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<b>let </b><font color="Maroon">A</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">D1</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>, <font color="Maroon">D2</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>



<b>consider </b><font color="Maroon">D</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:6"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span></span>)</span> &amp; <font color="Maroon">D</font> is <a href="goboard1.html#V1">increasing</a> )
 <b>by </b><i><a class="ref" href="goboard2.html#T21">GOBOARD2:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E4:6"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E7:6"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E8:6"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E9:6"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:6"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:6"/>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:6_1"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E3:6_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:6_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E5:6_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E6:6_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E7:6_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E8:6_1"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:6_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:6_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E12:6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6"/><b>then </b>
<font color="Maroon">D</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<a NAME="E14:6"/><i><font color="Green">E51</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E15:6"/>
<font color="Maroon">D1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T17">JORDAN7:17</a></i>;<br/>
<a NAME="E16:6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>hence </b><a NAME="E17:6"/>
<font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/>

<a NAME="E18:6"/><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E19:6"/>
<font color="Maroon">D2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T17">JORDAN7:17</a></i>;<br/>
<a NAME="E20:6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>hence </b><a NAME="E21:6"/>
<font color="Maroon">D2</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/>

<b>thus </b><a NAME="E22:6"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>


</div>
