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<div class="add">

<b>let </b><font color="Maroon">n0</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>;<br/>
<b>consider </b><font color="Maroon">NS</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:59"/><i><font color="Green">E27</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">NS</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E3:59"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">NS</font> is <a href="seqm_3.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a><br/>

<a NAME="E1:59_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</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="E2:59_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:59_1"/><b>then </b>
<font color="Maroon">NS</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:59_1"/>
<font color="Maroon">NS</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">NS</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/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/>
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">NS</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><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/>

<a NAME="E1:59_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E2:59_2"/>
<font color="Maroon">NS</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/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">NS</font> = <font color="Maroon">NS</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="frechet2.html#E9"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="seqm_3.html#T29">SEQM_3:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">NS</font>
;<br/>

<b>thus </b><a NAME="E6:59"/>
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">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 <b>by </b><i/>;<br/>


</div>
