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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/>

<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c2">aa</font> =  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>set </b><font color="Maroon" title="c3">bb</font> = <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> ;<br/>
<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> implies  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="E4">A3</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</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" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">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" title="E5">A4</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp;  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T48" title="MEASURE6:th.48">MEASURE6:48</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">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" title="E5">A5</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp;  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T49" title="MEASURE6:th.49">MEASURE6:49</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">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" title="E5">A6</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp;  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T50" title="MEASURE6:th.50">MEASURE6:50</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_3"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">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" title="E5">A7</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T51" title="MEASURE6:th.51">MEASURE6:51</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_1_1_1_4"><i><font color="Green" title="E5">A7</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">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" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:1"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  implies  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</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" title="E4">A8</font></i>: 
 <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</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" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">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" title="E5">A9</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T48" title="MEASURE6:th.48">MEASURE6:48</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_1"><i><font color="Green" title="E5">A9</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">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" title="E5">A10</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp;  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T49" title="MEASURE6:th.49">MEASURE6:49</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_2"><i><font color="Green" title="E5">A10</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">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" title="E5">A11</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T50" title="MEASURE6:th.50">MEASURE6:50</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_3"><i><font color="Green" title="E5">A11</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">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" title="E5">A12</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="measure6.html#T51" title="MEASURE6:th.51">MEASURE6:51</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:1_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="urysohn2.html#E1:1_2_1_1_4"><i><font color="Green" title="E5">A12</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">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" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1"/>
( (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  implies  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> implies  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


</div>
