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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">c1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">c2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</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">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><b>that </b><br/><a NAME="E1:2"/><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> )
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font>
 ;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K3">{.</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="setwiseo.html#K3">.}</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E5:2"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</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="Maroon">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>
 <b>and </b><br/><a NAME="E6:2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">c</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">c</font>
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green">E57</font></i>: 
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">C</font>  st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> &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">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <font color="Olive">B</font> holds <br/><font color="Maroon">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">c</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="Maroon">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">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a></i>;<br/>
<a NAME="E8:2"/>
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:2"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T70">ZFMISC_1:70</a></i>;<br/>
<a NAME="E10:2"/><b>then </b><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K3">{.</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="setwiseo.html#K3">.}</a></span>,<font color="Maroon">f</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c1</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c2</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c2</font></span>)</span>
<b>by </b><i>, </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="funct_2.html#K3">.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c2</font></span>)</span>
<b>by </b><i/>
;<br/>


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