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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">F</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; ( for <font color="Olive" title="b2">f</font> being <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> holds  <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1;<br/>
<b>consider </b><font color="Maroon" title="c3">S'</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">T'</font> =  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  as  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1 <b>by </b><i><a class="ref" href="waybel27.html#T31" target="_self" title="WAYBEL27:th.31">Th31</a>, <a class="ref" href="waybel26.html#T4" title="WAYBEL26:th.4">WAYBEL26:4</a></i>;<br/>
<a NAME="E3:51"/>
(  <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="c1">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #) <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">S'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S'</font> #) &amp;  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1 <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="c4">T'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T'</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E36">A1</font></i>:  <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> = 
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c3">S'</font>,<font color="Maroon" title="c4">T'</font>
<b>by </b><i><a class="ref" href="waybel27.html#T25" target="_self" title="WAYBEL27:th.25">Th25</a></i>
<br/>.= 
 <a href="waybel17.html#K2" title="WAYBEL17:func.2">SCMaps</a> <font color="Maroon" title="c3">S'</font>,<font color="Maroon" title="c4">T'</font>
<b>by </b><i><a class="ref" href="waybel27.html#T24" target="_self" title="WAYBEL27:th.24">Th24</a></i>
<br/>.= 
 <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c3">S'</font>,<font color="Maroon" title="c4">T'</font>
<b>by </b><i><a class="ref" href="waybel24.html#T38" title="WAYBEL24:th.38">WAYBEL24:38</a></i>
<br/>.= 
 <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c3">S'</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
<b>by </b><i><a class="ref" href="waybel26.html#D1" title="WAYBEL26:def.1">WAYBEL26:def 1</a></i>
;<br/>
<a NAME="E5:51"/><i><font color="Green" title="E37">A2</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51" title="YELLOW_9:th.51">YELLOW_9:51</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:51"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>and </b><br/><a NAME="E8:51"/><i><font color="Green" title="E39">A4</font></i>: 
for <font color="Olive" title="b1">V</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S'</font> holds  <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Olive" title="b1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S'</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E4:51"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="waybel26.html#T5" title="WAYBEL26:th.5">WAYBEL26:5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">G</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; ( for <font color="Olive" title="b1">f</font> being <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> holds  <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/>

<a NAME="E9:51"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c5">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E7:51"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E10:51"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:51"/>
<font color="Maroon" title="c6">F</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel27.html#E7:51"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="waybel_0.html#T68" title="WAYBEL_0:th.68">WAYBEL_0:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> holds  <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<b>let </b><font color="Maroon" title="c7">f</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E12:51"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:51"/><i><font color="Green" title="E42">A7</font></i>: 
( <font color="Maroon" title="c8">V</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> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">V</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E9:51"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c8">V</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S'</font> <b>by </b><i><a class="txt" href="waybel27.html#E5:51"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="waybel27.html#E14:51"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font> = 
<font color="Maroon" title="c9">V</font>
<b>by </b><i><a class="txt" href="waybel27.html#E9:51"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="waybel27.html#E10:51"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="waybel27.html#E14:51"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c9">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S'</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="waybel27.html#T13" target="_self" title="WAYBEL27:th.13">Th13</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="waybel27.html#E8:51"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="waybel27.html#E14:51"><i><font color="Green" title="E42">A7</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
