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

<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">E</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">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">d'</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="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D'</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>













<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>] <b>means </b><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">d'</font></span>)</span>;<br/>
<font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<font color="Maroon">d'</font> = 
<font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">E</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">d'</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T99">FINSEQ_2:99</a></i>
;<br/>
<a NAME="E2:24"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>]
 
;<br/>
<a NAME="E3:24"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><br/> for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">q</font>,<font color="Maroon">d'</font></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>,<font color="Maroon">d'</font> = 
<font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">d'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">q</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">d'</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/>
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:24"/>
<font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">p</font>,<font color="Maroon">d'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K2" target="_self">[:]</a> <font color="Maroon">q</font>,<font color="Maroon">d'</font></span>)</span>
 ;<br/>


</div>
