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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="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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>assume </b><a NAME="E1:41"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:41"/>
 <a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#R7">is_distributive_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="setwop_2.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseqop.html#T67">FINSEQOP:67</a></i>;<br/>
<a NAME="E3:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="setwop_2.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="ref" href="binop_1.html#D12">BINOP_1:def 12</a>, <a class="ref" href="finseqop.html#T65">FINSEQOP:65</a></i>;<br/>
<b>hence </b><a NAME="E4:41"/>
<span class="p1">(<span class="default"><a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="setwop_2.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="ref" href="setwop_2.html#T39" target="_self">Th39</a></i>;<br/>


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