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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<a NAME="E1:81"/><i><font color="Green" title="E58">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:81_1"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>and </b><br/><a NAME="E2:81_1"/><i><font color="Green" title="E59">A2</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 ;<br/><b>take </b><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">r</font>;<br/><b>thus </b><a NAME="E3:81_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:81_1"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c5">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E4:81_1"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>and </b><br/><a NAME="E5:81_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font>
 ;<br/><i><font color="Green" title="E60">A3</font></i>: <span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">x1</font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:81"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x2</font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:81"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E8:81_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:81_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="polynom5.html#E6:81_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">x</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <b>by </b><i><a class="txt" href="polynom5.html#E1:81"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="cfcont_1.html#T61" title="CFCONT_1:th.61">CFCONT_1:61</a></i>;<br/>


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