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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</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>and </b><br/><a NAME="E5:42"/><i><font color="Green">E24</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>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:42"/><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E7:42"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="setwop_2.html#E6:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E8:42"/><i><font color="Green">E27</font></i>: 
(  <a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:42"/><i><font color="Green">E28</font></i>: 
 <a href="setwop_2.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green">E20</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green">E21</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green">E20</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green">E21</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green">E20</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green">E21</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green">E22</font></i></a>, <a class="txt" href="setwop_2.html#E4:42"><i><font color="Green">E23</font></i></a>, <a class="txt" href="setwop_2.html#E5:42"><i><font color="Green">E24</font></i></a>, <a class="txt" href="setwop_2.html#E6:42"><i><font color="Green">E25</font></i></a>, <a class="txt" href="setwop_2.html#E8:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setwop_2.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E4:42"><i><font color="Green">E23</font></i></a>, <a class="txt" href="setwop_2.html#E6:42"><i><font color="Green">E25</font></i></a>, <a class="txt" href="setwop_2.html#E7:42"><i><font color="Green">E26</font></i></a>, <a class="txt" href="setwop_2.html#E8:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="setwop_2.html#E9:42"><i><font color="Green">E28</font></i></a>, <a class="txt" href="setwop_2.html#E31"><i><font color="Green">Lemma12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green">E20</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green">E21</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self">Def2</a></i>
;<br/>


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