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<b>let </b><font color="Maroon" title="c3">th1</font>, <font color="Maroon" title="c4">th2</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:122_1_2"/><i><font color="Green" title="E101">A4</font></i>: 
( <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">th1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>and </b><br/><a NAME="E2:122_1_2"/><i><font color="Green" title="E102">A5</font></i>: 
( <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">th2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/>

<a NAME="E3:122_1_2"/>
( <font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:122_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font> &amp; <font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4 )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:122_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E2:122_1_2_1"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 &amp; <font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E3:122_1_2_1"/>
<font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>

<a NAME="E4:122_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font> &amp; <font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4 )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:122_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:122_1_2_1"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 &amp; <font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E6:122_1_2_1"/>
<font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:122_1_2"/><b>then </b>
<font color="Maroon" title="c3">th1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">th2</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:122_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="txt" href="sin_cos.html#E2:122_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="ref" href="sin_cos.html#T77" target="_self" title="SIN_COS:th.77">Th77</a></i>;<br/>
<b>hence </b><a NAME="E5:122_1_2"/>
<font color="Maroon" title="c3">th1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">th2</font>
 ;<br/>


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