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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>thus </b><a NAME="E1:72"/>
( <font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a> implies for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <font color="Olive">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a>
 ;<br/>

<b>thus </b><a NAME="E2:72_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <font color="Olive">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:72_1_1"/>
( <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> )
 ;<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="E3:72_1_1"/><i><font color="Green">E31</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">x2</font> )
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, <a class="txt" href="fintopo6.html#E2:72"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E5:72_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">g2</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g2</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">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font>  st <font color="Olive">h</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">h</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">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">h</font> ) )
 <b>by </b><i><a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E6:72_1_1"/>
<font color="Maroon">g2</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E7:72_1_1"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <font color="Olive">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 ;<br/>


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


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

<b>assume </b><a NAME="E2:72"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <font color="Olive">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font>
 ;<br/>

<a NAME="E3:72"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> 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="Olive">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">x2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:72_2"/>
( <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> )
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:72_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:72"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E4:72_2"/>
( <font color="Maroon">g</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <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> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</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">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E5:72_2"/>
 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">x2</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:72"/>
<font color="Maroon">A</font> is <a href="fintopo6.html#V3" target="_self">arcwise_connected</a>
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a></i>;<br/>


</div>
