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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="mcart_1.html#K16">pr2</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">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:29_2_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:29_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:29_2_1_1"/><i><font color="Green">E12</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="E2:29_2_1_1"/><b>then </b><i><font color="Green">E13</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>7</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="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E3:29_2_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="mcart_1.html#K16">pr2</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>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:29_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_9.html#E1:29_2_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="msualg_9.html#E1:29_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="msualg_9.html#E3:29_2_1_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E6:29_2_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>7</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:29_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E7:29_2_1_1"/><b>then </b>
 <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>7</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:29_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:29_2_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<a NAME="E9:29_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_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> ; <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="E1:29_2_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:29_2_1_1_1"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>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="E4:29_2_1_1_1"/><i><font color="Green">E19</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>
 <b>and </b><br/><a NAME="E5:29_2_1_1_1"/><i><font color="Green">E20</font></i>: 
<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> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E1:29_2_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/>
<a NAME="E6:29_2_1_1_1"/><i><font color="Green">E21</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> <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>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_9.html#E4:29_2_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<a NAME="E7:29_2_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<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"><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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<a NAME="E8:29_2_1_1_1"/>
 <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>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E6:29_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:29_2_1_1_1"/><b>then </b><i><font color="Green">E23</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>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</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>7</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>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#E1:29_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<a NAME="E10:29_2_1_1_1"/>
<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>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</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>5</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#E6:29_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:29_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:29_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msualg_9.html#E5:29_2_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msualg_9.html#E7:29_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msualg_9.html#E9:29_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:29_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <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#E8:29_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E5:29_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> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:29_2_1"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
