<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c3">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c1">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="c4">f</font>
 <b>and </b><br/><a NAME="E3:69"/><i><font color="Green" title="E64">A3</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1"/>
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font> )
 </span><b>by </b><i><a class="txt" href="finseq_6.html#E3:69"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>suppose </b></a><a NAME="E1:69_1_1"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 ;<br/><div class="add"><a NAME="E2:69_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D2" title="FINSEQ_5:def.2">FINSEQ_5:def 2</a></i>;<br/><a NAME="E3:69_1_1"/><b>then </b><i><font color="Green" title="E66">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> ;<br/><a NAME="E5:69_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:69"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E6:69_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:69"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:69_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E2:69"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:69_1_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:69_1_1"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="finseq_6.html#T63" target="_self" title="FINSEQ_6:th.63">Th63</a></i>;<br/><b>hence </b><a NAME="E9:69_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:69_1_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2"><b>suppose </b></a><a NAME="E1:69_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 ;<br/><div class="add"><a NAME="E2:69_1_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:69"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="finseq_5.html#T41" title="FINSEQ_5:th.41">FINSEQ_5:41</a></i>;<br/><b>hence </b><a NAME="E3:69_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T66" target="_self" title="FINSEQ_6:th.66">Th66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
