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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0;<br/>
<b>thus </b>0 <a href="sin_cos.html#K4" target="_self">!c</a>  = 
<a href="sin_cos.html#K2" target="_self">Prod_complex_n</a>  <a href="comseq_1.html#K1">.</a> 0

<br/>.= 
1
<b>by </b><i><a class="ref" href="sin_cos.html#D4" target="_self">Def4</a></i>
;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><i><font color="Green">E10</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  = 
<a href="sin_cos.html#K2" target="_self">Prod_complex_n</a>  <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K2" target="_self">Prod_complex_n</a>  <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>

;<br/><b>thus </b><a NAME="E3:11_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:11_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="sin_cos.html#E2:11_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:11"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sin_cos.html#E2:11"><i><font color="Green">E7</font></i></a>, <a class="txt" href="sin_cos.html#E3:11"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  = 
<a href="sin_cos.html#K2" target="_self">Prod_complex_n</a>  <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K2" target="_self">Prod_complex_n</a>  <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>


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