<?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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> 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/>for <font color="Olive">b<sub>2</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>;<br/>
<a NAME="E1:48_1_1"/><i><font color="Green">E27</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="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:48_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pralg_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<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>assume </b><a NAME="E8:48_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E9:48_1_1_1"/>
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>hence </b><a NAME="E10:48_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E8:48_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<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="E3:48_1_1"/><i><font color="Green">E28</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</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#E1:48_1_1"><i><font color="Green">E27</font></i></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#E3:48_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:48_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="48_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="E1:48_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/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E3:48_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pralg_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:48_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:48_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:48_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:48_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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:48_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/>
<a NAME="E7:48_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> ,<span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><b>proof </b></a><div class="add">

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

<a NAME="E1:48_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pralg_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E2:48_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:48_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E4:48_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:48_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="pralg_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:48_1_1_3"/><b>then </b>
( <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> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:48_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"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> ,<span class="p1">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<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  <a href="pralg_1.html#K12" target="_self">Carrier</a> <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/>
<a NAME="E9:48_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_1_1_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:48_1_1_4"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="unialg_1.html#K3">signature</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">signature</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E3:48_1_1_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pralg_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:48_1_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:48_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:48_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:48_1_1_4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E9:48_1_1_4"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E10:48_1_1_4"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</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>10</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E11:48_1_1_4"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E8:48_1_1_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E12:48_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:48_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pralg_1.html#E4:48_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E13:48_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>11</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:48_1_1_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E16:48_1_1_4"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:48_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pralg_1.html#E3:48_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E17:48_1_1_4"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<b>hence </b><a NAME="E18:48_1_1_4"/>
 <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E2:48_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pralg_1.html#E9:48_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pralg_1.html#E11:48_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pralg_1.html#E15:48_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_1.html#E16:48_1_1_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="pralg_1.html#V4" target="_self">equal-arity</a>  <a href="pralg_1.html#M1" target="_self">ManySortedOperation</a> of  <a href="pralg_1.html#K12" target="_self">Carrier</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pralg_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E11:48_1_1"/>
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>hence </b><a NAME="E12:48_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E3:48_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
