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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font></span><br/>

<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</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#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">e</font> = <font color="Maroon" title="c3">x</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" title="EUCLID:th.67">EUCLID:67</a></span>;<br/>
<a NAME="E6:38"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Lim_inf</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan19.html#D4" title="JORDAN19:def.4">JORDAN19:def 4</a></span>;<br/>
<a NAME="E7:38"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T27" title="JORDAN2C:th.27">JORDAN2C:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">OO</font> =  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> as   <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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="jordan22.html#E36"><span class="lab"><font color="Green" title="E31">Lm4</font></span></a></span>;<br/><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c5">OO</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="jordan22.html#E36"><span class="lab"><font color="Green" title="E31">Lm4</font></span></a>, <a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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> <span class="kw"> such that </span><br/><a NAME="E5:38_1_1"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c6">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> 
 <span class="kw">and </span><br/><a NAME="E6:38_1_1"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c6">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:38_1_1"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c6">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:38_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="kurato_2.html#T14" title="KURATO_2:th.14">KURATO_2:14</a></span>;<br/><a NAME="E9:38_1_1"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T61" title="JORDAN6:th.61">JORDAN6:61</a></span>;<br/><a NAME="E10:38_1_1"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></span>;<br/><a NAME="E11:38_1_1"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E12:38_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c6">r</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38_1_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="jordan10.html#T19" title="JORDAN10:th.19">JORDAN10:19</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E13:38_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E11:38_1_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="txt" href="#E10:38_1_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E9:38_1_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:38_1_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="jordan10.html#K3" title="JORDAN10:func.3">UBD-Family</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan10.html#T14" title="JORDAN10:th.14">JORDAN10:14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:38_1_2"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E5:38_1_2"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan10.html#K3" title="JORDAN10:func.3">UBD-Family</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E6:38_1_2"/>
 <a href="jordan10.html#K3" title="JORDAN10:func.3">UBD-Family</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">m</font>)</span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan10.html#D1" title="JORDAN10:def.1">JORDAN10:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">m</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:38_1_2"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">OO</font> =  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</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="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="jordan22.html#E36"><span class="lab"><font color="Green" title="E31">Lm4</font></span></a></span>;<br/><a NAME="E10:38_1_2"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c7">OO</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="jordan22.html#E36"><span class="lab"><font color="Green" title="E31">Lm4</font></span></a>, <a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/><a NAME="E11:38_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E8:38_1_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="gobrd14.html#T36" title="GOBRD14:th.36">GOBRD14:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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> <span class="kw"> such that </span><br/><a NAME="E13:38_1_2"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c8">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> 
 <span class="kw">and </span><br/><a NAME="E14:38_1_2"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E16:38_1_2"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E13:38_1_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="kurato_2.html#T14" title="KURATO_2:th.14">KURATO_2:14</a></span>;<br/><span class="kw">thus </span><a NAME="E17:38_1_2"/>
contradiction
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2_1_1"/>
( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:38_1_2_1_1_1"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></span>;<br/><a NAME="E3:38_1_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T61" title="JORDAN6:th.61">JORDAN6:61</a></span>;<br/><a NAME="E4:38_1_2_1_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:38_1_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="sprect_3.html#T26" title="SPRECT_3:th.26">SPRECT_3:26</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E1:38_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E2:38_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E3:38_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:38_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">m</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T47" title="JORDAN1H:th.47">JORDAN1H:47</a></span>;<br/><a NAME="E3:38_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A24</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:38_1_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:38_1_2_1_1_2"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E5:38_1_2_1_1_2"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">k</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T47" title="JORDAN1H:th.47">JORDAN1H:47</a></span>;<br/><a NAME="E6:38_1_2_1_1_2"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:38_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A26</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c8">r</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_3.html#T26" title="SPRECT_3:th.26">SPRECT_3:26</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><a NAME="E8:38_1_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T61" title="JORDAN6:th.61">JORDAN6:61</a></span>;<br/><a NAME="E9:38_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:38_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:38_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E4:38_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="txt" href="#E7:38_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="txt" href="#E8:38_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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