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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E4:4"/><b>then </b><i><font color="Green">E7</font></i>: 
( not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K4">.}</a></span>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="setwiseo.html#K3">.}</a></span> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="setwiseo.html#K3">.}</a></span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="setwop_2.html#E4:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="setwop_2.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="setwop_2.html#T3" target="_self">Th3</a></i>
;<br/>


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