<?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>, <font color="Maroon">c<sub>3</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>4</sub></font>, <font color="Maroon">c<sub>5</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>6</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="pboole.html#K10">|]</a></span>;<br/>











<b>assume </b><a NAME="E1:33"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K1">doms</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<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="E2:33"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:33"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_9.html#K1" target="_self">Mpr1</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="mcart_1.html#K15">pr1</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_9.html#K2" target="_self">Mpr2</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="mcart_1.html#K16">pr2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_9.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:33"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="pboole.html#K10">|]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<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="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:33_1"/>
( <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:33"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:33"><i><font color="Green">E13</font></i></a>, <a class="ref" href="mssubfam.html#T17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E7:33"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="pboole.html#K10">|]</a></span>
 
<b>by </b><i><a class="ref" href="closure1.html#T3">CLOSURE1:3</a></i>;<br/>
<a NAME="E8:33"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<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>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<a NAME="E10:33"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<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>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pralg_1.html#D18">PRALG_1:def 18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<a NAME="E11:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K1">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:33"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E12:33"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mssubfam.html#T14">MSSUBFAM:14</a></i>;<br/>
<i><font color="Green">E19</font></i>: <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>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_9.html#E3:33"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pralg_1.html#D18">PRALG_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <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>7</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="msualg_9.html#E12:33"><i><font color="Green">E18</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_9.html#K2" target="_self">Mpr2</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_9.html#E4:33"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pralg_1.html#D18">PRALG_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <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>7</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="msualg_9.html#E12:33"><i><font color="Green">E18</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
;<br/>
<a NAME="E15:33"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_9.html#K1" target="_self">Mpr1</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_9.html#K2" target="_self">Mpr2</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E10:33"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_9.html#E13:33"><i><font color="Green">E19</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E16:33"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_9.html#K1" target="_self">Mpr1</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_9.html#K2" target="_self">Mpr2</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="closure1.html#K1">..</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:33"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pralg_1.html#D18">PRALG_1:def 18</a></i>;<br/>


</div>
