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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:60"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="measure5.html#NM1">Interval</a><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Olive">b<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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="E3:60_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:60_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E7:60_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="measure6.html#E5:60_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E8:60_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="measure5.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="measure5.html#K2">.[</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="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></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="E1:60_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:60_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:60_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></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="E6:60_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E5:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="measure6.html#E4:60_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>
<a NAME="E7:60_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E8:60_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:60_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:60_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="measure5.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:60_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>11</sub></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="E2:60_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="measure5.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></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="E5:60_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>
<a NAME="E6:60_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E7:60_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:60_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:60_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E5:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="measure6.html#E7:60_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:60_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:60_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> is <a href="measure5.html#V1">open_interval</a>
 <b>by </b><i><a class="txt" href="measure6.html#E7:60_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> )
 ;<br/>

<b>assume </b><a NAME="E3:60"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></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="E6:60"/>
<font color="Maroon">c<sub>4</sub></font> <a href="measure6.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="measure6.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E7:60"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 <b>by </b><i><a class="txt" href="measure6.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="measure6.html#E3:60"><i><font color="Green">E44</font></i></a></i>;<br/>


</div>
