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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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  )
 ;<br/>

<b>set </b><font color="Maroon">e</font> =  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">u</font> =  <a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font>;<br/>
<b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D20">BINOP_1:def 20</a><br/><b>let </b><font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<font color="Maroon">F</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span>,<font color="Maroon">d1</font></span>)</span>,<font color="Maroon">d2</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span></span>)</span>,<font color="Maroon">d2</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span>,<font color="Maroon">d2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, </i>
<br/>.= 
<font color="Maroon">F</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, </i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
<b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>
<b>hence </b><a NAME="E3:77"/>
<span class="p1">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6" target="_self">the_inverseOp_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseqop.html#E1:77"><i><font color="Green">E36</font></i></a>, </i>;<br/>


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