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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">x</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c3">C</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:37_1_1"/><b>then </b>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T17" title="TBSP_1:th.17">TBSP_1:17</a></i>;<br/><a NAME="E3:37_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">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> 
 ;<br/><div class="add"><a NAME="E2:37_1_2"/><i><font color="Green" title="E30">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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/><b>reconsider </b><font color="Maroon" title="c4">y</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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/><b>reconsider </b><font color="Maroon" title="c5">q</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> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">Q</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c7">J</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,1 as   <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> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">P</font> = <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> , <font color="Maroon" title="c9">K</font> = <font color="Maroon" title="c7">J</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <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> ;<br/><i><font color="Green" title="E31">A3</font></i>: <font color="Maroon" title="c9">K</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q1</font> where <font color="Olive" title="b1">q1</font> is   <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="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 </span>}</span> 
<b>by </b><i><a class="ref" href="jordan1k.html#T31" target="_self" title="JORDAN1K:th.31">Th31</a></i>
;<br/><a NAME="E8:37_1_2"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="jgraph_2.html#K2" title="JGRAPH_2:func.2">AffineMap</a> <font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="jgraph_2.html#T54" title="JGRAPH_2:th.54">JGRAPH_2:54</a></i>;<br/><a NAME="E9:37_1_2"/>
 <a href="jgraph_2.html#K2" title="JGRAPH_2:func.2">AffineMap</a> <font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="jordan1k.html#T36" target="_self" title="JORDAN1K:th.36">Th36</a></i>;<br/><a NAME="E10:37_1_2"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
 <a href="jgraph_2.html#K2" title="JGRAPH_2:func.2">AffineMap</a> <font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c1">r</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E8:37_1_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E11:37_1_2"/>
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K2" title="JGRAPH_2:func.2">AffineMap</a> <font color="Maroon" title="c1">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c1">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">J</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="jordan1k.html#T35" target="_self" title="JORDAN1K:th.35">Th35</a></i>;<br/><a NAME="E12:37_1_2"/><b>then </b>
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K2" title="JGRAPH_2:func.2">AffineMap</a> <font color="Maroon" title="c1">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c1">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">K</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E10:37_1_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="jordan1k.html#T5" target="_self" title="JORDAN1K:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E13:37_1_2"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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>
 <b>by </b><i><a class="txt" href="jordan1k.html#E2:37_1_2"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1k.html#E7:37_1_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="jordan2c.html#T61" title="JORDAN2C:th.61">JORDAN2C:61</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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