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

<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 <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<a NAME="E1:59_1_1"/><i><font color="Green">E77</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> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> &amp; <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="59_1_1_1"><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> ;<br/>

<b>assume </b><a NAME="E1:59_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E3:59_1_1_1"/><i><font color="Green">E78</font></i>: 
( <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="realset2.html#T11" target="_self">Th11</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="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>thus </b><a NAME="E5:59_1_1_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font>] )
 <b>by </b><i><a class="txt" href="realset2.html#E3:59_1_1_1"><i><font color="Green">E78</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:59_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>,<span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="realset2.html#E1:59_1_1"><i><font color="Green">E77</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>,<span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E4:59_1_1"/><i><font color="Green">E78</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> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>thus </b><a NAME="E5:59_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="realset2.html#E4:59_1_1"><i><font color="Green">E78</font></i></a></i>;<br/>


</div>
