<?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  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>22</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>;<br/>







<a NAME="E1:37"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="realset1.html#M3">DnT</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="realset1.html#K5">}</a></span> st <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> #) is   <a href="realset2.html#NM1">Group</a> &amp; ( for <font color="Olive">b<sub>6</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>7</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Olive">b<sub>6</sub></font><br/>for <font color="Olive">b<sub>8</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>6</sub></font> st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="realset1.html#K4">!</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/> <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> #) is   <a href="realset2.html#NM1">Group</a> ) &amp; ( for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>4</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>6</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <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"><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>4</sub></font> <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"><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">b<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <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"><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> ) ) )
 
<b>by </b><i><a class="ref" href="realset2.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>26</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>27</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>28</sub></font> being    <a href="realset1.html#M3">DnT</a> of <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>29</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">c<sub>27</sub></font></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">c<sub>27</sub></font></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="realset1.html#K4">!</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> holds <br/> <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> #) is   <a href="realset2.html#NM1">Group</a>
 ;<br/>
<a NAME="E5:37"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <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#E3:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self">Def9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="binop_1.html#NM2">BinOp</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">c<sub>21</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <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#E3:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="realset1.html#M3">DnT</a> of <font color="Maroon">c<sub>31</sub></font>, <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#E3:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>29</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"><font color="Maroon">c<sub>31</sub></font></span><a href="realset1.html#K5">}</a></span> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E10:37"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#R2">=</a>  <a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="ref" href="realset2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</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#K3" target="_self">odf</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="ref" href="realset2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E12:37"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green">E60</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="realset1.html#M3">DnT</a> of  <a href="realset2.html#K4" target="_self">ndf</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> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green">E60</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</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> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green">E60</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E15:37"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#R2">=</a>  <a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E11:37"><i><font color="Green">E61</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green">E62</font></i></a>, <a class="ref" href="realset2.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E16:37"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <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#E11:37"><i><font color="Green">E61</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green">E62</font></i></a>, <a class="ref" href="realset2.html#D13" target="_self">Def13</a></i>;<br/>
<b>assume </b><a NAME="E17:37"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</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">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <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>hence </b><a NAME="E18:37"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> #) is   <a href="realset2.html#NM1">Group</a>
 <b>by </b><i><a class="txt" href="realset2.html#E4:37"><i><font color="Green">E58</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green">E59</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green">E62</font></i></a>, <a class="txt" href="realset2.html#E15:37"><i><font color="Green">E63</font></i></a>, <a class="txt" href="realset2.html#E16:37"><i><font color="Green">E64</font></i></a></i>;<br/>


</div>
