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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E65">A1</font></i>: 
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font> is <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E66">A2</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><b>let </b><font color="Maroon" title="c4">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V3</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font>;<br/><b>assume </b><a NAME="E2:97_1"/><i><font color="Green" title="E67">A3</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> holds  <font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E3:97_1"/><i><font color="Green" title="E68">A4</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 ;<br/><b>set </b><font color="Maroon" title="c6">x</font> =  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> ;<br/><a NAME="E4:97_1"/><i><font color="Green" title="E69">A5</font></i>: 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T16" title="WAYBEL_5:th.16">WAYBEL_5:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">r</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c3">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c3">I</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/><a NAME="E6:97_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">f</font> =  <a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>set </b><font color="Maroon" title="c9">X</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #);<br/><a NAME="E8:97_1"/><i><font color="Green" title="E70">A6</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #) holds  <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #);<br/>

<a NAME="E1:97_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #)
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E2:97_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:97_1"/><i><font color="Green" title="E71">A7</font></i>: 
 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #) is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:97_1_2"/>
( <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">z</font> )
 ;<br/>

<b>thus </b><a NAME="E2:97_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">z</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E8:97_1"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:97_1"/>
 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #) is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a><br/>

<b>take </b>
<font color="Maroon" title="c10">x</font>
;<br/>

<b>thus </b><a NAME="E1:97_1_3"/>
( <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c11">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="waybel11.html#E8:97_1"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">X</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">f</font> #) as  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel11.html#E9:97_1"><i><font color="Green" title="E71">A7</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font>) <b>-&gt; </b> non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#L1" title="WAYBEL_0:struct.1">NetStr</a> of <font color="Maroon" title="c1">L</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c11">J</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font><b> such that </b><br/><a NAME="E13:97_1"/><i><font color="Green" title="E72">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> holds  <font color="Maroon" title="c11">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i><a NAME="E14:97_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c10">X</font> holds <br/><font color="Maroon" title="c11">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_1_4"/>
<font color="Maroon" title="c12">i</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="c10">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">i'</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">rFi</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">i'</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:97_1_4"/><i><font color="Green" title="E73">A9</font></i>: 
<font color="Maroon" title="c14">rFi</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:97_1"><i><font color="Green" title="E68">A4</font></i></a></i>;<br/>
<a NAME="E5:97_1_4"/>
<font color="Maroon" title="c11">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c13">i'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:97_1_4"/>
<font color="Maroon" title="c11">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E4:97_1_4"><i><font color="Green" title="E73">A9</font></i></a>, <a class="ref" href="waybel11.html#T20" target="_self" title="WAYBEL11:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">J</font> = <font color="Maroon" title="c11">J</font> as    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_6.html#T33" title="YELLOW_6:th.33">YELLOW_6:33</a></i>;<br/><a NAME="E16:97_1"/><i><font color="Green" title="E73">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_1_5"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c14">i'</font> = <font color="Maroon" title="c13">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<b>take </b><font color="Maroon" title="c15">R</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>;<br/>

<b>thus </b><a NAME="E3:97_1_5"/>
<font color="Maroon" title="c15">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:97_1_5"/>
<font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">R</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E74">A11</font></i>:  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font> = 
<font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="ref" href="yellow_7.html#T45" title="YELLOW_7:th.45">YELLOW_7:45</a></i>
<br/>.= 
 <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font>
<b>by </b><i><a class="txt" href="waybel11.html#E16:97_1"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>
;<br/><a NAME="E18:97_1"/><i><font color="Green" title="E75">A12</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E19:97_1"/><b>then </b><i><font color="Green" title="E76">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:97_1"/><i><font color="Green" title="E77">A14</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> holds  <font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c14">i'</font> = <font color="Maroon" title="c13">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<a NAME="E2:97_1_7"/><i><font color="Green" title="E78">A15</font></i>: 
<font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">Ji</font> = <font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> as  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:97_1_7"/>
<font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>
<a NAME="E5:97_1_7"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">Ji</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_7"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:97_1_7"/>
<font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:97_1"/><i><font color="Green" title="E78">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c14">i'</font> = <font color="Maroon" title="c13">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<a NAME="E2:97_1_8"/><i><font color="Green" title="E79">A17</font></i>: 
<font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c14">i'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">Ji</font> = <font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> as  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:97_1_8"/><i><font color="Green" title="E80">A18</font></i>: 
<font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E20:97_1"><i><font color="Green" title="E77">A14</font></i></a></i>;<br/>
<a NAME="E5:97_1_8"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
;<br/>
<a NAME="E6:97_1_8"/><b>then </b>
<font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">i</font> = 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_5.html#D1" title="WAYBEL_5:def.1">WAYBEL_5:def 1</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_8"><i><font color="Green" title="E79">A17</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>
<br/>.= 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c15">Ji</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c15">Ji</font>
<b>by </b><i><a class="ref" href="waybel11.html#T22" target="_self" title="WAYBEL11:th.22">Th22</a></i>
;<br/>
<a NAME="E8:97_1_8"/><b>then </b>
<font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E9:97_1_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_8"><i><font color="Green" title="E79">A17</font></i></a>, <a class="txt" href="waybel11.html#E4:97_1_8"><i><font color="Green" title="E80">A18</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:97_1"/><i><font color="Green" title="E79">A19</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:97_1"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/><a NAME="E23:97_1"/><b>then </b><i><font color="Green" title="E80">A20</font></i>: 
 <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E20:97_1"><i><font color="Green" title="E77">A14</font></i></a>, <a class="ref" href="yellow_6.html#T34" title="YELLOW_6:th.34">YELLOW_6:34</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">p</font></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : <font color="Olive" title="b1">p</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ;<br/><a NAME="E24:97_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" title="YELLOW_6:th.35">YELLOW_6:35</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">G</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">i</font>,$2</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>verum;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> $1</span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>($1,$2),<font color="Maroon" title="c1">L</font>;<br/><b>set </b><font color="Maroon" title="c14">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c15">D'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c16">E'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span> ;<br/><a NAME="E26:97_1"/><i><font color="Green" title="E81">A21</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> holds  <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/><b>let </b><font color="Maroon" title="c18">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>;<br/>



<a NAME="E1:97_1_9"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">j</font>,<font color="Maroon" title="c18">g</font>) iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:97_1_9_1"/>
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a></i>;<br/>
<a NAME="E2:97_1_9_1"/><b>then </b><i><font color="Green" title="E82">A22</font></i>: 
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:97_1_9_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">j</font>,<font color="Maroon" title="c18">g</font>)
 ;<br/><b>then consider </b><font color="Maroon" title="c20">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><b> such that </b><br/><a NAME="E3:97_1_9_1_1"/><i><font color="Green" title="E83">A23</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c18">g</font>
 <b>and </b><br/><a NAME="E4:97_1_9_1_1"/>
<font color="Maroon" title="c20">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c20">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">x</font>;<br/><b>reconsider </b><font color="Maroon" title="c23">i'</font> = <font color="Maroon" title="c20">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/><a NAME="E7:97_1_9_1_1"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ;<br/><a NAME="E8:97_1_9_1_1"/><b>then </b><i><font color="Green" title="E84">A24</font></i>: 
<font color="Maroon" title="c23">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E9:97_1_9_1_1"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E18:97_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="ref" href="waybel_5.html#T8" title="WAYBEL_5:th.8">WAYBEL_5:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c23">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E18:97_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="waybel11.html#E8:97_1_9_1_1"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="waybel_5.html#T9" title="WAYBEL_5:th.9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c23">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">y</font>
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_9_1"><i><font color="Green" title="E82">A22</font></i></a>, <a class="txt" href="waybel11.html#E3:97_1_9_1_1"><i><font color="Green" title="E83">A23</font></i></a>, <a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>
;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E5:97_1_9_1"/><i><font color="Green" title="E83">A25</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span>
 <b>and </b><br/><a NAME="E6:97_1_9_1"/><i><font color="Green" title="E84">A26</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 ;<br/>

<a NAME="E7:97_1_9_1"/><i><font color="Green" title="E85">A27</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E18:97_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="waybel11.html#E5:97_1_9_1"><i><font color="Green" title="E83">A25</font></i></a>, <a class="ref" href="waybel_5.html#T8" title="WAYBEL_5:th.8">WAYBEL_5:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">i'</font> = <font color="Maroon" title="c20">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> <b>by </b><i><a class="txt" href="waybel11.html#E5:97_1_9_1"><i><font color="Green" title="E83">A25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c21">i'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> ;<br/>
<a NAME="E10:97_1_9_1"/><i><font color="Green" title="E86">A28</font></i>: 
<font color="Maroon" title="c22">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">j</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:97_1"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<font color="Maroon" title="c19">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E18:97_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="waybel11.html#E6:97_1_9_1"><i><font color="Green" title="E84">A26</font></i></a>, <a class="txt" href="waybel11.html#E7:97_1_9_1"><i><font color="Green" title="E85">A27</font></i></a>, <a class="ref" href="waybel_5.html#T9" title="WAYBEL_5:th.9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c21">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">g</font>
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_9_1"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E12:97_1_9_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">j</font>,<font color="Maroon" title="c18">g</font>)
 <b>by </b><i><a class="txt" href="waybel11.html#E10:97_1_9_1"><i><font color="Green" title="E86">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:97_1_9"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">j</font>,<font color="Maroon" title="c18">g</font>),<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:97_1"/><i><font color="Green" title="E82">A29</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:97_1_10"/><i><font color="Green" title="E83">A30</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" title="YELLOW_6:th.35">YELLOW_6:35</a></i>;<br/>
<a NAME="E2:97_1_10"/><i><font color="Green" title="E84">A31</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span><br/> for <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><br/> for <font color="Olive" title="b3">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/> <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">p</font>),<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">g</font>),<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c18">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/><b>let </b><font color="Maroon" title="c19">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:97_1_10_1"/><i><font color="Green" title="E85">A32</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>

<a NAME="E2:97_1_10_1"/><i><font color="Green" title="E86">A33</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> #)
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c19">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E4:97_1_10_1"/>
<font color="Maroon" title="c20">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span>
 
;<br/>
<a NAME="E5:97_1_10_1"/><b>then </b><i><font color="Green" title="E87">A34</font></i>: 
<font color="Maroon" title="c20">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">i'</font> = <font color="Maroon" title="c18">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> ;<br/>
<a NAME="E7:97_1_10_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c10">X</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_1_10_1_1"/>
<font color="Maroon" title="c22">i</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="c10">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c23">ii</font> = <font color="Maroon" title="c22">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c24">i'</font> = <font color="Maroon" title="c23">ii</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<a NAME="E4:97_1_10_1_1"/><i><font color="Green" title="E88">A35</font></i>: 
<font color="Maroon" title="c12">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">R</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>;<br/>
<a NAME="E5:97_1_10_1_1"/>
<font color="Maroon" title="c20">g'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c23">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:97_1_10_1_1"><i><font color="Green" title="E88">A35</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
 <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon" title="c26">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c26">x</font>
;<br/>

<a NAME="E7:97_1_10_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">x</font>
 
;<br/>
<b>hence </b><a NAME="E8:97_1_10_1_1"/>
(  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c24">i'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c26">x</font> )
 <b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:97_1_10_1"/><b>then </b><i><font color="Green" title="E88">A36</font></i>: 
<font color="Maroon" title="c20">g'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">g'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E5:97_1_10_1"><i><font color="Green" title="E87">A34</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E9:97_1_10_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1_10_1_2"/>
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>)
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><b> such that </b><br/><a NAME="E3:97_1_10_1_2"/><i><font color="Green" title="E89">A37</font></i>: 
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c19">g</font>
 <b>and </b><br/><a NAME="E4:97_1_10_1_2"/><i><font color="Green" title="E90">A38</font></i>: 
<font color="Maroon" title="c23">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">i</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c24">j'</font> = <font color="Maroon" title="c23">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c25">q</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c19">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="yellow_6.html#T35" title="YELLOW_6:th.35">YELLOW_6:35</a></i>;<br/>
<a NAME="E7:97_1_10_1_2"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c24">j'</font>,<font color="Maroon" title="c20">g'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c21">i'</font>,<font color="Maroon" title="c20">g'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:97_1_10_1"><i><font color="Green" title="E88">A36</font></i></a>, <a class="txt" href="waybel11.html#E4:97_1_10_1_2"><i><font color="Green" title="E90">A38</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<a NAME="E8:97_1_10_1_2"/><b>then </b><i><font color="Green" title="E91">A39</font></i>: 
<font color="Maroon" title="c25">q</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">p</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:97_1_10_1"><i><font color="Green" title="E85">A32</font></i></a>, <a class="txt" href="waybel11.html#E2:97_1_10_1"><i><font color="Green" title="E86">A33</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E9:97_1_10_1_2"/>
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c25">q</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_10_1_2"><i><font color="Green" title="E89">A37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:97_1_10_1_2"/>
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>)
 <b>by </b><i><a class="txt" href="waybel11.html#E8:97_1_10_1_2"><i><font color="Green" title="E91">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:97_1_10_1"/><b>then </b><i><font color="Green" title="E89">A40</font></i>: 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>),<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3" title="WAYBEL_7:th.3">WAYBEL_7:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>] <b>means </b>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">i</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> = <font color="Maroon" title="c13">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $1,<font color="Maroon" title="c19">g</font>;<br/>
<a NAME="E11:97_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c22">A</font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>] <b>means </b>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">p</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> = <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $1;<br/>
<a NAME="E13:97_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c23">B</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E15:97_1_10_1"/>
<font color="Maroon" title="c23">B</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c22">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D2" title="YELLOW_4:def.2">YELLOW_4:def 2</a><br/>

<b>assume </b><a NAME="E1:97_1_10_1_3"/>
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span><b> such that </b><br/><a NAME="E3:97_1_10_1_3"/><i><font color="Green" title="E90">A41</font></i>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c25">q</font>
 <b>and </b><br/><a NAME="E4:97_1_10_1_3"/><i><font color="Green" title="E91">A42</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c25">q</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c26">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Maroon" title="c27">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c12">J</font></span>)</span><b> such that </b><br/><a NAME="E6:97_1_10_1_3"/><i><font color="Green" title="E92">A43</font></i>: 
<font color="Maroon" title="c25">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c27">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:97_1_10"><i><font color="Green" title="E83">A30</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">k'</font> = <font color="Maroon" title="c26">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> ;<br/>
<b>set </b><font color="Maroon" title="c29">a</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c19">g</font>;<br/>
<a NAME="E8:97_1_10_1_3"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c19">g</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">a</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c19">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c30">a</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c31">f'</font> = <font color="Maroon" title="c27">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c12">J</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:97_1_10_1_3"/><i><font color="Green" title="E93">A44</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c28">k'</font>,<font color="Maroon" title="c31">f'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c21">i'</font>,<font color="Maroon" title="c20">g'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:97_1_10_1"><i><font color="Green" title="E85">A32</font></i></a>, <a class="txt" href="waybel11.html#E2:97_1_10_1"><i><font color="Green" title="E86">A33</font></i></a>, <a class="txt" href="waybel11.html#E4:97_1_10_1_3"><i><font color="Green" title="E91">A42</font></i></a>, <a class="txt" href="waybel11.html#E6:97_1_10_1_3"><i><font color="Green" title="E92">A43</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E12:97_1_10_1_3"/><b>then </b>
<font color="Maroon" title="c18">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E13:97_1_10_1_3"/>
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c32">k'</font> = <font color="Maroon" title="c26">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<b>set </b><font color="Maroon" title="c33">N</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>;<br/>
<a NAME="E15:97_1_10_1_3"/><i><font color="Green" title="E94">A45</font></i>: 
<font color="Maroon" title="c12">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">g'k</font> = <font color="Maroon" title="c20">g'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c26">k</font>, <font color="Maroon" title="c35">f'k</font> = <font color="Maroon" title="c31">f'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c26">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E13:97_1"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/>
<a NAME="E17:97_1_10_1_3"/><i><font color="Green" title="E95">A46</font></i>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c35">f'k</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_10_1_3"><i><font color="Green" title="E90">A41</font></i></a>, <a class="txt" href="waybel11.html#E6:97_1_10_1_3"><i><font color="Green" title="E92">A43</font></i></a>, <a class="txt" href="waybel11.html#E15:97_1_10_1_3"><i><font color="Green" title="E94">A45</font></i></a>, <a class="ref" href="yellow_6.html#T36" title="YELLOW_6:th.36">YELLOW_6:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">kg</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c19">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="yellow_6.html#T35" title="YELLOW_6:th.35">YELLOW_6:35</a></i>;<br/>
<i><font color="Green" title="E96">A47</font></i>: <font color="Maroon" title="c30">a</font> = 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c36">kg</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c32">k'</font></span>)</span></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c34">g'k</font>
<b>by </b><i><a class="txt" href="waybel11.html#E15:97_1_10_1_3"><i><font color="Green" title="E94">A45</font></i></a>, <a class="ref" href="yellow_6.html#T36" title="YELLOW_6:th.36">YELLOW_6:36</a></i>
;<br/>
<a NAME="E20:97_1_10_1_3"/>
<font color="Maroon" title="c20">g'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c31">f'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E11:97_1_10_1_3"><i><font color="Green" title="E93">A44</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<a NAME="E21:97_1_10_1_3"/><b>then </b>
<font color="Maroon" title="c20">g'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c26">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c31">f'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c26">k</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E22:97_1_10_1_3"/>
<font color="Maroon" title="c30">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">b</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E15:97_1_10_1_3"><i><font color="Green" title="E94">A45</font></i></a>, <a class="txt" href="waybel11.html#E17:97_1_10_1_3"><i><font color="Green" title="E95">A46</font></i></a>, <a class="txt" href="waybel11.html#E19:97_1_10_1_3"><i><font color="Green" title="E96">A47</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:97_1_10_1"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>),<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>),<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T1" target="_self" title="WAYBEL11:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E17:97_1_10_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">p</font>),<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>),<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E10:97_1_10_1"><i><font color="Green" title="E89">A40</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:97_1_10"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1_10_2"/>
<font color="Maroon" title="c17">e</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#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span><b> such that </b><br/><a NAME="E3:97_1_10_2"/><i><font color="Green" title="E85">A48</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c18">p</font>),<font color="Maroon" title="c1">L</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c19">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Maroon" title="c20">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span><b> such that </b><br/><a NAME="E5:97_1_10_2"/><i><font color="Green" title="E86">A49</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c20">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a>, <a class="txt" href="waybel11.html#E1:97_1_10"><i><font color="Green" title="E83">A30</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E6:97_1_10_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c20">g</font>),<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_10"><i><font color="Green" title="E84">A31</font></i></a>, <a class="txt" href="waybel11.html#E3:97_1_10_2"><i><font color="Green" title="E85">A48</font></i></a>, <a class="txt" href="waybel11.html#E5:97_1_10_2"><i><font color="Green" title="E86">A49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:97_1_10_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:97_1_10"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Maroon" title="c19">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span><b> such that </b><br/><a NAME="E6:97_1_10"/><i><font color="Green" title="E85">A50</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font>),<font color="Maroon" title="c1">L</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">j</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:97_1"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="yellow_6.html#T35" title="YELLOW_6:th.35">YELLOW_6:35</a></i>;<br/>
<a NAME="E8:97_1_10"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c20">j</font>),<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:97_1_10"><i><font color="Green" title="E84">A31</font></i></a>, <a class="txt" href="waybel11.html#E6:97_1_10"><i><font color="Green" title="E85">A50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:97_1_10"/>
<font color="Maroon" title="c17">e</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#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>),<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E28:97_1"/><i><font color="Green" title="E83">A51</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>, <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="waybel11.html#S1" target="_self" title="WAYBEL11:sch.1">WAYBEL11:sch 1</a>(<a class="txt" href="waybel11.html#E26:97_1"><i><font color="Green" title="E81">A21</font></i></a>);<br/></i><a NAME="E29:97_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b2">g</font>) where <font color="Olive" title="b2">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>] </span>}</span>  iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:97_1_11"/>
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1_11_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span><b> such that </b><br/><a NAME="E3:97_1_11_1"/><i><font color="Green" title="E84">A52</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c18">g</font>
;<br/>

<b>thus </b><a NAME="E4:97_1_11_1"/><i><font color="Green" title="E85">A53</font></i>: 
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E19:97_1"><i><font color="Green" title="E76">A13</font></i></a></i>;<br/>

<b>thus </b><font color="Maroon" title="c17">y</font> = 
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c18">g</font></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_11_1"><i><font color="Green" title="E84">A52</font></i></a>, <a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">g</font>
<b>by </b><i><a class="txt" href="waybel11.html#E4:97_1_11_1"><i><font color="Green" title="E85">A53</font></i></a>, <a class="ref" href="waybel_5.html#D2" title="WAYBEL_5:def.2">WAYBEL_5:def 2</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:97_1_11"/><i><font color="Green" title="E84">A54</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:97_1_11"/><i><font color="Green" title="E85">A55</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_11"><i><font color="Green" title="E84">A54</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<font color="Maroon" title="c17">y</font> = 
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_11"><i><font color="Green" title="E84">A54</font></i></a>, <a class="txt" href="waybel11.html#E4:97_1_11"><i><font color="Green" title="E85">A55</font></i></a>, <a class="ref" href="waybel_5.html#D2" title="WAYBEL_5:def.2">WAYBEL_5:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E7:97_1_11"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E30:97_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">g</font>) where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:97_1"/><b>then </b><i><font color="Green" title="E84">A56</font></i>: 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E27:97_1"><i><font color="Green" title="E82">A29</font></i></a>, <a class="txt" href="waybel11.html#E28:97_1"><i><font color="Green" title="E83">A51</font></i></a>, <a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">x'</font> =  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>set </b><font color="Maroon" title="c18">X1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c19">X2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> ;<br/><a NAME="E33:97_1"/><i><font color="Green" title="E85">A57</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_12"><b>proof </b></a><div class="add">

<a NAME="E1:97_1_12"/><i><font color="Green" title="E86">A58</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> holds   <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/>

<b>set </b><font color="Maroon" title="c21">X3</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span> ;<br/>
<a NAME="E1:97_1_12_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span>  iff  ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">u</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:97_1_12_1_1_1"/>
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c23">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><b> such that </b><br/><a NAME="E3:97_1_12_1_1_1"/><i><font color="Green" title="E87">A59</font></i>: 
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c23">i</font>
 <b>and </b><br/><a NAME="E4:97_1_12_1_1_1"/>
<font color="Maroon" title="c23">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c24">u</font> = <font color="Maroon" title="c23">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c25">u</font> = <font color="Maroon" title="c24">u</font>;<br/><a NAME="E6:97_1_12_1_1_1"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ;<br/><b>hence </b><a NAME="E7:97_1_12_1_1_1"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E8:97_1_12_1_1_1"/>
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">u</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_12_1_1_1"><i><font color="Green" title="E87">A59</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c23">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:97_1_12_1_1"/><i><font color="Green" title="E87">A60</font></i>: 
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:97_1_12_1_1"/><i><font color="Green" title="E88">A61</font></i>: 
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">u</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c24">i</font> = <font color="Maroon" title="c23">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> <b>by </b><i><a class="txt" href="waybel11.html#E3:97_1_12_1_1"><i><font color="Green" title="E87">A60</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:97_1_12_1_1"/><i><font color="Green" title="E89">A62</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:97_1"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/>
<a NAME="E7:97_1_12_1_1"/>
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c24">i</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:97_1_12_1_1"><i><font color="Green" title="E88">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:97_1_12_1_1"/>
<font color="Maroon" title="c22">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel11.html#E6:97_1_12_1_1"><i><font color="Green" title="E89">A62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:97_1_12_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:97_1_12_1"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:97_1_12"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1_12_2"/>
<font color="Maroon" title="c20">u</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#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font><b> such that </b><br/><a NAME="E3:97_1_12_2"/><i><font color="Green" title="E87">A63</font></i>: 
<font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c21">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 ;<br/>
<a NAME="E4:97_1_12_2"/>
<font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:97_1_12"><i><font color="Green" title="E86">A58</font></i></a>, <a class="txt" href="waybel11.html#E3:97_1_12_2"><i><font color="Green" title="E87">A63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:97_1_12_2"/>
<font color="Maroon" title="c20">u</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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c20">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>consider </b><font color="Maroon" title="c21">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font>;<br/>
<b>assume </b><a NAME="E4:97_1_12"/>
<font color="Maroon" title="c20">u</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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E5:97_1_12"/><b>then </b>
 ex <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> st <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 
;<br/>
<a NAME="E6:97_1_12"/><b>then </b>
<font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c21">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:97_1_12"><i><font color="Green" title="E86">A58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:97_1_12"/>
<font color="Maroon" title="c20">u</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#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:97_1_13"/>
( <font color="Maroon" title="c20">u</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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span>  iff  ex <font color="Olive" title="b1">j'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> st <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )
 ;<br/><a NAME="E2:97_1_13"/><b>then </b>
( <font color="Maroon" title="c20">u</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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span>  iff <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )
 ;<br/><b>hence </b><a NAME="E3:97_1_13"/>
( <font color="Maroon" title="c20">u</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_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span> where <font color="Olive" title="b1">j'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span>  iff <font color="Maroon" title="c20">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b2">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">X</font> : verum </span>}</span> ,<font color="Maroon" title="c1">L</font> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">x'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="waybel11.html#E33:97_1"><i><font color="Green" title="E85">A57</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>
<br/>.= 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>
;<br/><a NAME="E36:97_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c10">X</font>
 ;<br/><a NAME="E37:97_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c10">X</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></i>;<br/><a NAME="E38:97_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E22:97_1"><i><font color="Green" title="E79">A19</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></i>;<br/><a NAME="E39:97_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:97"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="waybel11.html#E21:97_1"><i><font color="Green" title="E78">A16</font></i></a>, <a class="ref" href="yellow_6.html#D26" title="YELLOW_6:def.26">YELLOW_6:def 26</a></i>;<br/><a NAME="E40:97_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a>  <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c12">J</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E23:97_1"><i><font color="Green" title="E80">A20</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></i>;<br/><a NAME="E41:97_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel11.html#E31:97_1"><i><font color="Green" title="E84">A56</font></i></a>, <a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E42:97_1"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel11.html#E4:97_1"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel_5.html#T18" title="WAYBEL_5:th.18">WAYBEL_5:18</a></i>;<br/>


</div>
