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<div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E2:22"/><i><font color="Green">E17</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E18</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>then </b> <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="absvalue.html#K2" target="_self">sgn</a> 1
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:22_1"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<b>then </b> <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="absvalue.html#K2" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:22_2"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div>
