<?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="E1:21"/><i><font color="Green">E20</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  or  not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="measure5.html#T60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1"/>
( <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><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_1_2_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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E3:21_1_2_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> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><a href="supinf_2.html#K1">0.</a> ,<a href="supinf_2.html#K1">0.</a> </span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#T14">MEASURE5:14</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E4:21_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="measure5.html#D10">MEASURE5:def 10</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_2_1_1_2_1"/><i><font color="Green">E28</font></i>: 
( (  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or (  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 ;<br/>

<a NAME="E2:21_1_2_1_1_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span> )
 
<b>by </b><i>, <a class="ref" href="measure6.html#T52">MEASURE6:52</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="measure5.html#T13">MEASURE5:13</a>, <a class="ref" href="measure5.html#T14">MEASURE5:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:21_1_2_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> 
 <b>by </b><i>, <a class="ref" href="measure6.html#T52">MEASURE6:52</a></i>;<br/><a NAME="E4:21_1_2_1_1_2"/><i><font color="Green">E30</font></i>: 
( (  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span> ) &amp; ( <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">B</font> being   <a href="measure5.html#NM1">Interval</a><b> such that </b><br/><a NAME="E6:21_1_2_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E7:21_1_2_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:21_1_2_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:21_1_2_1_1_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font>
 ;<br/>
<b>thus </b><a NAME="E6:21_1_2_1_1_2_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, , <a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:21_1_2_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( (  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="measure6.html#K5">^^</a> </span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">B</font></span>)</span> ) &amp; ( <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 <b>by </b><i/>;<br/><a NAME="E9:21_1_2_1_1_2"/><i><font color="Green">E58</font></i>: 
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  is   <a href="real_1.html#NM1">Real</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_2_1_1_2_3"/><i><font color="Green">E59</font></i>: 
(  not  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> is   <a href="real_1.html#NM1">Real</a> or  not <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  is   <a href="real_1.html#NM1">Real</a> )
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1_2_1_1_2_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:21_1_2_1_1_2_3"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E6:21_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1_2_3_1_1"/>
(  not  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> is   <a href="real_1.html#NM1">Real</a> or  not <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  is   <a href="real_1.html#NM1">Real</a> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_1"/>
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> is  not   <a href="real_1.html#NM1">Real</a>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_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="E1:21_1_2_1_1_2_3_1_1_1_1_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_1_1_1_1"/>
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><b>hence </b><a NAME="E3:21_1_2_1_1_2_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_1_1_1_2"/>
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="urysohn2.html#E7:21_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="measure6.html#T54">MEASURE6:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_2_1_1_2_3_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  is  not   <a href="real_1.html#NM1">Real</a>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1_2_3_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_3_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="urysohn2.html#E7:21_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="measure6.html#T54">MEASURE6:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_2_1_1_2"/>
<font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2_3_1_1_2_1_1_2"/><b>then </b>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>;<br/><b>hence </b><a NAME="E3:21_1_2_1_1_2_3_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_2_1_1_2_3_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:21_1_2_1_1_2_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:21_1_2_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E12:21_1_2_1_1_2"/><i><font color="Green">E69</font></i>: 
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1_2_4_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span> )
 </span><b>by </b><i>, <a class="ref" href="measure6.html#T52">MEASURE6:52</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_4_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_4_1_1_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_4_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_4_1_1_2"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_4_1_1_3"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_4_1_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_4_1_1_3"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_4_1_1_4"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_4_1_1_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_4_1_1_4"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_2_1_1_2_4"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> &amp; <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> )
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1_2_5_1"/>
( <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> or  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_5_1_1"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_5_1_1"/>
<font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_5_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green">E20</font></i></a>, , <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_5_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_5_1_2"/><i><font color="Green">E70</font></i>: 
 <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2_5_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="measure5.html#K5">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>consider </b><font color="Maroon">s1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">t1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:21_1_2_1_1_2_5_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="measure6.html#K5">^^</a>  &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t1</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="urysohn2.html#E7:21_1_2_1_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:21_1_2_1_1_2_5_1_2"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E7:21_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, </i>;<br/><a NAME="E6:21_1_2_1_1_2_5_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t1</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E7:21_1_2_1_1_2_5_1_2"/><b>then </b>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  &amp;  <a href="measure6.html#K4">^^</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> <a href="measure6.html#K5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>then </b> <a href="measure5.html#K5">vol</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green">E20</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E9:21_1_2_1_1_2_5_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:21_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="integra2.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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