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<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">k</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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:79"/><i><font color="Green">E55</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>  holds  1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:79_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E2:79_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> ;<br/>
<b>consider </b><font color="Maroon">rseq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:79"/><i><font color="Green">E56</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>  holds  <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:79"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">rseq</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<a NAME="E1:79_2"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</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>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T18">NEWTON:18</a></i>
;<br/>
<a NAME="E2:79_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self">Def8</a>, , <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<a NAME="E3:79_2_1"/><i><font color="Green">E62</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:79_2_1_2"/>
( <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:79_2_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<i><font color="Green">E92</font></i>: <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:79_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:79_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="sin_cos.html#D9" target="_self">Def9</a>, <a class="ref" href="real_2.html#T146">REAL_2:146</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:79_2_1"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:79_2_1"/>
( <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2"/>
<font color="Maroon">rseq</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D13">SEQM_3:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/><b>then </b>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/>
<a NAME="E7:79"/><b>then </b>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="sin_cos.html#K5" target="_self">!</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:79"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">m</font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<a NAME="E9:79"/>
( <font color="Maroon">n</font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:79"/>
( <font color="Maroon">m</font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="sin_cos.html#K5" target="_self">!</a>  )
 <b>by </b><i><a class="ref" href="sin_cos.html#D4" target="_self">Def4</a>, <a class="ref" href="sin_cos.html#D5" target="_self">Def5</a>, <a class="ref" href="sin_cos.html#D8" target="_self">Def8</a>, <a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/>


</div>
