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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K1" target="_self">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:5_1_1_1_1"/>
 <a href="complex1.html#K1" target="_self">Re</a> <font color="Maroon">z</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_2"/>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 2, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:5_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:5_1_1_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="complex1.html#K1" target="_self">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:5_1_1_1_2"/>
0 <a href="hidden.html#R2">in</a> 2
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:5_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E7:5_1_1_1_2"/>
 <a href="complex1.html#K1" target="_self">Re</a> <font color="Maroon">z</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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