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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>







<a NAME="E1:29"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E2:29"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> implies  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="setlim_1.html#E1:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k11</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:29_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k11</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k11</font> )
 <b>by </b><i><a class="txt" href="setlim_1.html#E3:29_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E6:29_1"/>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">k</font>
 
<b>by </b><i><a class="txt" href="setlim_1.html#E5:29_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">k22</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:29_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k22</font></span>)</span>
 <b>by </b><i><a class="txt" href="setlim_1.html#E5:29_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E9:29_1"/>
<font color="Maroon">k22</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="setlim_1.html#E3:29_1"><i><font color="Green">E29</font></i></a>, </i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 ;<br/><a NAME="E4:29_2"/>
<font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:29_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="setlim_1.html#E1:29"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 ;<br/>


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