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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</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">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </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">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T17">TBSP_1:17</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <a href="xboole_0.html#K1">{}</a> 

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>

;<br/><b>hence </b><a NAME="E4:37_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is  <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">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">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_2"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>4</sub></font>,1 as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> , <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1 </span>}</span> 
<b>by </b><i><a class="ref" href="jordan1k.html#T31" target="_self">Th31</a></i>
;<br/><a NAME="E8:37_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T61">JORDAN2C:61</a></i>;<br/><a NAME="E9:37_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jgraph_2.html#T54">JGRAPH_2:54</a></i>;<br/><a NAME="E10:37_1_2"/>
 <a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan1k.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E11:37_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="txt" href="jordan1k.html#E9:37_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/><a NAME="E12:37_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:37_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan1k.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E13:37_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E11:37_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan1k.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E14:37_1_2"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is  <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1k.html#E2:37_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan1k.html#E8:37_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
