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

<span class="kw">let </span><font color="Maroon" title="c1">A</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#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">F</font> <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<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="c2">F</font> <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font>
 ; <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">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E2:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">A</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> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_8.html#D8" target="_self" title="FUNCT_8:def.8">Def8</a></span>;<br/>
<a NAME="E3:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c1">A</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D7" title="VALUED_1:def.7">VALUED_1:def 7</a></span>;<br/>
<a NAME="E4:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</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="E5:32"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font> is  <a href="funct_8.html#V6" title="FUNCT_8:attr.6">odd</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="funct_8.html#D8" target="_self" title="FUNCT_8:def.8">Def8</a></span>;<br/>
<a NAME="E6:32"/>
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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="c3">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="c3">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span><br/>

<a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c3">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"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">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"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="c3">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </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="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </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="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="valued_1.html#D7" title="VALUED_1:def.7">VALUED_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">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="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">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="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="funct_8.html#D6" target="_self" title="FUNCT_8:def.6">Def6</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">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="#E3:32_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1: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"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">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="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="valued_1.html#D7" title="VALUED_1:def.7">VALUED_1:def 7</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"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">A6</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="E6:32_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="c3">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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="E7:32"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</font> is  <a href="funct_8.html#V2" title="FUNCT_8:attr.2">with_symmetrical_domain</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c1">A</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="#E4:32"><span class="lab"><font color="Green" title="E15">A4</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="E8:32"/>
<font color="Maroon" title="c2">F</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  <a href="funct_8.html#R2" title="FUNCT_8:pred.2">is_odd_on</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E14">A3</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>
