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

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="funct_5.html#K4">uncurry</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E1:32_1_1"/><i><font color="Green">E14</font></i>: 
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/>
<a NAME="E2:32_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <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>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E16</font></i>: 
<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/><div class="add"><a NAME="E2:32_1_1_1_1"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p4">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T13" target="_self">Th13</a></i>;<br/><b>then 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="E4:32_1_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <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;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E5:32_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</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"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pralg_2.html#E4:32_1_1_1_1"><i><font color="Green">E17</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>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:32_1_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</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>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:32_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E7:32_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:32_1_1_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>9</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_1.html#NM1">Function</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_2.html#E7:32_1_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E3:32_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pralg_2.html#E2:32_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pralg_2.html#E1:32_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:32_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E7:32_1_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E5:32_1_1_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:32_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_5.html#T29">FUNCT_5:29</a></i>;<br/>
<a NAME="E6:32_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:32_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>12</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>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pralg_2.html#D18" target="_self">Def18</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:32_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pralg_2.html#E5:32_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pralg_2.html#E6:32_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pralg_2.html#E8:32_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:32_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E7:32_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2"/>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</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="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:32_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:32_1_1_1_2"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/><font color="Maroon">c<sub>7</sub></font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E4:32_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a>, <a class="ref" href="funct_6.html#T88">FUNCT_6:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:32_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>
;<br/><b>hence </b><a NAME="E6:32_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12" target="_self">OPER</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
