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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:17_1_1"/>
 <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:17_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="prob_1.html#D6">PROB_1:def 6</a></i>;<br/>
<a NAME="E4:17_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E3:17_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:17_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:17"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:17_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> 0
 
;<br/>
<b>hence </b><a NAME="E3:17_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_3"/><i><font color="Green">E20</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:17_3"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_3_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:17_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="prob_3.html#E1:17_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E3:17_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:17_3"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:17_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:17_3"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:17_3"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:17_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:17_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="prob_3.html#E3:17_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="prob_3.html#E4:17_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prob_3.html#E2:17"><i><font color="Green">E18</font></i></a>, <a class="txt" href="prob_3.html#E3:17"><i><font color="Green">E19</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:17"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div>
