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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_2.html#V1">disjoint_valued</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:38"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:38"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:38"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="prob_3.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prob_3.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E8:38"/>
<font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="prob_3.html#E7:38"><i><font color="Green">E37</font></i></a>, <a class="txt" href="prob_3.html#E1:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="prob_2.html#D4">PROB_2:def 4</a></i>;<br/>
<a NAME="E9:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:38"/>
contradiction
 <b>by </b><i><a class="txt" href="prob_3.html#E5:38"><i><font color="Green">E36</font></i></a>, <a class="txt" href="prob_3.html#E7:38"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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