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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <a href="sin_cos.html#K33" title="SIN_COS:func.33">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> &amp; <a href="sin_cos.html#K33" title="SIN_COS:func.33">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> implies <font color="Maroon" title="c3">th1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">th2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:123_1_2"/><span class="lab"><font color="Green" title="E97">A11</font></span>: 
<a href="sin_cos.html#K33" title="SIN_COS:func.33">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
 <span class="kw">and </span><br/><a NAME="E2:123_1_2"/><span class="lab"><font color="Green" title="E98">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:123_1_2"/><span class="lab"><font color="Green" title="E99">A13</font></span>: 
<a href="sin_cos.html#K33" title="SIN_COS:func.33">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
 <span class="kw">and </span><br/><a NAME="E4:123_1_2"/><span class="lab"><font color="Green" title="E100">A14</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">th1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">th2</font></span><br/>

<a NAME="E5:123_1_2"/><span class="lab"><font color="Green" title="E101">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">th1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E98">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E6:123_1_2"/><span class="lab"><font color="Green" title="E102">A16</font></span>: 
<font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:123_1_2"><span class="lab"><font color="Green" title="E98">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E7:123_1_2"/><span class="lab"><font color="Green" title="E103">A17</font></span>: 
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123_1_2"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E8:123_1_2"/><span class="lab"><font color="Green" title="E104">A18</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:123_1_2"><span class="lab"><font color="Green" title="E101">A15</font></span></a>, <a class="txt" href="#E7:123_1_2"><span class="lab"><font color="Green" title="E103">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E9:123_1_2"/><span class="lab"><font color="Green" title="E105">A19</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">th2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:123_1_2"><span class="lab"><font color="Green" title="E100">A14</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E10:123_1_2"/><span class="lab"><font color="Green" title="E106">A20</font></span>: 
<font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:123_1_2"><span class="lab"><font color="Green" title="E100">A14</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E11:123_1_2"/><span class="lab"><font color="Green" title="E107">A21</font></span>: 
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:123_1_2"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E12:123_1_2"/><span class="lab"><font color="Green" title="E108">A22</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:123_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="txt" href="#E11:123_1_2"><span class="lab"><font color="Green" title="E107">A21</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E13:123_1_2"/><span class="lab"><font color="Green" title="E109">A23</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123_1_2"><span class="lab"><font color="Green" title="E97">A11</font></span></a>, <a class="txt" href="#E3:123_1_2"><span class="lab"><font color="Green" title="E99">A13</font></span></a>, <a class="txt" href="#E8:123_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="txt" href="#E12:123_1_2"><span class="lab"><font color="Green" title="E108">A22</font></span></a>, <a class="ref" href="sin_cos.html#T77" target="_self" title="SIN_COS:th.77">Th77</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:123_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:123_1_2"><span class="lab"><font color="Green" title="E109">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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