<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="compts_1.html#NV4" title="COMPTS_1:NV.4">Hausdorff</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</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="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  holds <br/><font color="Olive" title="b2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Olive" title="b1">N</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>  <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"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<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="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  implies <font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 ; <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="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span></span><br/>

<a NAME="E2:77"/>
for <font color="Olive" title="b1">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span> holds  <font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span>; <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="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <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><br/>

<a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <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 <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">O</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E3:77_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a></span>;<br/>
<a NAME="E4:77_1"/>
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D20" title="YELLOW_6:def.20">YELLOW_6:def 20</a></span>;<br/>
<a NAME="E5:77_1"/><span class="kw">then </span>
<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="c5">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">s0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E7:77_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c6">s0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c2">N</font> <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="c5">O</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></span>;<br/>
<a NAME="E8:77_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# 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="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#D8" target="_self" title="WAYBEL_9:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">s1</font> = <font color="Maroon" title="c6">s0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">s1</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c7">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">j1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>; <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="c7">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j1</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:77_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c7">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">j1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>;<br/>
<a NAME="E12:77_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:77_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E8:77_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E10:77_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></span>;<br/>
<a NAME="E13:77_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E14:77_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
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="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#D8" target="_self" title="WAYBEL_9:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:77_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E13:77_1"><span class="lab"><font color="Green" title="E36">A7</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:77"/>
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K11" title="YELLOW_6:func.11">lim</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</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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