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

<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fin_topo.html#V1" target="_self">filled</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>



<b>thus </b><a NAME="E1:34"/>
( <font color="Maroon">A</font> is <a href="fin_topo.html#V5" target="_self">connected</a> implies for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">S</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">S</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S</font> holds <br/><font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</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"><font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> ) &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="fin_topo.html#V5" target="_self">connected</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E2:34_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font> = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<a NAME="E3:34_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E5:34_1"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:34_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/>( <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:34_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:34_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>

<b>thus </b><a NAME="E4:34_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">S</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E8:34_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:34_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E10:34_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>)
 <b>and </b><br/><a NAME="E11:34_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E12:34_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> holds <br/>( <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> )
 <b>from </b><i><a class="ref" href="fin_topo.html#S1" target="_self">FIN_TOPO:sch 1</a>(, , <a class="ref" href="fin_topo.html#T1" target="_self">Th1</a>);<br/></i>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E13:34_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E14:34_1"/>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E15:34_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</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"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>

<b>set </b><font color="Maroon">B</font> = <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>
<a NAME="E16:34_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="txt" href="fin_topo.html#E2:34_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>assume </b><a NAME="E18:34_1"/>
not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<a NAME="E19:34_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <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="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> implies <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E20:34_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><b>thus </b><a NAME="E2:34_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:34_1_3_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:34_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:34_1_3_1"/><b>then </b>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fin_topo.html#E1:34_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fin_topo.html#E8:34_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:34_1_3_1"/><b>then </b>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1_3_1"/>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="fin_topo.html#E6:34_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:34_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fin_topo.html#E5:34_1"><i><font color="Green">E30</font></i></a>, );<br/></i>
<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E24:34_1"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E26:34_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:34_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="fin_topo.html#E6:34_1"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E28:34_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E29:34_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E30:34_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="fin_topo.html#E10:34_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/>
<a NAME="E32:34_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E33:34_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="fin_topo.html#E4:34_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="fin_topo.html#E8:34_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fin_topo.html#E9:34_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>


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

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">S</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">S</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S</font> holds <br/><font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</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"><font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> ) &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S</font></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font><b> such that </b><a NAME="E3:34"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E5:34"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:34"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:34"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D18" target="_self">FIN_TOPO:def 18</a><br/>

<a NAME="E8:34"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:34"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E10:34"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green">E72</font></i>: <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>
<a NAME="E13:34"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">S</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E15:34"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E16:34"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E17:34"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</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"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E18:34"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E19:34"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> implies <font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> );<br/>
<a NAME="E20:34"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><b>thus </b><a NAME="E2:34_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_3_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:34_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> 
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:34_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9" target="_self">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:34_3_1"/>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:34"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E24:34"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E26:34"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:34"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:34"/><b>then </b>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E29:34"/><b>then </b>
<font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E30:34"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E31:34"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E32:34"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div>
