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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Olive" title="b2">X</font> holds <br/>( not <font color="Olive" title="b1">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Olive" title="b2">X</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Olive" title="b1">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Olive" title="b2">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> )</span><br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Olive" title="b1">X</font> holds <br/>( not <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Olive" title="b1">X</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">X</font> implies ( not <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> ) )</span><br/>





<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:66"/>
<font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>and </b><br/><a NAME="E5:66"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E6:66"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E5:66"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:66"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D13" target="_self" title="YELLOW_6:def.13">Def13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_6.html#V2" title="YELLOW_6:attr.2">full</a>  <a href="yellow_6.html#M1" title="YELLOW_6:mode.1">SubNetStr</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E9:66"/>
<font color="Maroon" title="c6">M</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">YELLOW_6:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">M</font> st <br/>( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/>

<a NAME="E1:66_1"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">M</font> )
 
;<br/>
<a NAME="E2:66_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T19" target="_self" title="YELLOW_6:th.19">Th19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:66_1"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:66_1"/><i><font color="Green" title="E37">A4</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c10">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E7:66_1"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c11">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">e</font>
 <b>and </b><br/><a NAME="E8:66_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E9:66_1"/>
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:66_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c12">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">M</font> <b>by </b><i><a class="ref" href="yellow_6.html#D13" target="_self" title="YELLOW_6:def.13">Def13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> )</span><br/>

<a NAME="E11:66_1"/>
( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">e</font> &amp; <font color="Maroon" title="c10">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">e</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E5:66_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="yellow_6.html#E7:66_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:66_1"/>
( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T21" target="_self" title="YELLOW_6:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:66"/>
<font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c3">X</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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