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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>











<b>assume </b><a NAME="E1:42"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>;<br/>
<a NAME="E2:42"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:42"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
 <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">l</font> = 
 <a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">l</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" target="_self">Replace</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">i</font>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:42"/>
 <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_7.html#K2" target="_self">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" target="_self">Swap</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>


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