<?xml version="1.0"?>
<div class="add">

<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">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</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>;<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/><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><b>that </b><br/><a NAME="E1:78"/><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> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  )
 <b>and </b><br/><a NAME="E2:78"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d2</font> or <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d2</font>,<font color="Maroon">d</font> )
 ;<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1"/>
( <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d2</font> or <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d2</font>,<font color="Maroon">d</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><b>suppose </b></a><a NAME="E1:78_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d2</font>
 ;<br/><div class="add"><b>thus </b><font color="Maroon">d1</font> = 
<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>,<font color="Maroon">d1</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</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">d</font></span>)</span>,<font color="Maroon">d</font></span>)</span>,<font color="Maroon">d1</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></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">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d2</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</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"><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">d</font></span>)</span>,<font color="Maroon">d</font></span>)</span>,<font color="Maroon">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</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>,<font color="Maroon">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<font color="Maroon">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_2"><b>suppose </b></a><a NAME="E1:78_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d2</font>,<font color="Maroon">d</font>
 ;<br/><div class="add"><b>thus </b><font color="Maroon">d1</font> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<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#E2:78"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d1</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<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">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</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> <font color="Maroon">d2</font>,<font color="Maroon">d</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">d</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></a>, <a class="txt" href="finseqop.html#E1:78_1_2"><i><font color="Green">E39</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> <font color="Maroon">d2</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<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">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</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> <font color="Maroon">d2</font>,<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#E2:78"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<font color="Maroon">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:78"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
