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

<span class="kw">let </span><font color="Maroon" title="c1">L1</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L1</font><br/> for <font color="Olive" title="b2">B1</font> being  <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L1</font> holds  <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">A</font></span>)</span> where <font color="Olive" title="b3">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b4">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B1</font> &amp; <font color="Olive" title="b4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B1</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B1</font> being  <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L1</font> holds  <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">A</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b3">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B1</font> be   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font></span><br/>





<a NAME="E1:33"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</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="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E2:33"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span>  or <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><br/>

<a NAME="E2:33_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ex <font color="Olive" title="b2">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">WU</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">WU</font> = <font color="Maroon" title="c4">WU</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<div><span class="lab"><font color="Green" title="E21">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">BL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">W</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b2">F</font></span>)</span></span>)</span> where <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">F</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span>}</span>  as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#T32" title="WAYBEL19:th.32">WAYBEL19:32</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">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="c2">T</font>;<br/><span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies for <font color="Olive" title="b1">pT</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> holds <br/> ex <font color="Olive" title="b2">cT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b2">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> &amp; <font color="Olive" title="b1">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">cT</font> &amp; <font color="Olive" title="b2">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c8">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">pT</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> holds <br/> ex <font color="Olive" title="b2">cT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b2">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> &amp; <font color="Olive" title="b1">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">cT</font> &amp; <font color="Olive" title="b2">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">pT</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> implies  ex <font color="Olive" title="b1">cT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> &amp; <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">cT</font> &amp; <font color="Olive" title="b1">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> ) )</span><br/><span class="kw">assume </span><a NAME="E4:33_2"/>
<font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cT</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> &amp; <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">cT</font> &amp; <font color="Olive" title="b1">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E6:33_2"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BL</font>
 <span class="kw">and </span><br/><a NAME="E7:33_2"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">and </span><br/><a NAME="E8:33_2"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">W</font>, <font color="Maroon" title="c12">FT</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E10:33_2"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c12">FT</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:33_2"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E12:33_2"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c12">FT</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/><a NAME="E13:33_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">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="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">pS</font> = <font color="Maroon" title="c9">pT</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">W1</font> = <font color="Maroon" title="c11">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a></span>;<br/><a NAME="E16:33_2"/>
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c7">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="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></span>;<br/><a NAME="E17:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c11">W</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"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_2"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="waybel14.html#T33" title="WAYBEL14:th.33">WAYBEL14:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">pL</font> = <font color="Maroon" title="c13">pS</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Olive" title="b1">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Olive" title="b1">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y1</font> );<br/><a NAME="E19:33_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#D8" title="WAYBEL23:def.8">WAYBEL23:def 8</a></span>;<br/><a NAME="E20:33_2"/>
<font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_2"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E10:33_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:33_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">and </span><br/><a NAME="E23:33_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">xS</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S</font><span class="kw"> such that </span><br/><a NAME="E25:33_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c17">xS</font>
 <span class="kw">and </span><br/><a NAME="E26:33_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c17">xS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:33_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">xL</font> = <font color="Maroon" title="c17">xS</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E13:33_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a></span>;<br/><a NAME="E28:33_2"/>
<font color="Maroon" title="c17">xS</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c13">pS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:33_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E25:33_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="waybel_3.html#T8" title="WAYBEL_3:th.8">WAYBEL_3:8</a></span>;<br/><a NAME="E29:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">xL</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c15">pL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E13:33_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="waybel_8.html#T8" title="WAYBEL_8:th.8">WAYBEL_8:8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">bL</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E31:33_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c19">bL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E32:33_2"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c18">xL</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">bL</font>
 <span class="kw">and </span><br/><a NAME="E33:33_2"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c19">bL</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c15">pL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:33_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="waybel23.html#T47" title="WAYBEL23:th.47">WAYBEL23:47</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">FL</font> = <font color="Maroon" title="c12">FT</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/><a NAME="E35:33_2"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c12">FT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/><a NAME="E36:33_2"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
not <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c12">FT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_2"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E10:33_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E37:33_2"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">FL</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">FL</font> implies  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c21">y</font>,<font color="Olive" title="b1">b</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_1"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">FL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c21">y</font>,<font color="Olive" title="b1">b</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">y1</font> = <font color="Maroon" title="c21">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<a NAME="E3:33_2_1"/>
not <font color="Maroon" title="c22">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:33_2"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E35:33_2"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E1:33_2_1"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E5:33_2_1"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
( <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Maroon" title="c23">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c22">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T46" title="WAYBEL23:th.46">WAYBEL23:46</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">b</font> = <font color="Maroon" title="c23">b1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c24">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c24">b</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c23">b1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">b</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Maroon" title="c23">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">y1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c22">y1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">b</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Maroon" title="c23">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c22">y1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:33_2_1"/>
( <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">b</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Maroon" title="c23">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Maroon" title="c23">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c22">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2_1"><span class="lab"><font color="Green" title="E43">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c21">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E39:33_2"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">FL</font>
 <span class="kw">and </span><br/><a NAME="E40:33_2"/><span class="lab"><font color="Green" title="E43">A26</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">FL</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E37:33_2"><span class="lab"><font color="Green" title="E41">A22</font></span></a>);<br/></span><a NAME="E41:33_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">f</font> <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 <font color="Maroon" title="c1">L1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">f</font> or <font color="Maroon" title="c22">z</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 <font color="Maroon" title="c1">L1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_2"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">z</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 <font color="Maroon" title="c1">L1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_2_2"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">f</font>
 <span class="kw">and </span><br/><a NAME="E4:33_2_2"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:33_2_2"/>
 ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Olive" title="b1">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Olive" title="b1">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="txt" href="#E3:33_2_2"><span class="lab"><font color="Green" title="E44">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_2_2"/>
<font color="Maroon" title="c22">z</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 <font color="Maroon" title="c1">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2_2"><span class="lab"><font color="Green" title="E45">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">FFL</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">f</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> ;<br/><a NAME="E43:33_2"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<font color="Maroon" title="c22">FFL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font> or <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_3"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_2_3"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">f</font>
 <span class="kw">and </span><br/><a NAME="E4:33_2_3"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:33_2_3"/>
 ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Olive" title="b1">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Olive" title="b1">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="txt" href="#E3:33_2_3"><span class="lab"><font color="Green" title="E45">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_2_3"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2_3"><span class="lab"><font color="Green" title="E46">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E44:33_2"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font> or <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_4"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">z1</font> = <font color="Maroon" title="c23">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E4:33_2_4"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
<font color="Maroon" title="c25">v1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">z1</font>
 <span class="kw">and </span><br/><a NAME="E5:33_2_4"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
<font color="Maroon" title="c25">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">FL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">b1</font>, <font color="Maroon" title="c27">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E7:33_2_4"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<font color="Maroon" title="c27">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v1</font>
 <span class="kw">and </span><br/><a NAME="E8:33_2_4"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c26">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">v1</font>
 <span class="kw">and </span><br/><a NAME="E9:33_2_4"/>
<font color="Maroon" title="c26">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E10:33_2_4"/>
not <font color="Maroon" title="c26">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font>
 <span class="kw">and </span><br/><a NAME="E11:33_2_4"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c26">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c27">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="txt" href="#E5:33_2_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a></span>;<br/>
<a NAME="E12:33_2_4"/>
<font color="Maroon" title="c26">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c24">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2_4"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E7:33_2_4"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E11:33_2_4"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/>
<a NAME="E13:33_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c26">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></span>;<br/>
<a NAME="E14:33_2_4"/>
<font color="Maroon" title="c26">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E5:33_2_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E8:33_2_4"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:33_2_4"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_2_4"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c23">cT</font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">bL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">cT</font> = <font color="Maroon" title="c23">cT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> &amp; <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">cT</font> &amp; <font color="Maroon" title="c24">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> )</span><br/><a NAME="E46:33_2"/>
( <font color="Maroon" title="c24">cT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Maroon" title="c19">bL</font>,<font color="Maroon" title="c22">FFL</font> &amp; <font color="Maroon" title="c22">FFL</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E47:33_2"/>
<font color="Maroon" title="c24">cT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:33_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E43:33_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">cT</font> &amp; <font color="Maroon" title="c24">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> )</span><br/><a NAME="E48:33_2"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">bL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:33_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="waybel_3.html#T12" title="WAYBEL_3:th.12">WAYBEL_3:12</a></span>;<br/><a NAME="E49:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A40</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">bL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:33_2"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="xboole_1.html#T35" title="XBOOLE_1:th.35">XBOOLE_1:35</a></span>;<br/><a NAME="E50:33_2"/>
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font> or  not <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font> or  not <font color="Maroon" title="c25">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_5"/>
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c25">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_2_5"/><span class="lab"><font color="Green" title="E47">A41</font></span>: 
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">f</font>
 <span class="kw">and </span><br/><a NAME="E4:33_2_5"/><span class="lab"><font color="Green" title="E48">A42</font></span>: 
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:33_2_5"/>
 ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">v</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; not <font color="Olive" title="b1">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> &amp; <font color="Olive" title="b1">b1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="txt" href="#E3:33_2_5"><span class="lab"><font color="Green" title="E47">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_2_5"/>
not <font color="Maroon" title="c25">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2_5"><span class="lab"><font color="Green" title="E48">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E51:33_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">pL</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FFL</font> )
 ;<br/><a NAME="E52:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A43</font></span>: 
not <font color="Maroon" title="c15">pL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/><a NAME="E53:33_2"/>
<font color="Maroon" title="c15">pL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">bL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:33_2"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="waybel_3.html#T8" title="WAYBEL_3:th.8">WAYBEL_3:8</a></span>;<br/><span class="kw">hence </span><a NAME="E54:33_2"/>
<font color="Maroon" title="c9">pT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">cT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:33_2"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font></span><br/><a NAME="E55:33_2"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font> or <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font> )</span><br/>

<a NAME="E1:33_2_6"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c17">xS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E13:33_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="yellow12.html#T13" title="YELLOW12:th.13">YELLOW12:13</a></span>;<br/>
<a NAME="E2:33_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A44</font></span>: 
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:33_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:33_2_6"/>
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font></span><br/>

<span class="kw">hence </span><a NAME="E4:33_2_6"/>
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E2:33_2_6"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E56:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c18">xL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c20">FL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E35:33_2"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/><a NAME="E57:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c19">bL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c22">FFL</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:33_2"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E58:33_2"/>
<font color="Maroon" title="c24">cT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_2"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:33"/>
<font color="Maroon" title="c5">WU</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_3"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">WU</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c8">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font><span class="kw"> such that </span><br/><a NAME="E3:33_3"/><span class="lab"><font color="Green" title="E22">A45</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">A</font>
 <span class="kw">and </span><br/><a NAME="E4:33_3"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E5:33_3"/>
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A1</font> = <font color="Maroon" title="c8">A</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<a NAME="E8:33_3"/>
(  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c10">a1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel31.html#T25" target="_self" title="WAYBEL31:th.25">Th25</a>, <a class="ref" href="waybel19.html#T40" title="WAYBEL19:th.40">WAYBEL19:40</a></span>;<br/>
<a NAME="E9:33_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A46</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></span>;<br/>
<font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c8">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E22">A45</font></span></a>, <a class="ref" href="yellow12.html#T13" title="YELLOW12:th.13">YELLOW12:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">A1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:33_3"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33_3"><span class="lab"><font color="Green" title="E23">A46</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:33"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" title="WAYBEL31:func.2">Way_Up</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Olive" title="b2">A</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="yellow_9.html#T32" title="YELLOW_9:th.32">YELLOW_9:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
