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<b>let </b><font color="Maroon">Y</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">X</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">Y</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E14</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="E2:46"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>

<a NAME="E3:46"/><i><font color="Green">E16</font></i>: 
(  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="txt" href="setwiseo.html#E2:46"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E4:46"/>
 ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span> &amp; ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></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"><font color="Olive">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="ref" href="setwiseo.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:46"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a></i>;<br/>


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