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<b>let </b><font color="Maroon">D</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">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E5:42"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d2</font>, <font color="Olive">d3</font>, <font color="Olive">d4</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d3</font>,<font color="Olive">d4</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d2</font>,<font color="Olive">d4</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E7:42"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E8:42"/><i><font color="Green">E61</font></i>: 
(  <a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">p</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">p</font></span>)</span> &amp;  <a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">q</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">q</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">E63</font></i>: 
 <a href="setwop_2.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font></span>)</span> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">p</font>,<font color="Maroon">e</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3" target="_self">dom</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</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">p</font>,<font color="Maroon">e</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" target="_self">[#]</a> <font color="Maroon">q</font>,<font color="Maroon">e</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , , , </i>
<br/>.= 
<font color="Maroon">F</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">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font></span>)</span>
<b>by </b><i>, , , , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i>, , , </i>
;<br/>


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