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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/>

<b>assume </b><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> ;<br/>
<a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="E1:1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>case </b></a><a NAME="E1:1_1_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span> &amp;  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T48">MEASURE6:48</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_1"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T53">MEASURE5:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><b>case </b></a><a NAME="E1:1_1_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> &amp;  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T49">MEASURE6:49</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_2"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T54">MEASURE5:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_3"><b>case </b></a><a NAME="E1:1_1_1_1_3"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span> &amp;  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T50">MEASURE6:50</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_3"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T55">MEASURE5:55</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_4"><b>case </b></a><a NAME="E1:1_1_1_1_4"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T51">MEASURE6:51</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_4"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T56">MEASURE5:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_1"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E4</font></i>: 
 <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="E1:1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_1"><b>case </b></a><a NAME="E1:1_2_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T48">MEASURE6:48</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_1"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T53">MEASURE5:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_2"><b>case </b></a><a NAME="E1:1_2_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> &amp;  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T49">MEASURE6:49</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_2"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T54">MEASURE5:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_3"><b>case </b></a><a NAME="E1:1_2_1_1_3"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T50">MEASURE6:50</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_3"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T55">MEASURE5:55</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_4"><b>case </b></a><a NAME="E1:1_2_1_1_4"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="measure6.html#T51">MEASURE6:51</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_4"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="measure5.html#T56">MEASURE5:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_2"/>
 <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1"/>
( (  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="measure5.html#K5">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:1"><i><font color="Green">E3</font></i></a></i>;<br/>


</div>
