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

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="realset2.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <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>;<br/>



<b>set </b><font color="Maroon">c<sub>23</sub></font> = <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>set </b><font color="Maroon">c<sub>24</sub></font> = <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="realset1.html#K4">!</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>,<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>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="realset1.html#K5">{</a><span class="default"><span class="p3">(<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>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="realset1.html#K4">!</a> <span class="p2">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<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>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> #) as  <a href="rlvect_1.html#V1">strict</a> <a href="realset2.html#NM1">Group</a> <b>by </b><i><a class="ref" href="realset2.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>26</sub></font> ;<br/>
<a NAME="E3:45"/>
<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="realset1.html#K1">||</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="realset1.html#K5">{</a><span class="default"><span class="p3">(<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>)</span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2">:]</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>by </b><i><a class="ref" href="realset1.html#T11">REALSET1:11</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>26</sub></font> <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>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> be    <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>;<br/>

<a NAME="E1:45_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></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"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<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>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1"/>
<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>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="realset2.html#E4:45"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <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> ;<br/>
<i><font color="Green">E67</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>28</sub></font>,<span class="p2">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="realset2.html#E5:45"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="ref" href="realset2.html#D1" target="_self">Def1</a></i>
;<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"><span class="p2">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>26</sub></font> <a href="binop_1.html#K1">.</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="realset2.html#E5:45"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="ref" href="realset2.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E9:45"/>
( <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>,<span class="p2">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</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"><span class="p2">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="realset2.html#E7:45"><i><font color="Green">E67</font></i></a></i>;<br/>


</div>
