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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>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="binop_1.html#V1">commutative</a> )
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green">E19</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E6:19"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="finsop_1.html#E3:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="finsop_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E23</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="finsop_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="finsop_1.html#E6:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E9:19"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E10:19"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E11:19"/><i><font color="Green">E25</font></i>: 
 <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</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="E12:19"/>
 <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="finsop_1.html#E5:19"><i><font color="Green">E20</font></i></a>, <a class="txt" href="finsop_1.html#E11:19"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>, <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="finsop_1.html#E8:19"><i><font color="Green">E23</font></i></a>, <a class="txt" href="finsop_1.html#E10:19"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<i><font color="Green">E26</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="finsop_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="finsop_1.html#E7:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<a NAME="E15:19"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<a NAME="E16:19"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="finsop_1.html#E3:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="finsop_1.html#E14:19"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="finsop_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="finsop_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T80">RELAT_1:80</a></i>
;<br/>
<b>hence </b><a NAME="E19:19"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="finsop_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="finsop_1.html#E5:19"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finsop_1.html#T8" target="_self">Th8</a></i>;<br/>


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