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

<b>let </b><font color="Maroon">A</font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:63"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">B</font> being  <a href="measure5.html#NM1">Interval</a><br/> for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">B</font> is <a href="measure5.html#V4">left_open_interval</a> holds <br/><font color="Olive">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Olive">B</font> is <a href="measure5.html#V4">left_open_interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</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/>
<b>assume </b><a NAME="E3:63_1"/>
<font color="Maroon">B</font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:63_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K3">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D8">MEASURE5:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font>;<br/>
<a NAME="E6:63_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="measure6.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E7:63_1"/>
<font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</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"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font></span>)</span></span><a href="measure5.html#K3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_1_1"/>
<font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font></span>)</span></span><a href="measure5.html#K3">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:63_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:63_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="measure6.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d</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="E6:63_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="ref" href="measure6.html#T18" target="_self">Th18</a>, <a class="ref" href="measure6.html#T22" target="_self">Th22</a>, <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>
<a NAME="E7:63_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:63_1_1_1"/>
<font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="measure6.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:63_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font></span>)</span></span><a href="measure5.html#K3">.]</a></span>
 <b>by </b><i><a class="ref" href="measure6.html#T23" target="_self">Th23</a>, <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c</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="E2:63_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font></span>)</span></span><a href="measure5.html#K3">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c</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="E5:63_1_1"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T21" target="_self">Th21</a>, <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>
<a NAME="E6:63_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:63_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:63_1_1"/>
<font color="Maroon">c1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:63_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure6.html#T18" target="_self">Th18</a>, <a class="ref" href="measure6.html#T22" target="_self">Th22</a>, <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:63_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:63_1"/>
<font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</font> is <a href="measure5.html#V4">left_open_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T19" target="_self">Th19</a>, <a class="ref" href="measure5.html#D8">MEASURE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:63"/>
( <font color="Maroon">x</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a> implies <font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">x</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">A</font> as   <a href="measure5.html#NM1">Interval</a> <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:63_2"/>
<font color="Maroon">y</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:63_2"/>
<font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T17" target="_self">Th17</a>, <a class="ref" href="measure6.html#T18" target="_self">Th18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
( <font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a> iff <font color="Maroon">x</font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">A</font> is <a href="measure5.html#V4">left_open_interval</a> )
 <b>by </b><i><a class="ref" href="measure6.html#T17" target="_self">Th17</a></i>;<br/>


</div>
