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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">p</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<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>assume </b><a NAME="E1:45"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:45"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D7">FINSEQ_4:def 7</a></i>;<br/>
<a NAME="E3:45"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <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><a class="ref" href="finseq_5.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <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_4.html#K7">|--</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <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="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:45_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_4.html#D7">FINSEQ_4:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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