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<a NAME="E1:81_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</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>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:81_1_1"/><span class="kw">then </span>
( ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">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="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Olive" title="b1">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="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  holds <br/><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) &amp;  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  is  <a href="funct_8.html#V1" title="FUNCT_8:attr.1">symmetrical</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_8.html#D1" target="_self" title="FUNCT_8:def.1">Def1</a>, <a class="ref" href="sin_cos.html#T33" title="SIN_COS:th.33">SIN_COS:33</a></span>;<br/>
<a NAME="E3:81_1_1"/><span class="kw">then </span>
(  <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  is  <a href="funct_8.html#V2" title="FUNCT_8:attr.2">with_symmetrical_domain</a>  &amp;  <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  is  <a href="funct_8.html#V5" title="FUNCT_8:attr.5">quasi_odd</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_8.html#D2" target="_self" title="FUNCT_8:def.2">Def2</a>, <a class="ref" href="funct_8.html#D6" target="_self" title="FUNCT_8:def.6">Def6</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:81_1_1"/>
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  is  <a href="funct_8.html#V6" title="FUNCT_8:attr.6">odd</a> 
 ; <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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