<?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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</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">L1</font> holds   <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c1">L1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <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">L1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c1">L1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">B1</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T74" onmouseover="rs('waybel23/T74')" onmouseout="rh()">WAYBEL23:74</a></span>;<br/>
<a NAME="E3:32"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</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">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</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">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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>  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="E1:32_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="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</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">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  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="E2:32_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">u</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="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:32_1"/>
<font color="Maroon" title="c4">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="c2">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">B0</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</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">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</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> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">B2</font> =  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">y</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/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">y</font> );<br/>
<a NAME="E5:32"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c8">y</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c7">z</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E3:32_2"/>
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c7">z</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:32_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c7">z</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c6">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="E7:32"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E8:32"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B0</font>
 <span class="kw">and </span><br/><a NAME="E9:32"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S6" onmouseover="rs('funct_1/S6')" onmouseout="rh()">FUNCT_1:sch 6</a>(<a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E25">A2</font></span></a>);<br/></span>
<a NAME="E10:32"/>
<font color="Maroon" title="c4">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">u</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="E3:32_3"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c8">u</font>
 <span class="kw">and </span><br/><a NAME="E4:32_3"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>
<a NAME="E5:32_3"/>
 ex <font color="Olive" title="b1">y</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="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b1">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E4:32_3"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E3:32_3"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E4:32_3"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="card_1.html#T12" onmouseover="rs('card_1/T12')" onmouseout="rh()">CARD_1:12</a></span>;<br/>
<a NAME="E12:32"/><span class="lab"><font color="Green" title="E29">A9</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" onmouseover="rs('yellow_9/D4')" onmouseout="rh()">YELLOW_9:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A10</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_4_1"/>
( <font color="Maroon" title="c1">L1</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  or <font color="Maroon" title="c1">L1</font> is  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_4_1_1"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c1">L1</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font></span><br/><div class="add"><a NAME="E2:32_4_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_4_1_1_1"/>
<font color="Maroon" title="c7">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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c7">z</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="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<a NAME="E3:32_4_1_1_1"/>
<font color="Maroon" title="c8">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">z1</font>
 
;<br/>
<a NAME="E4:32_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<font color="Maroon" title="c8">z1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T18" onmouseover="rs('waybel_0/T18')" onmouseout="rh()">WAYBEL_0:18</a></span>;<br/>
<a NAME="E5:32_4_1_1_1"/>
<font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:32_4_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>;<br/>
<a NAME="E6:32_4_1_1_1"/><span class="kw">then </span>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c8">z1</font> 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="waybel11.html#D5" onmouseover="rs('waybel11/D5')" onmouseout="rh()">WAYBEL11:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="E8:32_4_1_1_1"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E9:32_4_1_1_1"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon" title="c8">z1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E10:32_4_1_1_1"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c8">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_4_1_1_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<a NAME="E11:32_4_1_1_1"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c2">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>;<br/>
<a NAME="E12:32_4_1_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c8">z1</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_4_1_1_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="yellow_0.html#T35" onmouseover="rs('yellow_0/T35')" onmouseout="rh()">YELLOW_0:35</a></span>;<br/>
<a NAME="E13:32_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<font color="Maroon" title="c8">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" onmouseover="rs('waybel_0/T39')" onmouseout="rh()">WAYBEL_0:39</a></span>;<br/>
<a NAME="E14:32_4_1_1_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c9">A</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_4_1_1_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="yellow_2.html#T22" onmouseover="rs('yellow_2/T22')" onmouseout="rh()">YELLOW_2:22</a></span>;<br/>
<a NAME="E15:32_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_4_1_1_1"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="yellow_0.html#D3" onmouseover="rs('yellow_0/D3')" onmouseout="rh()">YELLOW_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:32_4_1_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_4_1_1_1"><span class="lab"><font color="Green" title="E33">A13</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:32_4_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font> &amp;  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="waybel_0.html#T50" onmouseover="rs('waybel_0/T50')" onmouseout="rh()">WAYBEL_0:50</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_4_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_4_1_2"/>
<font color="Maroon" title="c1">L1</font> is  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font></span><br/><div class="add"><a NAME="E2:32_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#NV10" title="STRUCT_0:NV.10">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/><a NAME="E3:32_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">B0</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel31.html#T23" target="_self" onmouseover="rs('waybel31/T23')" onmouseout="rh()">Th23</a>, <a class="ref" href="yellow15.html#T30" onmouseover="rs('yellow15/T30')" onmouseout="rh()">YELLOW15:30</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_4_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow15.html#T27" onmouseover="rs('yellow15/T27')" onmouseout="rh()">YELLOW15:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E14:32"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">T</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" onmouseover="rs('xboole_1/T2')" onmouseout="rh()">XBOOLE_1:2</a>, <a class="ref" href="yellow_0.html#T42" onmouseover="rs('yellow_0/T42')" onmouseout="rh()">YELLOW_0:42</a></span>;<br/>
<a NAME="E15:32"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">T</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font> </span> } </span> 
 
;<br/>
<a NAME="E16:32"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">T</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D27" onmouseover="rs('waybel_0/D27')" onmouseout="rh()">WAYBEL_0:def 27</a></span>;<br/>
<a NAME="E17:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="yellow_0.html#T26" onmouseover="rs('yellow_0/T26')" onmouseout="rh()">YELLOW_0:26</a>, <a class="ref" href="yellow_0.html#T42" onmouseover="rs('yellow_0/T42')" onmouseout="rh()">YELLOW_0:42</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">B2</font> =  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c4">B0</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="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font> &amp; <font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font> implies  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_5"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font>
 <span class="kw">and </span><br/><a NAME="E2:32_5"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font></span><br/><a NAME="E3:32_5"/>
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_5"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="waybel_0.html#D27" onmouseover="rs('waybel_0/D27')" onmouseout="rh()">WAYBEL_0:def 27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Y2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font><span class="kw"> such that </span><br/><a NAME="E5:32_5"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c10">Y2</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">and </span><br/><a NAME="E6:32_5"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c10">Y2</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><a NAME="E7:32_5"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_5"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="waybel_0.html#D27" onmouseover="rs('waybel_0/D27')" onmouseout="rh()">WAYBEL_0:def 27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y1</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font><span class="kw"> such that </span><br/><a NAME="E9:32_5"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c11">Y1</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">and </span><br/><a NAME="E10:32_5"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c11">Y1</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><a NAME="E11:32_5"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c11">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y2</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>;<br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> = 
<font color="Maroon" title="c8">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T41" onmouseover="rs('yellow_0/T41')" onmouseout="rh()">YELLOW_0:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c11">Y1</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c10">Y2</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E9:32_5"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E5:32_5"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="yellow12.html#T10" onmouseover="rs('yellow12/T10')" onmouseout="rh()">YELLOW12:10</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Y2</font></span>)</span>,<font color="Maroon" title="c2">T</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_5"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E6:32_5"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="yellow_2.html#T3" onmouseover="rs('yellow_2/T3')" onmouseout="rh()">YELLOW_2:3</a></span>
;<br/><a NAME="E13:32_5"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">B0</font> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">T</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_5"><span class="lab"><font color="Green" title="E38">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:32_5"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D27" onmouseover="rs('waybel_0/D27')" onmouseout="rh()">WAYBEL_0:def 27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c8">B2</font> = <font color="Maroon" title="c7">B2</font> as   <a href="waybel23.html#V2" title="WAYBEL23:attr.2">join-closed</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">by</span> <span class="lab"><a class="ref" href="waybel23.html#T18" onmouseover="rs('waybel23/T18')" onmouseout="rh()">WAYBEL23:18</a></span>;<br/>
<a NAME="E21:32"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">b</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="b3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b3">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font> implies  ex <font color="Olive" title="b1">b</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">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c10">y</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="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<a NAME="E2:32_6"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c4">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T50" onmouseover="rs('waybel_0/T50')" onmouseout="rh()">WAYBEL_0:50</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:32_6"/>
<font color="Maroon" title="c9">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</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">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<a NAME="E4:32_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T8" onmouseover="rs('waybel_3/T8')" onmouseout="rh()">WAYBEL_3:8</a></span>;<br/>
<a NAME="E5:32_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c12">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c11">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="yellow12.html#T13" onmouseover="rs('yellow12/T13')" onmouseout="rh()">YELLOW12:13</a></span>;<br/>
<a NAME="E6:32_6"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c11">x1</font> 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="waybel11.html#T36" onmouseover="rs('waybel11/T36')" onmouseout="rh()">WAYBEL11:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">u</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="E8:32_6"/><span class="lab"><font color="Green" title="E34">A27</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E9:32_6"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<font color="Maroon" title="c12">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">u</font>
 <span class="kw">and </span><br/><a NAME="E10:32_6"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c13">u</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="c11">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_6"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">b</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c13">u</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="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">b</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c14">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<a NAME="E12:32_6"/>
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_6"><span class="lab"><font color="Green" title="E34">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:32_6"/>
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_6"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c14">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<a NAME="E14:32_6"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c11">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c11">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T11" onmouseover="rs('waybel_3/T11')" onmouseout="rh()">WAYBEL_3:11</a></span>;<br/>
<a NAME="E15:32_6"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>;<br/>
<a NAME="E16:32_6"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c11">x1</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c13">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_6"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="txt" href="#E14:32_6"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="yellow_0.html#T35" onmouseover="rs('yellow_0/T35')" onmouseout="rh()">YELLOW_0:35</a></span>;<br/>
<a NAME="E17:32_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c13">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" onmouseover="rs('waybel_0/T39')" onmouseout="rh()">WAYBEL_0:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:32_6"/>
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="yellow_0.html#T1" onmouseover="rs('yellow_0/T1')" onmouseout="rh()">YELLOW_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font></span><br/>

<a NAME="E19:32_6"/>
<font color="Maroon" title="c13">u</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_6"><span class="lab"><font color="Green" title="E34">A27</font></span></a>, <a class="ref" href="yellow_8.html#T10" onmouseover="rs('yellow_8/T10')" onmouseout="rh()">YELLOW_8:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:32_6"/>
<font color="Maroon" title="c14">b</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E9:32_6"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="ref" href="waybel14.html#T26" onmouseover="rs('waybel14/T26')" onmouseout="rh()">WAYBEL14:26</a>, <a class="ref" href="waybel_8.html#T8" onmouseover="rs('waybel_8/T8')" onmouseout="rh()">WAYBEL_8:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c8">B2</font> is    <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="waybel23.html#T47" onmouseover="rs('waybel23/T47')" onmouseout="rh()">WAYBEL23:47</a></span>;<br/>
<a NAME="E23:32"/>
<font color="Maroon" title="c8">B2</font> is  <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="waybel23.html#D8" onmouseover="rs('waybel23/D8')" onmouseout="rh()">WAYBEL23:def 8</a></span>;<br/>
<a NAME="E24:32"/><span class="kw">then </span>
 <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c1">L1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E22:32"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="waybel31.html#T1" target="_self" onmouseover="rs('waybel31/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:32"/>
 <a href="waybel31.html#K1" title="WAYBEL31:func.1">CLweight</a> <font color="Maroon" title="c1">L1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E28">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
