<?xml version="1.0"?>
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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compts_1.html#D5" title="COMPTS_1:def.5">COMPTS_1:def 5</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>)  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  or  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>) st <br/>( <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>3</sub></font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c3">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  or  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>) st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">P</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> 
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
( <font color="Maroon" title="c3">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ; <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>) st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">P</font> as   <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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">z1</font> = <font color="Maroon" title="c2">p</font> as    <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> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">d</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">K1</font> =  <a href="seq_4.html#K23" title="SEQ_4:func.23">Ball</a> <font color="Maroon" title="c5">z1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>, <font color="Maroon" title="c8">K2</font> =  <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">K1</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K1" title="COMPLSP1:func.1">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>) st <br/>( <font color="Maroon" title="c7">K1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c7">K1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">K2</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="c7">K1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">K2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">K2</font> &amp; <font color="Maroon" title="c7">K1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">K2</font> )</span><br/>

<a NAME="E6:8"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="seq_4.html#K23" title="SEQ_4:func.23">Ball</a> <font color="Maroon" title="c5">z1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T129" title="SEQ_4:th.129">SEQ_4:129</a></span>;<br/>
<a NAME="E7:8"/>
 <a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> is  <a href="seq_4.html#V1" title="SEQ_4:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="seq_4.html#T139" title="SEQ_4:th.139">SEQ_4:139</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:8"/>
( <font color="Maroon" title="c7">K1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">K2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="complsp1.html#T6" target="_self" title="COMPLSP1:th.6">Th108</a></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="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">K2</font> &amp; <font color="Maroon" title="c7">K1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">K2</font> )</span><br/>

<a NAME="E9:8"/>
( <font color="Maroon" title="c4">A</font> is  <a href="seq_4.html#V2" title="SEQ_4:attr.2">closed</a>  &amp; not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="complsp1.html#T7" target="_self" title="COMPLSP1:th.7">Th110</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:8"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="seq_4.html#T134" title="SEQ_4:th.134">SEQ_4:134</a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:8"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T128" title="SEQ_4:th.128">SEQ_4:128</a>, <a class="ref" href="seq_4.html#T138" title="SEQ_4:th.138">SEQ_4:138</a></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="c7">K1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">K2</font></span><br/>

<span class="kw">assume </span><a NAME="E12:8"/>
<font color="Maroon" title="c7">K1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">K2</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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</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><span class="kw"> such that </span><br/><a NAME="E14:8"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="seq_4.html#K23" title="SEQ_4:func.23">Ball</a> <font color="Maroon" title="c5">z1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="seq_4.html#K25" title="SEQ_4:func.25">Ball</a> <font color="Maroon" title="c4">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<a NAME="E15:8"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T136" title="SEQ_4:th.136">SEQ_4:136</a></span>;<br/>
<a NAME="E17:8"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:8"/><span class="kw">then </span>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">x</font></span>)</span></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> <span class="p1">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T127" title="SEQ_4:th.127">SEQ_4:127</a></span>;<br/>
<a NAME="E19:8"/><span class="kw">then </span>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c9">x</font></span>)</span></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="c9">x</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K24" title="SEQ_4:func.24">dist</a> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="seq_4.html#T135" title="SEQ_4:th.135">SEQ_4:135</a></span>; <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>
