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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>, <font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>



<a NAME="E1:20"/><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E18</font></i>:  <a href="finseq_8.html#K2" target="_self">smid</a> <font color="Maroon">g</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<b>then </b> <a href="finseq_8.html#K2" target="_self">smid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="finseq_8.html#K2" target="_self">smid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i/>
;<br/>
<a NAME="E5:20"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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_8.html#K3" target="_self">ovlpart</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<i><font color="Green">E33</font></i>:  <a href="finseq_8.html#K3" target="_self">ovlpart</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font> = 
 <a href="finseq_8.html#K2" target="_self">smid</a> <font color="Maroon">g</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/>
<a NAME="E8:20"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1

;<br/>
<b>then </b><i><font color="Green">E38</font></i>:  <a href="finseq_8.html#K2" target="_self">smid</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/>
<a NAME="E11:20"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E12:20"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
 <a href="finseq_8.html#K2" target="_self">smid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T26">FINSEQ_5:26</a></i>
;<br/>
<a NAME="E14:20"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:20"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
 <a href="finseq_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span> = 
 <a href="finseq_8.html#K2" target="_self">smid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_8.html#E1:20"><i><font color="Green">E17</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T26">FINSEQ_5:26</a></i>
;<br/>
<b>hence </b><a NAME="E17:20"/>
(  <a href="finseq_8.html#K3" target="_self">ovlpart</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="finseq_8.html#K3" target="_self">ovlpart</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_8.html#K1" target="_self">^</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div>
