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<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> ] <b>means </b><span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</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#K4" target="_self">ndf</a> <font color="Maroon">F</font>;<br/>
<a NAME="E1:47_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:47_1_1_1"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E4:47_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1"/>
 ex <font color="Olive">C</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font>, <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">C</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font>, <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:47_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> holds <br/><span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font>
 ;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E5:47_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> holds  <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div>
