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<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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Olive">n</font>) ) &amp; ( <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  ) );<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<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="E3:3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="sin_cos.html#K1" target="_self">CHK</a> <font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>) = 
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>)

;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="sin_cos.html#K1" target="_self">CHK</a> <font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>) = 
0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>)
<b>by </b><i/>
<br/>.= 
0

;<br/><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="sin_cos.html#K1" target="_self">CHK</a> <font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>) as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E7:3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E2:3_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:3_1"/><i><font color="Green">E61</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>and </b><br/><a NAME="E5:3_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:3_1_2"/><b>then </b>
 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#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Olive">n</font>) ) &amp; ( <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:3_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> 
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="comseq_1.html#NM1">Complex_Sequence</a> <b>by </b><i>, <a class="ref" href="comseq_1.html#T1">COMSEQ_1:1</a></i>;<br/>
<b>take </b><font color="Maroon">seq</font> = <font color="Maroon">f</font>;<br/>

<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="E8:3_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="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Olive">l</font>) ) &amp; ( <font color="Olive">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> implies <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:3_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">n</font>) ) &amp; ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> implies <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/>


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