<?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 <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:38_1_1"/><i><font color="Green">E21</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> <font color="Maroon">c<sub>1</sub></font> holds <br/>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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:38_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <font color="Maroon">c<sub>1</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>3</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="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="pralg_1.html#E1:38_1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="relset_1.html#M2">M5</a>(<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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</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> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>) = <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:38_1_1"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E5:38_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:38_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:38_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E5:38_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E3:38_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E2:38_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E5:38_1_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is  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"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="unialg_1.html#T2">UNIALG_1:2</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="pralg_1.html#M1" target="_self">ManySortedOperation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pralg_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E11:38_1_1"/>
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</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/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Olive">b<sub>7</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>6</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Olive">b<sub>8</sub></font> holds <br/>for <font color="Olive">b<sub>9</sub></font> being 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 <font color="Olive">b<sub>7</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Olive">b<sub>7</sub></font><br/>for <font color="Olive">b<sub>10</sub></font> being 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 <font color="Olive">b<sub>8</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Olive">b<sub>8</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>10</sub></font> holds <br/> <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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="E1:38_1_1_4"/><i><font color="Green">E24</font></i>: 
( <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> &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>7</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E5:38_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E3:38_1_1_4"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:38_1_1_4"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be  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 <font color="Maroon">c<sub>16</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>16</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be  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 <font color="Maroon">c<sub>17</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>17</sub></font>;<br/>



<b>assume </b><a NAME="E5:38_1_1_4"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/>

<a NAME="E6:38_1_1_4"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E5:38_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pralg_1.html#E1:38_1_1_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_1.html#E3:38_1_1_4"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E3:38_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E3:38_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>20</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">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="unialg_1.html#T2">UNIALG_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>21</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">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="ref" href="unialg_1.html#T2">UNIALG_1:2</a></i>;<br/>
<a NAME="E11:38_1_1_4"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:38_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E14:38_1_1_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>16</sub></font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="txt" href="pralg_1.html#E13:38_1_1_4"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E16:38_1_1_4"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E6:38_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pralg_1.html#E11:38_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pralg_1.html#E13:38_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E17:38_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E18:38_1_1_4"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="pralg_1.html#E13:38_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:38_1_1_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E21:38_1_1_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>17</sub></font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="txt" href="pralg_1.html#E20:38_1_1_4"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E23:38_1_1_4"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E6:38_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pralg_1.html#E11:38_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pralg_1.html#E20:38_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E24:38_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E25:38_1_1_4"/>
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E18:38_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pralg_1.html#E20:38_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:38_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pralg_1.html#V4" target="_self">equal-arity</a>
 <b>by </b><i><a class="ref" href="pralg_1.html#D16" target="_self">Def16</a></i>;<br/>


</div>
