<?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>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E19">A1</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">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" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or  not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</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" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="urysohn2.html#T5" target="_self" title="URYSOHN2:th.5">Th5</a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E21">A3</font></i>: 
not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</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" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x</font> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a></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" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="urysohn2.html#T4" target="_self" title="URYSOHN2:th.4">Th4</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E3:21_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <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"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> ,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a>, <a class="ref" href="supinf_2.html#T19" title="SUPINF_2:th.19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E4:21_1_2_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="measure5.html#D10" title="MEASURE5:def.10">MEASURE5:def 10</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E22">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2"/><i><font color="Green" title="E23">A6</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &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="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &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="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="E24">A7</font></i>: 
( (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &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="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &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="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 ;<br/>

<a NAME="E2:21_1_2_1_1_2_1"/>
( <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> or <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> or <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> or <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#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="measure6.html#T52" title="MEASURE6:th.52">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" title="E21">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="measure5.html#T13" title="MEASURE5:th.13">MEASURE5:13</a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a>, <a class="ref" href="xxreal_1.html#T20" title="XXREAL_1:th.20">XXREAL_1:20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:21_1_2_1_1_2"/><i><font color="Green" title="E24">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#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#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="measure6.html#T52" title="MEASURE6:th.52">MEASURE6:52</a></i>;<br/><a NAME="E4:21_1_2_1_1_2"/><i><font color="Green" title="E25">A9</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>  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#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="urysohn2.html#T1" target="_self" title="URYSOHN2:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> as   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ;<br/><a NAME="E6:21_1_2_1_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</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" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:21_1_2_1_1_2_2"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2_2"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<a NAME="E4:21_1_2_1_1_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
;<br/>
<b>hence </b><a NAME="E5:21_1_2_1_1_2_2"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:21_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
( (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">B</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="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</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="c4">B</font></span>)</span> ) &amp; ( <font color="Maroon" title="c4">B</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="c4">B</font> implies  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c4">B</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="ref" href="urysohn2.html#T1" target="_self" title="URYSOHN2:th.1">Th1</a></i>;<br/><a NAME="E8:21_1_2_1_1_2"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="E28">A15</font></i>: 
(  not  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or  not <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1_2_1_1_2_3"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2_3"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/>
<a NAME="E5:21_1_2_1_1_2_3"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E6:21_1_2_1_1_2_3"/><b>then </b><i><font color="Green" title="E30">A17</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or  not <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2_3"><i><font color="Green" title="E28">A15</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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> is  not   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="E31">A18</font></i>: 
 <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4" title="XXREAL_0:def.4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_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"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2_3_1_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_1_1_1"/>
 <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="supinf_2.html#T7" title="SUPINF_2:th.7">SUPINF_2:7</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2_3_1_1_1_1_2"><b>case </b></a><a NAME="E1:21_1_2_1_1_2_3_1_1_1_1_2"/>
 <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2_3"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2_3"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="measure6.html#T54" title="MEASURE6:th.54">MEASURE6:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></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" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  is  not   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="E31">A19</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4" title="XXREAL_0:def.4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" 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="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or <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="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2_3_1_1_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" 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="supinf_1.html#K2" title="SUPINF_1:func.2">-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#E3:21_1_2_1_1_2_3"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2_3"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="measure6.html#T54" title="MEASURE6:th.54">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" 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="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_3_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="c5">s</font>, <font color="Maroon" title="c6">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:21_1_2_1_1_2"/><i><font color="Green" title="E28">A20</font></i>: 
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  &amp; <font color="Maroon" title="c5">s</font> <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/><a NAME="E11:21_1_2_1_1_2"/><i><font color="Green" title="E29">A21</font></i>: 
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">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" 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> or <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> or <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> or <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> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="measure6.html#T52" title="MEASURE6:th.52">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" 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>
 ;<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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="urysohn2.html#T17" target="_self" title="URYSOHN2:th.17">Th17</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" 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>
 ;<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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="urysohn2.html#T16" target="_self" title="URYSOHN2:th.16">Th16</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" 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>
 ;<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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="urysohn2.html#T15" target="_self" title="URYSOHN2:th.15">Th15</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" 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>
 ;<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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="urysohn2.html#T18" target="_self" title="URYSOHN2:th.18">Th18</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" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c4">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">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" 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> or  <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>  )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E3:21_1_2_1_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" 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 class="add"><b>hence </b><a NAME="E2:21_1_2_1_1_2_5_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="urysohn2.html#E6:21_1_2_1_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="urysohn2.html#E11:21_1_2_1_1_2"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="urysohn2.html#T1" target="_self" title="URYSOHN2:th.1">Th1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E30">A22</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 class="add"><a NAME="E2:21_1_2_1_1_2_5_1_2"/><b>then </b><i><font color="Green" title="E31">A23</font></i>: 
 <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> <font color="Maroon" title="c6">t</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:21_1_2_1_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c8">t1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/> <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">s</font></span>)</span>
<b>by </b><i><a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="urysohn2.html#E7:21_1_2_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="urysohn2.html#E10:21_1_2_1_1_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="urysohn2.html#E11:21_1_2_1_1_2"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="urysohn2.html#E1:21_1_2_1_1_2_5_1_2"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="urysohn2.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="urysohn2.html#E2:21_1_2_1_1_2_5_1_2"><i><font color="Green" title="E31">A23</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:21_1_2_1_1_2_5_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">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="E13:21_1_2_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">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" title="c2">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
