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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font>;<br/>
<b>consider </b><font color="Maroon">XN</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:47"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:47_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 ;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:47_1_1"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:47_1_1"/><b>then </b>
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:47_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E5:47_1_1"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E6:47_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a></i>;<br/><b>thus </b><a NAME="E7:47_1_1"/>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">seq1</font> = <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>;<br/><a NAME="E3:47_1"/>
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T47">SEQM_3:47</a></i>;<br/><b>then consider </b><font color="Maroon">Nseq</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E5:47_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:47_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:47_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:47_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E6:47_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:47_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E10:47_1_2"/>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:47"><i><font color="Green">E46</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:47_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 ;<br/><b>take </b><font color="Maroon">l</font> = <font color="Maroon">n1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2_1"/>
not <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:47_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:47_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:47_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E15:47_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E16:47_1_2"/>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E17:47_1_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E18:47_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:47_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) );<br/><a NAME="E7:47_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> );<br/>
<a NAME="E1:47_1_3"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E2:47_1_3"/>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:47_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E5:47_1_3"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:47_1_3"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font>
 ;<br/>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E7:47_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i><a class="txt" href="seq_4.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="txt" href="seq_4.html#E1:47_1"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">0'</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:47_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font>
 <b>and </b><br/><a NAME="E11:47_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="ref" href="seq_4.html#T8" target="_self">Th8</a>);<br/></i><a NAME="E12:47_1"/><i><font color="Green">E102</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E13:47_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:47_1"/><i><font color="Green">E104</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:47_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:47_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E3:47"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_1_5"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</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> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47_1"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E2:47_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</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> 1</span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="seq_4.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><b>take </b><font color="Maroon">Nseq1</font> = <font color="Maroon">Nseq</font> <a href="seqm_3.html#K4">*</a> <font color="Maroon">f</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E2:47_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <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> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><a NAME="E3:47_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <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> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><a NAME="E4:47_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font></span>)</span> <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> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/><b>hence </b><a NAME="E5:47_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font></span>)</span> <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> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/></div><b>end;</b></div><a NAME="E20:47_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font> is <a href="seqm_3.html#V4">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D14">SEQM_3:def 14</a></i>;<br/><b>hence </b><a NAME="E21:47_1"/>
<font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_2"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; not <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/>( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">l</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) );<br/><a NAME="E2:47_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:47_2_1"/><i><font color="Green">E109</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E2:47_2_1"/>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:47_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font>
 <b>and </b><br/><a NAME="E5:47_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E6:47_2_1"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font>
 ;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">l</font>;<br/>

<b>thus </b><a NAME="E7:47_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="seq_4.html#T8" target="_self">Th8</a>, <a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:47_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font> &amp; not <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">0'</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:47_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font>
 <b>and </b><br/><a NAME="E8:47_2"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>);<br/></i><a NAME="E9:47_2"/><i><font color="Green">E116</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E10:47_2"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:47_2"/><i><font color="Green">E118</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="seq_4.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:47_2_2"/>
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:47_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:47_2_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a></i>;<br/><b>thus </b><a NAME="E4:47_2_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font>;<br/><a NAME="E1:47_2_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a></i>;<br/><div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:47_2_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>]
 ;<br/><a NAME="E2:47_2_3_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:47_2_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E3:47_2_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/></div><b>end;</b></div><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_2_4"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</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> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:47_2_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</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> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="seq_4.html#E2:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><b>take </b><font color="Maroon">Nseq</font> = <font color="Maroon">f</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:47_2_5"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XN</font>
 <b>by </b><i/>;<br/><a NAME="E2:47_2_5"/>
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:47_2_5"/><b>then </b>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:47_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E5:47_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font></span>)</span> <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> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/></div><b>end;</b></div><a NAME="E18:47_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font> is <a href="seqm_3.html#V1">increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><a NAME="E19:47_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T23">SEQM_3:23</a></i>;<br/><b>hence </b><a NAME="E20:47_2"/>
<font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
 ex <font color="Olive">Nseq</font> being <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> st <font color="Maroon">seq</font> <a href="seqm_3.html#K3">*</a> <font color="Olive">Nseq</font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i/>;<br/>


</div>
