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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">T</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> holds <br/>for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st not <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b1">N</font> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> st <br/>( <font color="Olive" title="b3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b4">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b3">Y</font>  holds not <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b4">Z</font> ) )</span><br/><span class="kw">let </span><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">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> implies for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font> holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b3">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b2">Y</font>  holds not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b3">Z</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:109"/>
<font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font> holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b3">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b2">Y</font>  holds not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b3">Z</font> ) )</span><br/>

<a NAME="E2:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 ex <font color="Olive" title="b1">M</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D14" target="_self" title="YELLOW_6:def.14">Def14</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font> implies  ex <font color="Olive" title="b1">Y</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b2">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">Y</font>  holds not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b2">Z</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:109"/>
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b2">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">Y</font>  holds not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b2">Z</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font><span class="kw"> such that </span><br/><a NAME="E5:109"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
not <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></span>;<br/>
<a NAME="E6:109"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="waybel_0.html#T9" title="WAYBEL_0:th.9">WAYBEL_0:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">V</font></span>)</span> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T31" target="_self" title="YELLOW_6:th.31">Th31</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">Y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">Y</font>  holds not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b1">Z</font> ) )</span><br/>

<a NAME="E8:109"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D13" target="_self" title="YELLOW_6:def.13">Def13</a></span>;<br/>
<a NAME="E9:109"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:109"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D14" target="_self" title="YELLOW_6:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">Y</font>  holds not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b1">Z</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">Z</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c6">Z</font></span><br/>

<span class="kw">assume </span><a NAME="E11:109"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c6">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E12:109"/><span class="kw">then </span>
<font color="Maroon" title="c6">Z</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></span>;<br/>
<a NAME="E13:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T27" target="_self" title="YELLOW_6:th.27">Th27</a>, <a class="ref" href="yellow_6.html#T28" target="_self" title="YELLOW_6:th.28">Th28</a></span>;<br/>
<a NAME="E14:109"/>
<font color="Maroon" title="c5">Y</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T32" target="_self" title="YELLOW_6:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:109"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:109"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="waybel_0.html#T10" title="WAYBEL_0:th.10">WAYBEL_0:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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