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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">l</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>











<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c6">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c6">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">F</font> =  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font>;<br/>
<a NAME="E2:42"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> &amp; <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#T7" target="_self" title="FINSEQ_7:th.7">Th7</a></i>;<br/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c6">l</font> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="finseq_7.html#E2:42"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#T32" target="_self" title="FINSEQ_7:th.32">Th32</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#T32" target="_self" title="FINSEQ_7:th.32">Th32</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_7.html#T23" target="_self" title="FINSEQ_7:th.23">Th23</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#T35" target="_self" title="FINSEQ_7:th.35">Th35</a></i>
<br/>.= 
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="finseq_7.html#E3:42"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="finseq_7.html#T35" target="_self" title="FINSEQ_7:th.35">Th35</a></i>
<br/>.= 
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c6">l</font></span>)</span>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:42"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c6">l</font></span>)</span>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="finseq_7.html#T23" target="_self" title="FINSEQ_7:th.23">Th23</a></i>
;<br/>
<b>hence </b><a NAME="E5:42"/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c6">l</font></span>)</span>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">j</font>
 ;<br/>


</div>
