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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:32"/><i><font color="Green">E21</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:32_1"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E1:32"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<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="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Olive">b<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E5:32_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span>
 <b>and </b><br/><a NAME="E2:32_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:32_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<a NAME="E4:32_1_1"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T19" target="_self">Th19</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a>, <a class="ref" href="tops_1.html#T47">TOPS_1:47</a></i>;<br/>
<b>hence </b><a NAME="E5:32_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_1"/><i><font color="Green">E25</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="32_1_2"><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="E1:32_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:32_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</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>
 ;<br/>

<b>assume </b><a NAME="E3:32_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:32_1_2_1"/><b>then </b>
(  <a href="finseq_1.html#K2">Seg</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"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:32_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="tdlat_2.html#E3:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><a NAME="E3:32_1_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/><a NAME="E4:32_1_2_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/> <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tdlat_2.html#E3:32_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
<br/>.= 
 <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
<br/>.= 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tdlat_2.html#E3:32_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tdlat_2.html#T20" target="_self">Th20</a></i>
;<br/><b>hence </b><a NAME="E7:32_1_2_1"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_2_2"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</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> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
;<br/><a NAME="E3:32_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/><a NAME="E5:32_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/><a NAME="E7:32_1_2_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E8:32_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E3:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:32_1_2_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:32_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:32_1_2_2"/>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</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="tdlat_2.html#E1:32_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:32_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E3:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:32_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:32_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:32_1_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_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#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E9:32_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E14:32_1_2_2"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="tdlat_2.html#T19" target="_self">Th19</a></i>;<br/><i><font color="Green">E35</font></i>:  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tdlat_2.html#E2:32_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tdlat_2.html#E13:32_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T46">TOPS_1:46</a></i>
;<br/><a NAME="E16:32_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E9:32_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/><a NAME="E17:32_1_2_2"/><b>then </b>
(  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_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#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tdlat_2.html#E9:32_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_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> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><i><font color="Green">E37</font></i>:  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="tdlat_2.html#E16:32_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tdlat_2.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
<br/>.= 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_2.html#E16:32_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
;<br/><a NAME="E20:32_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="tdlat_2.html#E3:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E21:32_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E22:32_1_2_2"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E1:32_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tdlat_2.html#E3:32_1_2"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E23:32_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E15:32_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tdlat_2.html#E19:32_1_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E24:32_1_2_2"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E14:32_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>;<br/><b>hence </b><a NAME="E25:32_1_2_2"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tdlat_2.html#E2:32_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tdlat_2.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_1_2"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:32_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_1"/><i><font color="Green">E26</font></i>: 
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="tdlat_2.html#E5:32_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="tdlat_2.html#E6:32_1"><i><font color="Green">E25</font></i></a>);<br/></i>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:32_1_3"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tdlat_2.html#E4:32_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><b>hence </b><a NAME="E3:32_1_3"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E7:32_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:32_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tdlat_2.html#E4:32_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E3:32_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E4:32_1_4"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tdlat_2.html#T19" target="_self">Th19</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a>, <a class="ref" href="tops_1.html#T47">TOPS_1:47</a></i>;<br/><b>hence </b><a NAME="E5:32_1_4"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:32_1_5"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>hence </b><a NAME="E3:32_1_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:32_1"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E8:32_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tdlat_2.html#E9:32_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
