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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2 implies ( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</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>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) )</span><br/>

<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E117">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</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>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )</span><br/>

<a NAME="E2:115"/><b>then </b>
( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</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>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="sincos10.html#T108" target="_self" title="SINCOS10:th.108">Th108</a></i>;<br/>
<a NAME="E3:115"/><b>then </b>
( ( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</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>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) or <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> or  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:115"/>
( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</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>  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> &amp;  <a href="sincos10.html#K12" title="SINCOS10:func.12">arccosec2</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <b>by </b><i><a class="txt" href="sincos10.html#E1:115"><i><font color="Green" title="E117">A1</font></i></a>, <a class="ref" href="sincos10.html#T32" target="_self" title="SINCOS10:th.32">Th32</a>, <a class="ref" href="sincos10.html#T92" target="_self" title="SINCOS10:th.92">Th92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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