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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#R6" target="_self">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T37" target="_self">Th37</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K2" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T34" target="_self">Th34</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T38" target="_self">Th38</a></i>
;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E2:43"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6" target="_self">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


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