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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font>;<br/>
<a NAME="E1:5_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is    <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="lattice7.html#T1" target="_self" title="LATTICE7:th.1">Th1</a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<a NAME="E2:5_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</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="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>
<a NAME="E2:5_1_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>]
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:5_1_1"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:5_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:5_1_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="lattice7.html#E1:5_1_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="lattice7.html#E2:5_1_1_1"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">k</font>
;<br/>

<b>thus </b><a NAME="E6:5_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> ) )
 <b>by </b><i><a class="txt" href="lattice7.html#E4:5_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:5_1_1"/><i><font color="Green" title="E4">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> ) )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c3">k</font>
;<br/>

<b>thus </b><a NAME="E5:5_1_1"/>
(  ex <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> st <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="lattice7.html#M1" title="LATTICE7:mode.1">Chain</a> of  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> ) )
 <b>by </b><i><a class="txt" href="lattice7.html#E4:5_1_1"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/>


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