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<b>let </b><font color="Maroon">C</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  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>;<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">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">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>













<b>assume </b><a NAME="E1:20"/><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> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">F</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  &amp; <font color="Maroon">G</font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">F</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> = <font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span>;<br/>
<a NAME="E2:20"/>
<font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span> <a href="binop_1.html#R7">is_distributive_wrt</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseqop.html#T55">FINSEQOP:55</a></i>;<br/>
<a NAME="E3:20"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></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="Olive">d1</font>,<font color="Olive">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"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">d2</font></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="binop_1.html#D12">BINOP_1:def 12</a>, <a class="ref" href="finseqop.html#T73">FINSEQOP:73</a></i>;<br/>
<b>hence </b><a NAME="E4:20"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">d</font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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