5"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_5_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_5_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_1_1_1_5_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E7:10_1_1_1_1_1_5_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_1_1_1_5_1_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:10_1_1_1_1_1_5_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>
<a NAME="E11:10_1_1_1_1_1_5_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:10_1_1_1_1_1_5_1_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_1_1_1_5_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_1_1_1_5_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_1_1_1_5_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:10_1_1_1_1_1_5_1_2"/>
<font color="Maroon">q</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_1_1_1_5_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E6:10_1_1_1_1_1_5_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E7:10_1_1_1_1_1_5_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_5_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q3</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1_5_1_2_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:10_1_1_1_1_1_5_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_5_1_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_5_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1_5_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1_5_1_2_1_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:10_1_1_1_1_1_5_1_2_1_1_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , , <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_1_1_1_5_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_1_1_1_5_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1_1_1_1_1_5_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_1_1_1_1_1_5_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_5_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:10_1_1_1_1_1_5"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_1_1_1_5"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1_6"><b>case </b></a><a NAME="E1:10_1_1_1_1_1_6"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_1_1_6"/><b>then </b>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="ref" href="measure5.html#T14">MEASURE5:14</a></i>;<br/><a NAME="E3:10_1_1_1_1_1_6"/><b>then </b>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:10_1_1_1_1_1_6"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_1_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>case </b></a><a NAME="E1:10_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_2"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3"><b>case </b></a><a NAME="E1:10_1_1_3"/><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_3_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:10_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="ref" href="measure5.html#T14">MEASURE5:14</a></i>;<br/><a NAME="E3:10_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:10_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><a NAME="E6:10_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_3_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E9:10_1_1_3_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_3_1_1_2_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E7:10_1_1_3_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_3_1_1_2_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:10_1_1_3_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:10_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:10_1_1_3_1_1_2_1_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_3_1_1_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_3_1_1_2_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_2_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:10_1_1_3_1_1_2_1_2"/>
<font color="Maroon">q</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_2_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E6:10_1_1_3_1_1_2_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E7:10_1_1_3_1_1_2_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_2_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q3</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_2_1_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:10_1_1_3_1_1_2_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_2_1_2_1_1"/>
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_2_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_2_1_2_1_1_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:10_1_1_3_1_1_2_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E5:10_1_1_3_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_3_1_1_2_1_2_1_1_1"/>
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_3_1_1_2_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1_1_3_1_1_2_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_1_1_3_1_1_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_3_1_1_2_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:10_1_1_3_1_1_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_3_1_1_3"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, , , , <a class="ref" href="measure6.html#T35">MEASURE6:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_4"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_4"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_4"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E6:10_1_1_3_1_1_4"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_3_1_1_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E9:10_1_1_3_1_1_4"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E11:10_1_1_3_1_1_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E12:10_1_1_3_1_1_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_4_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">1o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">1r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">1o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">1r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">1o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1r</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">2o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">2r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">2o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">2r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">2o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">2r</font> )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E11:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:10_1_1_3_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font> is   <a href="supinf_1.html#NM3">R_eal</a> &amp; <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font> is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">1o1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">1r1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E15:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">1o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1o</font> &amp; <font color="Maroon">1r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">1r</font> )
 ;<br/>
<a NAME="E16:10_1_1_3_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font> is   <a href="supinf_1.html#NM3">R_eal</a> &amp; <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font> is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">2o1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">2r1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E18:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">2o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2o</font> &amp; <font color="Maroon">2r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">2r</font> )
 ;<br/>
<a NAME="E19:10_1_1_3_1_1_4_1_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">1o1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">1r1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:10_1_1_3_1_1_4_1_1"/>
<font color="Maroon">2o1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">2r1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E21:10_1_1_3_1_1_4_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, , , <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, , <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, , , , , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_3_1_1_4_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_4_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:10_1_1_3_1_1_4_1_2"/>
<font color="Maroon">q</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_4_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E6:10_1_1_3_1_1_4_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E7:10_1_1_3_1_1_4_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_4_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q3</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_4_1_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:10_1_1_3_1_1_4_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_4_1_2_1_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_4_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_4_1_2_1_1_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:10_1_1_3_1_1_4_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E5:10_1_1_3_1_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, , , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_3_1_1_4_1_2_1_1_1"/>
<font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_3_1_1_4_1_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_4_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_4_1_2_1_1_2"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_4_1_2_1_1_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">ri</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:10_1_1_3_1_1_4_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_3_1_1_4_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_3_1_1_4_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1_1_3_1_1_4_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_1_1_3_1_1_4_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_3_1_1_4_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">g</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:10_1_1_3_1_1_4"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E14:10_1_1_3_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E15:10_1_1_3_1_1_4"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="urysohn2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E16:10_1_1_3_1_1_4"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_5"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_5"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_5"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_5"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><a NAME="E6:10_1_1_3_1_1_5"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_3_1_1_5"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E9:10_1_1_3_1_1_5"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_5_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_5_1_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_3_1_1_5_1_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E7:10_1_1_3_1_1_5_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_3_1_1_5_1_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:10_1_1_3_1_1_5_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:10_1_1_3_1_1_5_1_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:10_1_1_3_1_1_5_1_1"/>
<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_3_1_1_5_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_3_1_1_5_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:10_1_1_3_1_1_5_1_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:10_1_1_3_1_1_5_1_2"/>
<font color="Maroon">q</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_3_1_1_5_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E6:10_1_1_3_1_1_5_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>;<br/>
<a NAME="E7:10_1_1_3_1_1_5_1_2"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_5_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">q3</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_5_1_2_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:10_1_1_3_1_1_5_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_5_1_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1_1_5_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_3_1_1_5_1_2_1_1_1"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:10"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">o</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_3_1_1_5_1_2_1_1_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">ri</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">o</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:10_1_1_3_1_1_5_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E5:10_1_1_3_1_1_5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="txt" href="urysohn2.html#E4:10"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_3_1_1_5_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_3_1_1_5_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1_1_3_1_1_5_1_2_1"/>
<font color="Maroon">q</font> <a href="real_1.html#K6">/</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E2:10"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_1_1_3_1_1_5_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_3_1_1_5_1"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:10_1_1_3_1_1_5"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_3_1_1_5"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i><a class="ref" href="urysohn2.html#T1" target="_self">Th1</a>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_3_1_1_6"><b>case </b></a><a NAME="E1:10_1_1_3_1_1_6"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:10_1_1_3_1_1_6"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i>, <a class="ref" href="measure5.html#T14">MEASURE5:14</a></i>;<br/><a NAME="E3:10_1_1_3_1_1_6"/><b>then </b>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:10_1_1_3_1_1_6"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_1_3"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:10"/>
<font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/>


</div>
