<?xml version="1.0"?>
<div class="add">

<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="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  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="E1:101"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<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> 
 ; <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="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">B</font></span><br/>

<a NAME="E2:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E3:101"/><span class="lab"><font color="Green" title="E31">A3</font></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> <font color="Maroon" title="c1">B</font> holds <br/><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_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="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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> implies <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1"/><span class="lab"><font color="Green" title="E32">A4</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="Red">::  thesis: </font></span></a><span class="hide"> <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<a NAME="E2:101_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <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>
 
<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></span>;<br/>
<span class="kw">then </span><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<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> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<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="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T33" title="SIN_COS:th.33">SIN_COS:33</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<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="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T224" title="XCMPLX_1:th.224">XCMPLX_1:224</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:101_1"/>
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</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>
<a NAME="E4:101"/>
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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> &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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</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="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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> implies <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_2"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:101_2"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <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> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></span>
<br/>.= 
<a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:101_2"/>
<span class="p1">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</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>
<a NAME="E5:101"/><span class="kw">then </span>
( <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font> is  <a href="funct_8.html#V2" title="FUNCT_8:attr.2">with_symmetrical_domain</a>  &amp; <a href="fdiff_9.html#K2" title="FDIFF_9:func.2">cosec</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">B</font> 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="txt" href="#E2:101"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <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></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101"/>
 <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="txt" href="#E1:101"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_8.html#D8" target="_self" title="FUNCT_8:def.8">Def8</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>
