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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <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="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> &amp; <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span> &amp; <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font> implies <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font></span><br/><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</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="c1">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/><a NAME="E10:24_1"/><span class="lab"><font color="Green" title="E50">A10</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="c2">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_1_1"/>
( <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font></span><br/><div class="add"><a NAME="E2:24_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:24_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">r1</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> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></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="#E9:24_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/><a NAME="E4:24_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<font color="Maroon" title="c1">r1</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> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:24_1_1_1_1"/>
<font color="Maroon" title="c2">r2</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> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></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="#E10:24_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>;<br/><a NAME="E6:24_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">r2</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> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:24_1_1_1_1"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E4:24_1_1_1_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="sincos10.html#T16" target="_self" title="SINCOS10:th.16">Th16</a>, <a class="ref" href="rfunct_2.html#T44" title="RFUNCT_2:th.44">RFUNCT_2:44</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font></span><br/><div class="add"><a NAME="E2:24_1_1_1_2"/><span class="kw">then </span>
 <a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="sin_cos6.html#T30" title="SIN_COS6:th.30">SIN_COS6:30</a></span>;<br/><a NAME="E3:24_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a></span>;<br/><a NAME="E4:24_1_1_1_2"/>
<a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="sincos10.html#E7"><span class="lab"><font color="Green" title="E7">Lm4</font></span></a>, <a class="ref" href="comptrig.html#T23" title="COMPTRIG:th.23">COMPTRIG:23</a></span>;<br/><a NAME="E5:24_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A15</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_1_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></span>;<br/><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> = 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E1:24_1_1_1_2"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a>, <a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a></span>
<br/>.= 
1

;<br/><span class="kw">hence </span><a NAME="E7:24_1_1_1_2"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E5:24_1_1_1_2"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:24_1"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:24"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> is  <a href="valued_0.html#V8" title="VALUED_0:attr.8">decreasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T44" title="RFUNCT_2:th.44">RFUNCT_2:44</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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