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

<b>set </b><font color="Maroon">pr</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">ca</font> =  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font>;<br/>
<b>set </b><font color="Maroon">ct</font> = <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>   st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( (  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span> ) &amp; (  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">w</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Olive">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K18" target="_self">~</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Olive">p</font></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Olive">w</font></span>)</span> ) );<br/>
<b>set </b><font color="Maroon">aa</font> = <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> </span>}</span> ;<br/>
<a NAME="E1:47_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:47_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_1_1_1"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:47_1_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span>;<br/><i><font color="Green">E39</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E3:47_1_1_1_1_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_1.html#E1:47_1_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:47_1_1_1_1_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E5:47_1_1_1_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E6:47_1_1_1_1_1_1_2"/><b>then </b>
 <a href="unialg_1.html#K3">arity</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> = 
0 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/>
<a NAME="E8:47_1_1_1_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:47_1_1_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E10:47_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:47_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/><b>assume </b><a NAME="E5:47_1_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E6:47_1_1_1_1_1_1"/>
( (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span> ) &amp; (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">w</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Olive">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K18" target="_self">~</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Olive">w</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:47_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Z</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">p</font> =  <a href="pralg_1.html#K18" target="_self">~</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">w</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span>;<br/><i><font color="Green">E42</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E5:47_1_1_1_1_1_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1_1_1_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:47_1_1_1_1_1_2_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:47_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:47_1_1_1_1_1_2_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:47_1_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:47_1_1_1_1_1_2_2"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="funct_5.html#D3">FUNCT_5:def 3</a></i>;<br/>
<i><font color="Green">E57</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> = 
 <a href="funct_5.html#K2">proj2</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Z</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
 <a href="funct_5.html#K2">proj2</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">j</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E10:47_1_1_1_1_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1_1_1_1_1_2_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K18" target="_self">~</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:47_1_1_1_1_1_2_2_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:47_1_1_1_1_1_2_2_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#D1" target="_self">Def1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:47_1_1_1_1_1_2_2_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">w</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>;<br/>
<a NAME="E11:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:47_1_1_1_1_1_2_2_2"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E13:47_1_1_1_1_1_2_2_2"/>
<font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:47_1_1_1_1_1_2_2_2"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) )
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E15:47_1_1_1_1_1_2_2_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E16:47_1_1_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:47_1_1_1_1_1_2_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a>, <a class="txt" href="pralg_1.html#E4:47_1_1_1_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></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/>
<a NAME="E13:47_1_1_1_1_1_2_2"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:47_1_1_1_1_1_2_2"/>
 <a href="unialg_1.html#K3">arity</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> </span>}</span> 

;<br/>
<a NAME="E16:47_1_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:47_1_1_1_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E18:47_1_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:47_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/><b>assume </b><a NAME="E7:47_1_1_1_1_1_2"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E8:47_1_1_1_1_1_2"/>
( (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span> ) &amp; (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">w</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Olive">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K18" target="_self">~</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">l</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Olive">w</font></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:47_1_1"/><i><font color="Green">E70</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E4:47_1_1"/>
 <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E6:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E7:47_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T128">FINSEQ_2:128</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E9:47_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> is <a href="unialg_1.html#V1">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>, <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D1">UNIALG_1:def 1</a><br/>

<b>assume </b><a NAME="E1:47_1_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></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/>
<a NAME="E3:47_1_1_2"/>
(  ex <font color="Olive">w</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> ) &amp;  ex <font color="Olive">w</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:47_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:47_1_1"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>, <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a><br/>

<b>assume </b><a NAME="E1:47_1_1_3"/><i><font color="Green">E73</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></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/>
<a NAME="E3:47_1_1_3"/>
 ex <font color="Olive">w</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:47_1_1_3"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:47_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E12:47_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K20" target="_self">ComAr</a> <font color="Maroon">O</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>

<b>assume </b><a NAME="E13:47_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>hence </b><a NAME="E14:47_1_1"/>
( (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K21" target="_self">EmptySeq</a> <font color="Maroon">B</font></span>)</span> ) &amp; (  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">w</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Olive">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="pralg_1.html#K18" target="_self">~</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K17" target="_self">uncurry</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="pralg_1.html#K16" target="_self">..</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K19" target="_self">curry</a> <font color="Olive">w</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
