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

<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> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/>
<a NAME="E1:3_1_1"/><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E2:3_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E3:3_1_1"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:3_1_1"/><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:3_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>] )
 ;<br/>

<a NAME="E2:3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:3_1_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E5:3_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:3_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pralg_1.html#E4:3_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E6:3_1_1_1"/><i><font color="Green">E13</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="pralg_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <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>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <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="E9:3_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pralg_1.html#E4:3_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E6:3_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E10:3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:3_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E11:3_1_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:3_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pralg_1.html#E4:3_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E6:3_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E13:3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:3_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E14:3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E15:3_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E3:3_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pralg_1.html#E4:3_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E5:3_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E11:3_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:3_1_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:3_1_1_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 ;<br/>

<a NAME="E2:3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:3_1_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E5:3_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:3_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E4:3_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E8:3_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</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>] ) )
 <b>and </b><br/><a NAME="E9:3_1_1"/><i><font color="Green">E12</font></i>: 
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="partfun1.html#S2">PARTFUN1:sch 2</a>(<a class="txt" href="pralg_1.html#E5:3_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pralg_1.html#E6:3_1_1"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E10:3_1_1"/><i><font color="Green">E13</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:3_1_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:3_1_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E8:3_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></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="E2:3_1_1_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:3_1_1_3"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>assume </b><a NAME="E1:3_1_1_3_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>hence </b><a NAME="E2:3_1_1_3_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E5:3_1_1_3"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:3_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E8:3_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E2:3_1_1_3"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:3_1_1"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
;<br/>
<a NAME="E12:3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></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">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">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="E13:3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="partfun1.html#T30">PARTFUN1:30</a></i>;<br/>
<a NAME="E15:3_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="unialg_1.html#V1">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>; <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="E1:3_1_1_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:3_1_1_4"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E10:3_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:3_1_1_4"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:3_1_1_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pralg_1.html#E2:3_1_1_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:3_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E4:3_1_1_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="pralg_1.html#E6:3_1_1_4"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="unialg_1.html#V2">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>; <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="E1:3_1_1_5"/><i><font color="Green">E15</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:3_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pralg_1.html#E10:3_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:3_1_1_5"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></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="E6:3_1_1_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:3_1_1_5"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pralg_1.html#E4:3_1_1_5"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_1_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E10:3_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></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"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="pralg_1.html#E10:3_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pralg_1.html#E15:3_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E18:3_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E10:3_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E19:3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>hence </b><a NAME="E20:3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E9:3_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


</div>
