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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E3:1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E4:1"/><b>then </b><i><font color="Green">E4</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E5:1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E6:1"/><b>then </b><i><font color="Green">E5</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<a NAME="E7:1"/><b>then </b><i><font color="Green">E6</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:1"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="integra4.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="integra4.html#E7:1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E10:1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="integra4.html#E8:1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E11:1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="integra1.html#T8">INTEGRA1:8</a></i>;<br/>
<a NAME="E12:1"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E13:1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra4.html#E4:1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="integra4.html#E10:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>


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