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<div class="add">

<b>let </b><font color="Maroon">c<sub>21</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>22</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>21</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>5</sub></font></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="realset1.html#K4">!</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:26"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</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"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
<font color="Maroon">c<sub>22</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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E2:26"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:26"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E2:26"><i><font color="Green">E49</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font>;<br/>

<a NAME="E1:26_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="realset2.html#E5:26"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> holds <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<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>21</sub></font>;<br/>

<a NAME="E1:26_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_2"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:26"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<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>21</sub></font>;<br/>

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

<b>thus </b><a NAME="E1:26_3"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:26"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font>;<br/>

<a NAME="E1:26_4"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_4"/>
<font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:26"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> #) is   <a href="realset2.html#NM1">Group</a>
 <b>by </b><i><a class="txt" href="realset2.html#E6:26"><i><font color="Green">E51</font></i></a>, <a class="txt" href="realset2.html#E7:26"><i><font color="Green">E52</font></i></a>, <a class="txt" href="realset2.html#E8:26"><i><font color="Green">E53</font></i></a>, <a class="ref" href="realset2.html#D1" target="_self">Def1</a>, <a class="ref" href="realset2.html#D2" target="_self">Def2</a>, <a class="ref" href="realset2.html#D3" target="_self">Def3</a>, <a class="ref" href="realset2.html#D4" target="_self">Def4</a></i>;<br/>


</div>
