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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/>(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">s</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K3">|^</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">s</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K3">|^</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span> ) ) )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E8:24"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E9:24"/><i><font color="Green">E54</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:24"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:24"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K3">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, </i>;<br/>


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