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

<a NAME="E1:31"/><i><font color="Green">E18</font></i>: 
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.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="measure5.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:31_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.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="measure5.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:31_2"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:31"/><i><font color="Green">E22</font></i>: 
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K4">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:31_3"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:31_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K4">.[</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#D4">MEASURE5:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/><i><font color="Green">E25</font></i>: 
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:31_4"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:31_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K3">.]</a></span>
 <b>by </b><i><a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:31"/>
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">P</font> =  <a href="supinf_1.html#K3">REAL</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<a NAME="E7:31"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K3">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.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="measure5.html#K2">.[</a></span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:31"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:31"/><b>then </b>
<font color="Maroon">P</font> is <a href="measure5.html#V1">open_interval</a>
 
<b>by </b><i><a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:31"/>
(  <a href="supinf_1.html#K3">REAL</a>  is   <a href="measure5.html#NM1">Interval</a> &amp;  <a href="supinf_1.html#K3">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.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="measure5.html#K2">.[</a></span> &amp;  <a href="supinf_1.html#K3">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K1">.]</a></span> &amp;  <a href="supinf_1.html#K3">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K4">.[</a></span> &amp;  <a href="supinf_1.html#K3">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="measure5.html#K3">.]</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:31"><i><font color="Green">E19</font></i></a>, , , <a class="txt" href="measure6.html#E3:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure5.html#D9">MEASURE5:def 9</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
