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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/>for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st not <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/> <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">B</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> implies for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/> <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">A</font> )</span><br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/> <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">A</font></span><br/>

<b>let </b><font color="Maroon" title="c3">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green" title="E39">A2</font></i>: 
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E3:51"/><i><font color="Green" title="E40">A3</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E1:51"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="jordan1k.html#T41" target="_self" title="JORDAN1K:th.41">Th41</a></i>;<br/>
<b>assume </b><a NAME="E4:51"/>
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:51"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E3:51"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:51"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E8:51"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T46" target="_self" title="JORDAN1K:th.46">Th46</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P</font> = <font color="Maroon" title="c2">B</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E11:51"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E1:51"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1k.html#E7:51"><i><font color="Green" title="E42">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E13:51"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E14:51"/><i><font color="Green" title="E46">A9</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:51"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1k.html#E7:51"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">q'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<a NAME="E16:51"/><i><font color="Green" title="E47">A10</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1k.html#E2:51"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1k.html#E11:51"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="jordan1k.html#T22" target="_self" title="JORDAN1K:th.22">Th22</a></i>;<br/>
<a NAME="E17:51"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E18:51"/><b>then </b><i><font color="Green" title="E49">A12</font></i>: 
 <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="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E13:51"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<b>then </b> <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c4">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan1k.html#T28" target="_self" title="JORDAN1K:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85" title="XCMPLX_1:th.85">XCMPLX_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="jordan1k.html#E16:51"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E20:51"/><b>then </b><i><font color="Green" title="E50">A13</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c4">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
 
<b>by </b><i><a class="txt" href="jordan1k.html#E13:51"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="real_2.html#T200" title="REAL_2:th.200">REAL_2:200</a></i>;<br/>
<a NAME="E21:51"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp;  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c4">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">q1</font> )
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E22:51"/><b>then </b><i><font color="Green" title="E51">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E20:51"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E52">A15</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">q1</font> )
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><a NAME="E3:51_2"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:51_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><b>hence </b><a NAME="E4:51_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1k.html#E2:51"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1k.html#E14:51"><i><font color="Green" title="E46">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E24:51"/><b>then </b>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E13:51"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/>
<a NAME="E25:51"/><b>then </b>
<font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan1k.html#E17:51"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="real_2.html#T118" title="REAL_2:th.118">REAL_2:118</a></i>;<br/>
<a NAME="E26:51"/><b>then </b><i><font color="Green" title="E52">A16</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1k.html#T27" target="_self" title="JORDAN1K:th.27">Th27</a></i>;<br/>
<a NAME="E27:51"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1k.html#E18:51"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/>
<a NAME="E28:51"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E16:51"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="jordan1k.html#E26:51"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="real_2.html#T145" title="REAL_2:th.145">REAL_2:145</a></i>;<br/>
<b>hence </b><a NAME="E29:51"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1k.html#E5:51"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1k.html#E8:51"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="jordan1k.html#E14:51"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="jordan1k.html#E22:51"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="jordan1k.html#T50" target="_self" title="JORDAN1K:th.50">Th50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
