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<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">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font>;<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>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="E1:71"/><i><font color="Green">E27</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> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">g</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">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="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> );<br/>
<a NAME="E2:71"/><i><font color="Green">E31</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><a class="txt" href="fintopo6.html#E7:71"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E3:71"/>
 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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>then consider </b><font color="Maroon">k0</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="E5:71"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k0</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="Maroon">k0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">g0</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E7:71"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g0</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g0</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">g0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g0</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:71"/>
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">g0</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>, </i>;<br/>
<b>hence </b><a NAME="E9:71"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> st <br/>( <font color="Olive">g</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">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="Olive">g</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="Olive">g</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/>;<br/>


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