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

<b>let </b><font color="Maroon" title="c1">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Olive" title="b1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span><br/><b>let </b><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <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="c4">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="setlim_2.html#D8" target="_self" title="SETLIM_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="setlim_2.html#D8" target="_self" title="SETLIM_2:def.8">Def8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="setlim_2.html#K8" title="SETLIM_2:func.8">(\)</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
