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

<b>let </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font>, <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font> holds <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</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#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font> holds <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</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>23</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#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<a NAME="E3:47_1_2"/>
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>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font> holds <br/><font color="Maroon">c<sub>22</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">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<a NAME="E2:47_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:47_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="realset2.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E2:47_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green">E68</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="realset2.html#E3:47_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="realset2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>24</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="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="realset2.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green">E68</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="realset2.html#E3:47_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="realset2.html#T6" target="_self">Th6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div>
