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<b>let </b><font color="Maroon" title="c1">X</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:()"><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#V10" title="WAYBEL_0:attr.10">eventually-directed</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font> holds <br/> (  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Olive" title="b1">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> ) ) implies <font color="Maroon" title="c1">X</font> is <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a> )</span><br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E37">A1</font></i>: 
for <font color="Olive" title="b1">N</font> being <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font> holds <br/> (  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Olive" title="b1">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a></span><br/>

<b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel25.html#D4" target="_self" title="WAYBEL25:def.4">WAYBEL25:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">D</font>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">V</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">V</font> ) )</span><br/>

<a NAME="E2:69"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<a NAME="E3:69"/>
 <a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c2">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c2">D</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel17.html#D4" title="WAYBEL17:def.4">WAYBEL17:def 4</a></i>;<br/>
<a NAME="E4:69"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="yellow14.html#T2" title="YELLOW14:th.2">YELLOW14:2</a></i>;<br/>
<b>hence </b><a NAME="E5:69"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">D</font>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E2:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="waybel_9.html#D3" title="WAYBEL_9:def.3">WAYBEL_9:def 3</a></i>; <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">V</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">V</font></span><br/>

<b>let </b><font color="Maroon" title="c3">V</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c2">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">V</font> )</span><br/>

<b>assume </b><a NAME="E6:69"/><i><font color="Green" title="E40">A4</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">V</font></span><br/>

<a NAME="E7:69"/><i><font color="Green" title="E41">A5</font></i>: 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c3">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E9">Lm1</font></i></a></i>;<br/>
<a NAME="E9:69"/>
<font color="Maroon" title="c4">Z</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E42">A6</font></i>:  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> = 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>

;<br/>
<a NAME="E2:69_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c1">X</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>
<a NAME="E3:69_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T77" title="TOPS_3:th.77">TOPS_3:77</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1"/>
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="#E1:69_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a> <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>
<a NAME="E10:69"/><b>then </b>
 <a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="#E7:69"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span><b> such that </b><br/><a NAME="E12:69"/><i><font color="Green" title="E42">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c6">a</font> = <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/><a NAME="E1:69_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c2">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E2:69_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font></span><br/><b>thus </b><a NAME="E3:69_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="#E12:69"><i><font color="Green" title="E42">A7</font></i></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="E14:69"/>
<font color="Maroon" title="c2">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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>
