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<span class="kw">let </span><font color="Maroon" title="c1">B</font> be   <a href="funct_8.html#V1" title="FUNCT_8:attr.1">symmetrical</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:102"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">B</font></span><br/>

<a NAME="E2:102"/>
<font color="Maroon" title="c1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="membered.html#D9" title="MEMBERED:def.9">MEMBERED:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c2">x</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><br/>

<span class="kw">assume </span><a NAME="E1:102_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</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><br/>

<a NAME="E2:102_1"/><span class="kw">then </span>
<a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E1:102"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/>
<a NAME="E3:102_1"/><span class="kw">then </span>
 not <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:102_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E5:102_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="sin_cos.html#T24" title="SIN_COS:th.24">SIN_COS:24</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:102_1"/>
<font color="Maroon" title="c2">x</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="ref" href="rfunct_1.html#D2" title="RFUNCT_1:def.2">RFUNCT_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:102"/>
 <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_8.html#T83" target="_self" title="FUNCT_8:th.83">Th83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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