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<a NAME="E1:31_1_1"/>
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>22</sub></font> being  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>23</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="realset1.html#M3">DnT</a> of <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>26</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>22</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>24</sub></font></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E50</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>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>thus </b><a NAME="E4:31_1_1"/>
<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>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="realset2.html#E3:31_1_1"><i><font color="Green">E50</font></i></a></i>;<br/>


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