<?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="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<a NAME="E1:61"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a><br/> for <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">B</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> holds <br/><font color="Olive" title="b2">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Olive" title="b1">B</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">B</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">y</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E3:61_1"/>
<font color="Maroon" title="c3">B</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:61_1"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c3">B</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"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c8">b</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E7:61_1"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">t</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E5:61_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/>
<a NAME="E8:61_1"/>
<font color="Maroon" title="c5">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</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"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">t</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">t</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c11">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c13">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:61_1_1_1"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">d</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:61_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">d1</font> = <font color="Maroon" title="c13">d</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:61_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">d1</font> &amp; <font color="Maroon" title="c14">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c14">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E5:61_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="measure6.html#E4:61_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E7:61_1_1_1"/><b>then </b><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c9">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c14">d1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c14">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">t</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/>
<a NAME="E8:61_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c14">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:61_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:61_1_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">t</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:61_1_1_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:61_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">t</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c11">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">c1</font> = <font color="Maroon" title="c12">c</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E5:61_1_1"/>
( <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">c1</font> &amp; <font color="Maroon" title="c13">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c8">b</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:61_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E6:61_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c13">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/>
<a NAME="E7:61_1_1"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c13">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/>
<a NAME="E8:61_1_1"/>
<font color="Maroon" title="c13">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:61_1_1"/><b>then </b>
( <font color="Maroon" title="c12">c</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E5:61_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="measure6.html#E7:61_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:61_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:61_1"/>
<font color="Maroon" title="c4">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="measure6.html#E7:61_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61"/>
( <font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> implies <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_2"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">B</font> = <font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> as   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> <b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:61_2"/>
<font color="Maroon" title="c4">y</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="measure6.html#T59" target="_self" title="MEASURE6:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E5:61_2"/>
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="measure6.html#E1:61"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="measure6.html#E1:61_2"><i><font color="Green" title="E45">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> iff <font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:61"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>


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