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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</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/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:38_1_1"/><i><font color="Green">E18</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> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</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>7</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>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:38_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>hence </b><a NAME="E4:38_1_1_1"/>
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</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">E19</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</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> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</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="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="pralg_2.html#E1:38_1_1"><i><font color="Green">E18</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_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>8</sub></font> holds <br/><font color="Maroon">c<sub>8</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>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green">E20</font></i>: 
<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/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_2_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_2"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:38_1_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2_1_2"/><i><font color="Green">E21</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_2"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:38_1_1_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</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   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="pralg_2.html#M2" target="_self">MSAlgebra-Family</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E4:38_1_1_3"/>
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> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <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 class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <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="38_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/>
<a NAME="E3:38_1_1_3_1"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:38_1_1_3_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E22</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font>;<br/>
<a NAME="E6:38_1_1_3_1"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:38_1_1_3_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_3_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E9:38_1_1_3_1"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:38_1_1_3_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E26</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_3_1_3"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3_1"><b>suppose </b></a><a NAME="E1:38_1_1_3_1_3_1"/><i><font color="Green">E27</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E3:38_1_1_3_1_3_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/><a NAME="E4:38_1_1_3_1_3_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <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/><b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E5:38_1_1_3_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</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="E1:38_1_1_3_1_3_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_1_2"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pralg_2.html#T11" target="_self">Th11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>
<a NAME="E8:38_1_1_3_1_3_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</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="E11:38_1_1_3_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span>
 
;<br/>
<a NAME="E13:38_1_1_3_1_3_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:38_1_1_3_1_3_1_2"/>
<span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E15:38_1_1_3_1_3_1_2"/><b>then </b>
<span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</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"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E16:38_1_1_3_1_3_1_2"/><b>then </b>
<span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 
;<br/>
<a NAME="E17:38_1_1_3_1_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E13:38_1_1_3_1_3_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E18:38_1_1_3_1_3_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:38_1_1_3_1_3_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E7:38_1_1_3_1_3_1"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/><a NAME="E8:38_1_1_3_1_3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:38_1_1_3_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</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="E1:38_1_1_3_1_3_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_1_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_1_3"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <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="E7:38_1_1_3_1_3_1_3"/>
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> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_1_3_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_1_3_1"/><i><font color="Green">E36</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_3_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> = 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pralg_2.html#T11" target="_self">Th11</a></i>
;<br/>
<a NAME="E7:38_1_1_3_1_3_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_1_3_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_1_3_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pralg_2.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:38_1_1_3_1_3_1_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:38_1_1_3_1_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E2:38_1_1_3_1_3_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3_2"><b>suppose </b></a><a NAME="E1:38_1_1_3_1_3_2"/><i><font color="Green">E27</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E3:38_1_1_3_1_3_2"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E4:38_1_1_3_1_3_2"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_3_1_3_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
  <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="38_1_1_3_1_3_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</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="E1:38_1_1_3_1_3_2_2_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_1"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_1"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_1"/><b>then </b><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green">E35</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:38_1_1_3_1_3_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <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="E6:38_1_1_3_1_3_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_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> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</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="38_1_1_3_1_3_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green">E39</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_2_2_1_1_2"/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E13:38_1_1_3_1_3_2_2_1_1_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:38_1_1_3_1_3_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_2_2_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>21</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="E2:38_1_1_3_1_3_2_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:38_1_1_3_1_3_2_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:38_1_1_3_1_3_2_2_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then 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="E3:38_1_1_3_1_3_2_2_2"/><i><font color="Green">E33</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>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:38_1_1_3_1_3_2_2_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> &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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_2"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <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="E11:38_1_1_3_1_3_2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</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="E1:38_1_1_3_1_3_2_2_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green">E39</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_2_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_2_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span>
 
;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_2_2"/>
<span class="p1"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E12:38_1_1_3_1_3_2_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="pralg_2.html#K7" target="_self">.|</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:38_1_1_3_1_3_2_2_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:38_1_1_3_1_3_2_2_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2"/>
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> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_3"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_2_3"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_3"/><i><font color="Green">E38</font></i>: 
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_3"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_3"/><b>then </b><i><font color="Green">E39</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>27</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_3"/>
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> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:38_1_1_3_1_3_2_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_3_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green">E46</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_2_2_3_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>30</sub></font></span>)</span> <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/>
<b>hence </b><a NAME="E13:38_1_1_3_1_3_2_2_3_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_2_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_2_2_3"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E13:38_1_1_3_1_3_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E14:38_1_1_3_1_3_2_2"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<b>hence </b><a NAME="E15:38_1_1_3_1_3_2_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E12:38_1_1_3_1_3_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/><a NAME="E5:38_1_1_3_1_3_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:38_1_1_3_1_3_2_3"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then 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="E3:38_1_1_3_1_3_2_3"/><i><font color="Green">E31</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:38_1_1_3_1_3_2_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_3"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pralg_2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_3"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="pralg_2.html#K8" target="_self">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_3"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_3"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E13:38_1_1_3_1_3_2_3"/><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pralg_2.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E14:38_1_1_3_1_3_2_3"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <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="E15:38_1_1_3_1_3_2_3"/>
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> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_3_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_3_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_3_1"/><i><font color="Green">E38</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self">Def16</a></i>;<br/>
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E6:38_1_1_3_1_3_2_3_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" target="_self">Carrier</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pralg_2.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:38_1_1_3_1_3_2_3"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pralg_2.html#E13:38_1_1_3_1_3_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pralg_2.html#E14:38_1_1_3_1_3_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:38_1_1_3_1_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E2:38_1_1_3_1_3_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E6:38_1_1_3"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</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="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" target="_self">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:38_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K11" target="_self">SORTS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>


</div>
