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<div class="add">

<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</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>assume </b><a NAME="E1:66"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">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">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>, <font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font>  holds <br/>(  not <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) ) )
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:66_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:66_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:66_1_1"/>
<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="fintopo6.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> <b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E43</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">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:66_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:66_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E5:66_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:66_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E60</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">E71</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:66_1_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</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="E3:66_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E4:66_1_3"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:66_1_3"/><b>then </b>
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E6:66_1_3"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E6:66_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:66_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E8:66_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <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> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:66_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">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">G</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_1_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="fin_topo.html#K9">^b</a>  &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:66_1_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="fin_topo.html#K9">^b</a>  &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E6:66_1_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:66_1_4"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, </i>;<br/><b>consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E10:66_1_4"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> ) )
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E12:66_1_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> ;<br/><i><font color="Green">E100</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E15:66_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E16:66_1_4"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">g</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><a NAME="E17:66_1_4"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:66_1_4"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:66_1_4"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, </i>;<br/><i><font color="Green">E105</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</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><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><a NAME="E21:66_1_4"/>
<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="fintopo6.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E22:66_1_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E23:66_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, , , <a class="ref" href="fintopo6.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:66_1"/>
not <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i>, , <a class="txt" href="fintopo6.html#E6:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
( <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a> implies <font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">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">A</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:66_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/><b>consider </b><font color="Maroon">p0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P</font>;<br/><a NAME="E5:66_2"/>
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p0</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E7:66_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">p1</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>consider </b><font color="Maroon">q0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>;<br/><a NAME="E9:66_2"/>
<font color="Maroon">q0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q0</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E11:66_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">q1</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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_2_1"/>
<font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a>
 ;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:66_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">f</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:66_2_1"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, </i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> );<br/><a NAME="E5:66_2_1"/><i><font color="Green">E111</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i>, , </i>;<br/><a NAME="E6:66_2_1"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E7:66_2_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><b>then consider </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:66_2_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i0</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">u0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E11:66_2_1"/>
<font color="Maroon">i0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E12:66_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">i0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E13:66_2_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i0</font> <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">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:66_2_1"/><i><font color="Green">E116</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E117</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">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:66_2_1_1"/><b>then </b>
<font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</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#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E16:66_2_1"/>
<font color="Maroon">i0</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">f</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:66_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:66_2_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">z0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E20:66_2_1"/>
<font color="Maroon">z0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">u0</font>
 <b>by </b><i>, , , <a class="ref" href="fintopo6.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E21:66_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z0</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E22:66_2_1"/><b>then </b>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z0</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E23:66_2_1"/><b>then </b>
<font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> 
 ;<br/><b>hence </b><a NAME="E24:66_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><b>hence </b><a NAME="E13:66_2"/>
not <font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66"/>
( <font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a> implies <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a> )
 ;<br/>


</div>
