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

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span>;<br/>
<a NAME="E1:45_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:45_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:45_1_1"/><i><font color="Green">E24</font></i>: 
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:45_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:45_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:45_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:45_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:45_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="group_7.html#E2:45_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:45_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_7.html#E1:45_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="group_7.html#E2:45_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E6:45_1_1"/>
<span class="p1"><a href="finseq_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K5">*&gt;</a></span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self">Def2</a></i>;<br/>


</div>
