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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<a NAME="E1:10_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E1:10_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:10_1_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 <b>and </b><br/><a NAME="E4:10_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_1_1_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:10_1_1_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="pcomps_2.html#E3:10_1_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:10_1_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E7:10_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="pcomps_2.html#E4:10_1_1_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>thus </b><a NAME="E8:10_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="pcomps_2.html#E3:10_1_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pcomps_2.html#E6:10_1_1_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:10_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/></div><b>end;</b></div>
<a NAME="E2:10_1_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E5:10_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E7:10_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">F<sub>2</sub></font>()</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:10_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:10_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="pcomps_2.html#E1:10_1"><i><font color="Green">E4</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E7:10_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E8:10_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">F<sub>2</sub></font>()</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">F<sub>2</sub></font>()</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="pcomps_2.html#E4:10_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="pcomps_2.html#E7:10_1"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
;<br/><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E10:10_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E4:10_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:10_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_3"/><i><font color="Green">E10</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>
;<br/>
<b>hence </b><a NAME="E3:10_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="pcomps_2.html#E1:10_1_3"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:10_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="pcomps_2.html#E10:10_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pcomps_2.html#E11:10_1"><i><font color="Green">E9</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E13:10_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:10_1_4"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E7:10_1"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E4:10_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="pcomps_2.html#E2:10_1_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:10_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_5"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:10_1_5"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_5_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="pcomps_2.html#E2:10_1_5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_5_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E7:10_1"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pcomps_2.html#E1:10_1_5_1_1_1"><i><font color="Green">E15</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_5_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_5_1_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:10_1_5_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E1:10_1_5_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:10_1_5_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:10_1_5_1_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E2:10_1_5_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:10_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:10_1_5_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E8:10_1_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1_5_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E9:10_1_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <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">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E10:10_1_5_1_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:10_1_5_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <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">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pcomps_2.html#E5:10_1_5_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E12:10_1_5_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E10:10_1_5_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1_5_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1_5_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pcomps_2.html#E1:10_1_5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pcomps_2.html#E3:10_1_5_1_1_2"><i><font color="Green">E17</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:10_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="pcomps_2.html#E13:10_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pcomps_2.html#E14:10_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E16:10_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:10_1_6"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>);<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>];<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>];<br/>
<a NAME="E2:10_1_6"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E1:10_1_6_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) implies for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_6_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:10_1_6_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E3:10_1_6_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E15:10_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_6_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="pcomps_2.html#E1:10_1_6_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="pcomps_2.html#E2:10_1_6_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:10_1_6_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E3:10_1_6_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E4:10_1_6_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E15:10_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_6_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="pcomps_2.html#E2:10_1_6_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="pcomps_2.html#E3:10_1_6_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_1_6"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="pcomps_2.html#E2:10_1_6"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:10_1_6"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:10_1"/><i><font color="Green">E15</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S6">FRAENKEL:sch 6</a>(<a class="txt" href="pcomps_2.html#E16:10_1"><i><font color="Green">E14</font></i></a>);<br/></i>
<b>then </b><i><font color="Green">E16</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>
;<br/>
<a NAME="E19:10_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E2:10_1_8"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:10_1_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pcomps_2.html#E12:10_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_2.html#E2:10_1_8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <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">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E7:10_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pcomps_2.html#E18:10_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <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">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E19:10_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <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">c<sub>2</sub></font> <a href="pcomps_2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pcomps_2.html#E5:10_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pcomps_2.html#E17:10_1"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/>


</div>
