<?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>assume </b><a NAME="E1:15"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E16</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/>

<b>assume </b><a NAME="E3:15"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:15"/><i><font color="Green">E18</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:15_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<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="txt" href="urysohn2.html#E2:15_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E6:15_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn2.html#E2:15_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn2.html#E5:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> implies ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  holds <br/>(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:15_2"/><i><font color="Green">E20</font></i>: 
 <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> 
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="measure6.html#T52">MEASURE6:52</a></i>;<br/>
<a NAME="E3:15_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  holds <br/>(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 ;<br/>

<a NAME="E2:15_2_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:15_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E4:15_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E6:15_2_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E7:15_2_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:15_2_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:15_2_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E7:15_2_1_1_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn2.html#E4:15_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:15_2_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:15_2_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn2.html#E7:15_2_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E12:15_2_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E9:15_2_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:15_2_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E11:15_2_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:15_2_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="supinf_1.html#NM3">R_eal</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font> is   <a href="supinf_1.html#NM3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E16:15_2_1_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/>
<a NAME="E17:15_2_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E13:15_2_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="urysohn2.html#E16:15_2_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E18:15_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E4:15_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E9:15_2_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="urysohn2.html#E12:15_2_1_1_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:15_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn2.html#E6:15_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn2.html#E4:15_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="urysohn2.html#E11:15_2_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="urysohn2.html#E16:15_2_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="urysohn2.html#E17:15_2_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_2_1_1_1"/>
<span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:15_2_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E4:15_2_1_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:15_2_1_1_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<a NAME="E4:15_2_1_1_1_2_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1_1_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:15_2_1_1_1_2_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E6:15_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E4:15_2_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E5:15_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E6:15_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_2_1_1_1_2_1_1"/>
 <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</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="urysohn2.html#E1:15_2_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:15_2_1_1_1_2_1_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:15_2_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:15_2_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E5:15_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2_1_1_1_2_1_1_2"/>
 <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1_1_1_2_1_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  &amp; <font color="Maroon">c<sub>12</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="urysohn2.html#E3:15_2_1_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="urysohn2.html#E1:15_2_1_1_1_2_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:15_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:15_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="integra2.html#D2">INTEGRA2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:15_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:15_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="urysohn2.html#E1:15_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E9:15_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:15_2_1_1"/><b>then </b>
( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E4:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E6:15_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:15_2_1_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E6:15_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn2.html#E7:15_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="measure6.html#D4">MEASURE6:def 4</a>, <a class="ref" href="measure6.html#D5">MEASURE6:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1"/>
(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:15_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a> )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="urysohn2.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn2.html#E2:15_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/>
<a NAME="E5:15_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  holds <br/>(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="urysohn2.html#E4:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn2.html#E3:15_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="measure6.html#T49">MEASURE6:49</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K5">^^</a>  holds <br/>(  <a href="measure6.html#K4">^^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure6.html#K5">^^</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 ;<br/>


</div>
