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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> is  <a href="yellow_6.html#V6" title="YELLOW_6:attr.6">(DIVERGENCE)</a> </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">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_6.html#D25" title="YELLOW_6:def.25">YELLOW_6:def 25</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <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">L</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> or  ex <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>2</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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="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">L</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> or  ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<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">L</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</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">b<sub>1</sub></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">b<sub>1</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/>

<a NAME="E3:28"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 ex <font color="Olive" title="b1">N1</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">L</font> st <br/>( <font color="Olive" title="b1">N1</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">N1</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">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></span>;<br/>
<a NAME="E4:28"/>
not for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></span>;<br/>
<a NAME="E5:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</font></span>: 
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> or  ex <font color="Olive" title="b1">p</font> being  <a href="waybel28.html#V1" title="WAYBEL28:attr.1">greater_or_equal_to_id</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c2">N</font> st not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K2" title="WAYBEL28:func.2">*</a> <font color="Olive" title="b1">p</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel28.html#T15" target="_self" title="WAYBEL28:th.15">Th15</a></span>;<br/>
<a NAME="E6:28"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_1"/>
( ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> ) or ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> ) or  ex <font color="Olive" title="b1">M</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">M</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">L</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Olive" title="b1">M</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="waybel28.html#T11" target="_self" title="WAYBEL28:th.11">Th11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</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">b<sub>1</sub></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">b<sub>1</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">N9</font> = <font color="Maroon" title="c2">N</font> 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#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">N9</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="c4">N9</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">L</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">N9</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:28_1_1"/>
<font color="Maroon" title="c4">N9</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></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">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">N9</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span><br/><span class="kw">given </span><font color="Maroon" title="c5">M2</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">N9</font><span class="kw"> such that </span><a NAME="E5:28_1_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M2</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</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="E6:28_1_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel21.html#T37" title="WAYBEL21:th.37">WAYBEL21:37</a></span>;<br/><a NAME="E7:28_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c5">M2</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></span>;<br/><a NAME="E8:28_1_1"/>
<font color="Maroon" title="c5">M2</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:28_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E9:28_1_1"/><span class="kw">then </span>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:28_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:28_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:28_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_2"/>
( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</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">b<sub>1</sub></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">b<sub>1</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><div class="add"><a NAME="E2:28_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c3">x</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D7" title="WAYBEL11:def.7">WAYBEL11:def 7</a></span>;<br/><a NAME="E3:28_1_2"/><span class="kw">then </span>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="waybel11.html#D8" title="WAYBEL11:def.8">WAYBEL11:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">M</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E5:28_1_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c4">M</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">L</font>
 <span class="kw">and </span><br/><a NAME="E6:28_1_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="yellow_6.html#D25" title="YELLOW_6:def.25">YELLOW_6:def 25</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">M</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="c4">M</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">L</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><span class="kw">thus </span><a NAME="E7:28_1_2"/>
<font color="Maroon" title="c4">M</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></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">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span><br/><a NAME="E8:28_1_2"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">M1</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E1:28_1_2_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:28_1_2_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</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="E3:28_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c5">M1</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:28_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">M11</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">L</font> st <br/>( <font color="Olive" title="b1">M11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">M11</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">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></span>;<br/>
<a NAME="E5:28_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A15</font></span>: 
not <font color="Maroon" title="c3">x</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E1:28_1_2_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="waybel11.html#D8" title="WAYBEL11:def.8">WAYBEL11:def 8</a></span>;<br/>
<a NAME="E6:28_1_2_1"/>
<font color="Maroon" title="c5">M1</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></span>;<br/>
<a NAME="E7:28_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_2_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E3:28_1_2_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:28_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_2_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="waybel11.html#D7" title="WAYBEL11:def.7">WAYBEL11:def 7</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:28_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3"><span class="kw">suppose </span></a><a NAME="E1:28_1_3"/>
 ex <font color="Olive" title="b1">M</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">M</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">L</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Olive" title="b1">M</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">b<sub>1</sub></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">b<sub>1</sub></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">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">M</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E3:28_1_3"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c4">M</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">L</font>
 <span class="kw">and </span><br/><a NAME="E4:28_1_3"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">M</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="c4">M</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">L</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> ) )</span><br/><span class="kw">thus </span><a NAME="E5:28_1_3"/>
<font color="Maroon" title="c4">M</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_3"><span class="lab"><font color="Green" title="E22">A16</font></span></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">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span><br/><a NAME="E6:28_1_3"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">M1</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E1:28_1_3_1"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
<font color="Maroon" title="c5">M1</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></span>;<br/>
<a NAME="E2:28_1_3_1"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
(  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c4">M</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Maroon" title="c4">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel28.html#T1" target="_self" title="WAYBEL28:th.1">Th1</a>, <a class="ref" href="waybel21.html#T37" title="WAYBEL21:th.37">WAYBEL21:37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:28_1_3_1"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</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="E4:28_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">M1</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:28_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_3_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E1:28_1_3_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:28_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E2:28_1_3_1"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:28_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
