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<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> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:29"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E30</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="E2:29_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="prob_3.html#E1:29_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E4:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_3.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prob_3.html#E4:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_3.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E4:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_3.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prob_3.html#E4:29_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_3.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29"/>
 <a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="prob_2.html#V1">disjoint_valued</a>
 <b>by </b><i><a class="ref" href="prob_3.html#D4" target="_self">Def4</a></i>;<br/>


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