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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:9"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">fs1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:9"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">fs1</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">fs1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">k</font>, <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) &amp; ( for <font color="Olive">i1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i1</font> &amp; <font color="Olive">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> &amp; not <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:9"/>
for <font color="Olive">i1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i1</font> &amp; <font color="Olive">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> holds <br/><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i1</font>,<font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R1" target="_self">are_adjacent1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E30</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> )
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</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">fs1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E4:9_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E5:9_1"/>
( <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:9_1"/>
<font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>,<font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R1" target="_self">are_adjacent1</a> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
 ex <font color="Olive">fs1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">fs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Olive">fs1</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">fs1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">fs1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">k</font>, <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">fs1</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) &amp; ( for <font color="Olive">i1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i1</font> &amp; <font color="Olive">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">fs1</font> holds <br/><font color="Olive">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i1</font>,<font color="Olive">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R1" target="_self">are_adjacent1</a>  ) )
 <b>by </b><i/>;<br/>


</div>
