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

<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">Y</font> being <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a> <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span>  st ( for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Olive" title="b2">N</font>,<font color="Olive" title="b3">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> ) holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Olive" title="b2">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be  <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a> <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-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#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> ) holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Olive" title="b1">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> ) implies  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span><br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</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#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<b>set </b><font color="Maroon" title="c4">m</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>;<br/>
<b>set </b><font color="Maroon" title="c5">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:79"/><i><font color="Green" title="E45">A2</font></i>: 
 <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="c2">Y</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</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="c2">Y</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="c2">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">fo</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19" title="WAYBEL24:th.19">WAYBEL24:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">fo</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/>
<a NAME="E5:79"/><i><font color="Green" title="E46">A3</font></i>: 
 <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 <font color="Maroon" title="c3">N</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="c3">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:79"/><i><font color="Green" title="E47">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E7:79"/>
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>set </b><font color="Maroon" title="c9">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> ;<br/>
<a NAME="E2:79_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">A</font> where <font color="Olive" title="b2">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b3">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> iff <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  )</span><br/>

<b>thus </b><a NAME="E1:79_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1_1_1"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> </span><br/>

<a NAME="E2:79_1_1_1"/><b>then </b><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:79_1_1_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">NET</font> =  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>
<a NAME="E4:79_1_1_1"/><i><font color="Green" title="E51">A8</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E29"><i><font color="Green" title="E22">Lm4</font></i></a></i>;<br/>
<a NAME="E5:79_1_1_1"/>
 <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E6:79_1_1_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E7:79_1_1_1"/><b>then </b><i><font color="Green" title="E53">A10</font></i>: 
 <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"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></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="c3">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">N</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D3" target="_self" title="WAYBEL25:def.3">Def3</a></i>;<br/>
<a NAME="E8:79_1_1_1"/>
 <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">W</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> as  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="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_2.html#T18" title="WAYBEL_2:th.18">WAYBEL_2:18</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="waybel25.html#T41" target="_self" title="WAYBEL25:th.41">Th41</a></i>
<br/>.= 
 <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> <font color="Maroon" title="c13">W</font>

;<br/>
<a NAME="E11:79_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">V</font>
 
<b>by </b><i><a class="txt" href="#E1:79_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E2:79_1_1_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="waybel25.html#D4" target="_self" title="WAYBEL25:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:79_1_1_1"/><i><font color="Green" title="E54">A11</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W</font>
 <b>and </b><br/><a NAME="E14:79_1_1_1"/><i><font color="Green" title="E55">A12</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:79_1_1_1"/><i><font color="Green" title="E56">A13</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E17:79_1_1_1"/><i><font color="Green" title="E57">A14</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="#E13:79_1_1_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="#E7:79_1_1_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="#E16:79_1_1_1"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/>
<i><font color="Green" title="E58">A15</font></i>: <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="#E6:79_1_1_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="waybel25.html#D3" target="_self" title="WAYBEL25:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="#E4:79_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T21" title="WAYBEL24:th.21">WAYBEL24:21</a></i>;<br/>
<a NAME="E21:79_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">g</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">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:79_1_1_1"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font>
 
<b>by </b><i><a class="txt" href="#E14:79_1_1_1"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="#E17:79_1_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="#E19:79_1_1_1"><i><font color="Green" title="E58">A15</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E23:79_1_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E24:79_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E22:79_1_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>assume </b><a NAME="E2:79_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c11">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:79_1_1"/><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z</font>
 <b>and </b><br/><a NAME="E5:79_1_1"/><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:79_1_1"/><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c11">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font>
 <b>and </b><br/><a NAME="E8:79_1_1"/><i><font color="Green" title="E52">A20</font></i>: 
 ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> )
 <b>by </b><i><a class="txt" href="#E5:79_1_1"><i><font color="Green" title="E50">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font>, <font color="Maroon" title="c14">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span><b> such that </b><br/><a NAME="E10:79_1_1"/><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">i</font>
 <b>and </b><br/><a NAME="E11:79_1_1"/><i><font color="Green" title="E54">A22</font></i>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="#E8:79_1_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/>
<a NAME="E12:79_1_1"/><i><font color="Green" title="E55">A23</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 
<b>by </b><i><a class="txt" href="#E4:79_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E7:79_1_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="#E11:79_1_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E13:79_1_1"/><i><font color="Green" title="E56">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">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">X</font>
 
<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>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E4:79_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="#E7:79_1_1"><i><font color="Green" title="E51">A19</font></i></a></i>;<br/>
<a NAME="E15:79_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E1:79_1_1_2"/>
 <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:79_1_1_2"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel25.html#T31" target="_self" title="WAYBEL25:th.31">Th31</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>
<a NAME="E16:79_1_1"/><b>then </b><i><font color="Green" title="E57">A25</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <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 <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T26" target="_self" title="WAYBEL25:th.26">Th26</a></i>;<br/>
<a NAME="E17:79_1_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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 <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="#E5:79"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:79_1_1"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">fo</font>
 
<b>by </b><i><a class="txt" href="#E10:79_1_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="#E16:79_1_1"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="waybel25.html#T40" target="_self" title="WAYBEL25:th.40">Th40</a></i>;<br/>
<a NAME="E19:79_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fo</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E21:79_1_1"/><i><font color="Green" title="E58">A26</font></i>: 
<font color="Maroon" title="c16">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E22:79_1_1"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">O</font>
 <b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>
<a NAME="E23:79_1_1"/>
<font color="Maroon" title="c8">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">O</font>
 
<b>by </b><i><a class="txt" href="#E1:79_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E12:79_1_1"><i><font color="Green" title="E55">A23</font></i></a>, <a class="txt" href="#E22:79_1_1"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E25:79_1_1"/><i><font color="Green" title="E60">A28</font></i>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E26:79_1_1"/><i><font color="Green" title="E61">A29</font></i>: 
( <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font> )
 
<b>by </b><i><a class="txt" href="#E25:79_1_1"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E27:79_1_1"/><b>then </b>
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="#E21:79_1_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E28:79_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="#E13:79_1_1"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="#E26:79_1_1"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</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>
<a NAME="E3:79_1"/><b>then </b><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E4:79_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  or <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:79_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:79_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp;  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:79_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">M</font> = <font color="Maroon" title="c10">M</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:79_1"/>
<font color="Maroon" title="c11">M</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> or <font color="Maroon" title="c12">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:79_1_3"/>
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>then consider </b><font color="Maroon" title="c13">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:79_1_3"/><i><font color="Green" title="E50">A31</font></i>: 
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A</font>
 <b>and </b><br/><a NAME="E4:79_1_3"/><i><font color="Green" title="E51">A32</font></i>: 
 ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c14">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font>, <font color="Maroon" title="c15">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span><b> such that </b><br/><a NAME="E6:79_1_3"/><i><font color="Green" title="E52">A33</font></i>: 
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">i</font>
 <b>and </b><br/><a NAME="E7:79_1_3"/><i><font color="Green" title="E53">A34</font></i>: 
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="#E4:79_1_3"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/>
<a NAME="E8:79_1_3"/><i><font color="Green" title="E54">A35</font></i>: 
 <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 <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> #)
 
;<br/>
<b>consider </b><font color="Maroon" title="c16">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span><b> such that </b><br/><a NAME="E10:79_1_3"/><i><font color="Green" title="E55">A36</font></i>: 
( <font color="Maroon" title="c15">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">g'</font> &amp; <font color="Maroon" title="c16">g'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="#E6:79_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">g''</font> = <font color="Maroon" title="c16">g'</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E8:79_1_3"><i><font color="Green" title="E54">A35</font></i></a>, <a class="txt" href="#E10:79_1_3"><i><font color="Green" title="E55">A36</font></i></a>, <a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</a></i>;<br/>
<a NAME="E12:79_1_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E13:79_1_3"/><b>then </b>
<font color="Maroon" title="c17">g''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:79_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E14:79_1_3"/>
<font color="Maroon" title="c12">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:79_1_3"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="#E7:79_1_3"><i><font color="Green" title="E53">A34</font></i></a>, <a class="txt" href="#E10:79_1_3"><i><font color="Green" title="E55">A36</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="E8:79_1"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:79_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</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="E8:79"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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 <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="#E6:79"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</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>
