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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="waybel_0.html#L1" target="_self">NetStr</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="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>thus </b><a NAME="E1:30"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1" target="_self">is_eventually_in</a> <font color="Maroon">c<sub>3</sub></font> implies  not <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R2" target="_self">is_often_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E1:30_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" target="_self">WAYBEL_0:def 11</a><br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12" target="_self">WAYBEL_0:def 12</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:30_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:30_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E1:30_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:30"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12" target="_self">WAYBEL_0:def 12</a><br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" target="_self">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:30"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E4:30"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E2:30"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:30"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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