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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">z1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D15" target="_self" title="COMPLSP1:def.15">COMPLSP1:def 15</a><br/>







<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="complsp1.html#K17" title="COMPLSP1:func.17">Ball</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c2">r1</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c2">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>;<br/>

<a NAME="E2:81"/>
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">r1</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:81"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="complsp1.html#T80" target="_self" title="COMPLSP1:th.80">Th80</a></i>;<br/>
<b>hence </b><a NAME="E3:81"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>;<br/>

<a NAME="E4:81"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="complsp1.html#K9" title="COMPLSP1:func.9">+</a> <font color="Maroon" title="c6">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T45" target="_self" title="COMPLSP1:th.45">Th45</a></i>;<br/>
<a NAME="E5:81"/><b>then </b><i><font color="Green" title="E51">A2</font></i>: 
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="complsp1.html#K9" title="COMPLSP1:func.9">+</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T69" target="_self" title="COMPLSP1:th.69">Th69</a></i>;<br/>
<b>assume </b><a NAME="E6:81"/>
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 ;<br/>

<a NAME="E7:81"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:81"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="complsp1.html#K9" title="COMPLSP1:func.9">+</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>
 
<b>by </b><i><a class="txt" href="complsp1.html#E5:81"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:81"/>
<font color="Maroon" title="c4">x</font> <a href="complsp1.html#K9" title="COMPLSP1:func.9">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="complsp1.html#K17" title="COMPLSP1:func.17">Ball</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c2">r1</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T80" target="_self" title="COMPLSP1:th.80">Th80</a></i>;<br/>


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