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<a NAME="E1:5_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is  <a href="funct_8.html#V1" title="FUNCT_8:attr.1">symmetrical</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:5_1_1_1"/>
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is  <a href="funct_8.html#V1" title="FUNCT_8:attr.1">symmetrical</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:5_1_1_1"/><span class="kw">then </span>
 ex <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 <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  not  <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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></span>;<br/>
<span class="kw">hence </span><a NAME="E3:5_1_1_1"/>
contradiction
 ; <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>
<a NAME="E2:5_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:5_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive">b<sub>1</sub></font> is  <a href="xcmplx_0.html#NV2" title="XCMPLX_0:NV.2">zero</a>  holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="funct_8.html#V2" title="FUNCT_8:attr.2">with_symmetrical_domain</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="funct_8.html#D2" target="_self" title="FUNCT_8:def.2">Def2</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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