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<b>let </b><font color="Maroon">Omega</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">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="dynkin.html#K3" target="_self">followed_by</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">Omega</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:19_1"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E2:19_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E3:19_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:19_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) )
 ;<br/>


</div><b>end;</b></div><a NAME="E3:19_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">r</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="prob_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E4:19_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 implies <font color="Maroon">r</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:19_1"/>
<font color="Maroon">r</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">r</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="dynkin.html#K3" target="_self">followed_by</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">Omega</font></span>)</span> is <a href="prob_2.html#V1">disjoint_valued</a>
 <b>by </b><i/>;<br/>


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