<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">r1</font> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">r1</font> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:162"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_4.html#D17" target="_self" title="SEQ_4:def.17">SEQ_4:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> implies  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:162"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> ) )</span><br/>

<a NAME="E3:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A2</font></span>: 
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T136" target="_self" title="SEQ_4:th.136">Th90</a></span>;<br/>
<span class="kw">take </span><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">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:162"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>  st <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> implies <font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:162"/>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font></span><br/>

<a NAME="E6:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A3</font></span>: 
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:162"/>
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="seq_4.html#T132" target="_self" title="SEQ_4:th.132">Th86</a></span>;<br/>
<a NAME="E8:162"/><span class="kw">then </span>
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:162"/>
<font color="Maroon" title="c4">x</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
