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

<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/>
not <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V3" title="FINTOPO6:attr.3">arcwise_connected</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font>  holds <br/>(  not <font color="Olive" title="b1">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> or  not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> or  not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> ) ) )
 <b>by </b><i><a class="ref" href="fintopo6.html#D7" target="_self" title="FINTOPO6:def.7">Def7</a></i>;<br/><a NAME="E4:66_1"/><i><font color="Green" title="E39">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b2">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b2">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E3:66_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:66_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b2">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green" title="E40">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:66_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:66_1_2"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:66_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E5:66_1_2"/><i><font color="Green" title="E41">A6</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> ) )
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E6:66_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="fintopo6.html#E5:66_1_2"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E41">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_3"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:66_1_3"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E3:66_1_3"/><b>then </b><i><font color="Green" title="E43">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E4:66_1_3"/><i><font color="Green" title="E44">A10</font></i>: 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:66_1_3"/><b>then </b>
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E6:66_1_3"/><b>then </b>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="fintopo6.html#E3:66_1_3"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="fintopo6.html#E4:66_1_3"><i><font color="Green" title="E44">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E7:66_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:66_1_3"><i><font color="Green" title="E42">A8</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:66_1"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:66_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/><a NAME="E9:66_1"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_4"/>
<font color="Maroon" title="c5">G</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:66_1_4"/><i><font color="Green" title="E44">A13</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:66_1_4"/><i><font color="Green" title="E45">A14</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1_4"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E6:66_1_4"/><i><font color="Green" title="E46">A15</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">G</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_1_4"><i><font color="Green" title="E44">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:66_1_4"/><i><font color="Green" title="E47">A16</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:66_1_4"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E10:66_1_4"/><i><font color="Green" title="E48">A17</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> ) )
 <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_1_4"><i><font color="Green" title="E47">A16</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E12:66_1_4"/><i><font color="Green" title="E49">A18</font></i>: 
( <font color="Maroon" title="c10">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E10:66_1_4"><i><font color="Green" title="E48">A17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font> ;<br/><i><font color="Green" title="E50">A19</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><a NAME="E15:66_1_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">z2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:66"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="fintopo6.html#E8:66_1_4"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="fintopo6.html#E10:66_1_4"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E16:66_1_4"/><b>then </b><i><font color="Green" title="E51">A20</font></i>: 
( <font color="Maroon" title="c11">g</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp; <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:66_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="fintopo6.html#T44" target="_self" title="FINTOPO6:th.44">Th44</a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><a NAME="E17:66_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:66_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/><a NAME="E18:66_1_4"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:66_1_4"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:66_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><i><font color="Green" title="E53">A22</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
;<br/><a NAME="E21:66_1_4"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:66_1_4"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="fintopo6.html#E6:66_1_4"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E22:66_1_4"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:66_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="fintopo6.html#E20:66_1_4"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E23:66_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:66_1_4"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="fintopo6.html#E6:66_1_4"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="fintopo6.html#E14:66_1_4"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="fintopo6.html#E16:66_1_4"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="fintopo6.html#E19:66_1_4"><i><font color="Green" title="E52">A21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:66_1"/>
not <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:66_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="fintopo6.html#E7:66_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="fintopo6.html#E8:66_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="fintopo6.html#E9:66_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
( <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> implies <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V3" title="FINTOPO6:attr.3">arcwise_connected</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E3:66_2"/><i><font color="Green" title="E38">A23</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c3">P</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">p0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P</font>;<br/><a NAME="E5:66_2"/>
<font color="Maroon" title="c5">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">p0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/><a NAME="E7:66_2"/><i><font color="Green" title="E39">A24</font></i>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">q0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">Q</font>;<br/><a NAME="E9:66_2"/>
<font color="Maroon" title="c7">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c7">q0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/><a NAME="E11:66_2"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_2_1"/>
<font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V3" title="FINTOPO6:attr.3">arcwise_connected</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E3:66_2_1"/><i><font color="Green" title="E41">A26</font></i>: 
( <font color="Maroon" title="c9">f</font> is <a href="fintopo6.html#V2" title="FINTOPO6:attr.2">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E7:66_2"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="fintopo6.html#E11:66_2"><i><font color="Green" title="E40">A25</font></i></a>, <a class="ref" href="fintopo6.html#D7" target="_self" title="FINTOPO6:def.7">Def7</a></i>;<br/><a NAME="E4:66_2_1"/><i><font color="Green" title="E42">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> );<br/><a NAME="E5:66_2_1"/><i><font color="Green" title="E43">A28</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fintopo6.html#E3:66_2_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="fintopo6.html#E4:66_2_1"><i><font color="Green" title="E42">A27</font></i></a></i>;<br/><a NAME="E6:66_2_1"/><i><font color="Green" title="E44">A29</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font>
 ;<br/><b>consider </b><font color="Maroon" title="c10">i0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:66_2_1"/><i><font color="Green" title="E45">A30</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">i0</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i0</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="fintopo6.html#E6:66_2_1"><i><font color="Green" title="E44">A29</font></i></a>, <a class="txt" href="fintopo6.html#E5:66_2_1"><i><font color="Green" title="E43">A28</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c11">u0</font> = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a></i>;<br/><a NAME="E10:66_2_1"/>
<font color="Maroon" title="c10">i0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fintopo6.html#E3:66_2_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E11:66_2_1"/><b>then </b><i><font color="Green" title="E46">A31</font></i>: 
<font color="Maroon" title="c10">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E12:66_2_1"/><b>then </b><i><font color="Green" title="E47">A32</font></i>: 
<font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:66_2_1"/><i><font color="Green" title="E48">A33</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><i><font color="Green" title="E49">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/><a NAME="E2:66_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i0</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="txt" href="fintopo6.html#E12:66_2_1"><i><font color="Green" title="E47">A32</font></i></a>, <a class="txt" href="fintopo6.html#E13:66_2_1"><i><font color="Green" title="E48">A33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E15:66_2_1"/>
<font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E12:66_2_1"><i><font color="Green" title="E47">A32</font></i></a>, <a class="txt" href="fintopo6.html#E13:66_2_1"><i><font color="Green" title="E48">A33</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:66_2_1"/><b>then </b><i><font color="Green" title="E50">A35</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:66_2_1"/><b>then </b><i><font color="Green" title="E51">A36</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fintopo6.html#E3:66_2_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="fintopo6.html#E14:66_2_1"><i><font color="Green" title="E49">A34</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">z0</font> = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo6.html#E16:66_2_1"><i><font color="Green" title="E50">A35</font></i></a></i>;<br/><a NAME="E19:66_2_1"/>
<font color="Maroon" title="c12">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">u0</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="txt" href="fintopo6.html#E11:66_2_1"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="fintopo6.html#D6" target="_self" title="FINTOPO6:def.6">Def6</a></i>;<br/><a NAME="E20:66_2_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c12">z0</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:66"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E21:66_2_1"/><b>then </b>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c12">z0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E8:66_2_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E22:66_2_1"/><b>then </b>
<font color="Maroon" title="c12">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 ;<br/><b>hence </b><a NAME="E23:66_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:66_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fintopo6.html#E17:66_2_1"><i><font color="Green" title="E51">A36</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><b>hence </b><a NAME="E13:66_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V3" title="FINTOPO6:attr.3">arcwise_connected</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66"/>
( <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V3" title="FINTOPO6:attr.3">arcwise_connected</a> implies <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> )
 ;<br/>


</div>
