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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:95"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D15" target="_self">COMPLSP1:def 15</a><br/>

<b>assume </b><a NAME="E2:95"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="complsp1.html#K19" target="_self">Ball</a> <font color="Maroon">A</font>,<font color="Maroon">r1</font>
 ;<br/>

<b>take </b><font color="Maroon">r</font> = <font color="Maroon">r1</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span>;<br/>

<a NAME="E3:95"/>
 <a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T45" target="_self">Th45</a>, <a class="ref" href="complsp1.html#T37" target="_self">Th37</a></i>;<br/>
<b>hence </b><a NAME="E4:95"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>

<b>let </b><font color="Maroon">z</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/>

<a NAME="E5:95"/><i><font color="Green">E102</font></i>: 
 <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T44" target="_self">Th44</a>, <a class="ref" href="complsp1.html#T31" target="_self">Th31</a></i>;<br/>
<b>assume </b><a NAME="E6:95"/>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E7:95"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:95"/><b>then </b>
 <a href="complsp1.html#K18" target="_self">dist</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T46" target="_self">Th46</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:95"/>
<font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="complsp1.html#K19" target="_self">Ball</a> <font color="Maroon">A</font>,<font color="Maroon">r1</font>
 ;<br/>


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