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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span>;<br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="msualg_9.html#K1" target="_self">Mpr1</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:31"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</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>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:31_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:31_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:31_1"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:31"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E5:31_1"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:31_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:31_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_9.html#E7:31_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E9:31_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>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:31_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E10:31_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></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>9</sub></font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E11:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E8:31_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E12:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E9:31_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>
<a NAME="E13:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:31_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E16:31_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msualg_9.html#E3:31_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msualg_9.html#E4:31_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<a NAME="E17:31_1"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:31_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msualg_9.html#E5:31_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E19:31_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="msualg_9.html#E2:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="msualg_9.html#E15:31_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msualg_9.html#E16:31_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="msualg_9.html#E17:31_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E21:31_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E15:31_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msualg_9.html#E19:31_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:31"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_9.html#K1" target="_self">Mpr1</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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>6</sub></font>
 ;<br/>


</div>
