<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be  non <a href="xboole_0.html#V1">empty</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/>

<b>thus </b><a NAME="E1:16"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a>  implies ( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being non <a href="xboole_0.html#V1">empty</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> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><a NAME="E4:16_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E1:16_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal2.html#T4" target="_self">Th4</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:16_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E7:16_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E8:16_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:16_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:16_1"/><i><font color="Green">E41</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:16_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E12:16_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E13:16_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<a NAME="E14:16_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E15:16_1"/><i><font color="Green">E46</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:16_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E6:16_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="topreal1.html#K2">R^2-unit_square</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/>
<a NAME="E18:16_1"/><i><font color="Green">E48</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</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> ;<br/>
<a NAME="E20:16_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E5:16_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topreal2.html#E13:16_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E21:16_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:16_1"/><i><font color="Green">E51</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E23:16_1"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K2">R^2-unit_square</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K2">R^2-unit_square</a>  )
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being  non <a href="xboole_0.html#V1">empty</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> such that </b><br/><a NAME="E25:16_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 <b>and </b><br/><a NAME="E26:16_1"/><i><font color="Green">E53</font></i>: 
(  <a href="topreal1.html#K2">R^2-unit_square</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="topreal2.html#E20:16_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topreal2.html#T1" target="_self">Th1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<a NAME="E27:16_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E28:16_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</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="pre_topc.html#D9">PRE_TOPC:def 9</a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E29:16_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="pre_topc.html#K2">[#]</a> <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#K15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> as  non <a href="xboole_0.html#V1">empty</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="topreal2.html#E27:16_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<i><font color="Green">E55</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> = 
<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>

;<br/>
<i><font color="Green">E56</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> = 
<a href="topreal1.html#K2">R^2-unit_square</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E33:16_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E31:16_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E35:16_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E36:16_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E37:16_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>
 
;<br/>
<a NAME="E38:16_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T4">TOPMETR:4</a></i>;<br/>
<a NAME="E39:16_1"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E31:16_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E40:16_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E41:16_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E27:16_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E42:16_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E43:16_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_3"/>
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <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> such that </b><br/><a NAME="E3:16_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:16_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E5:16_1_3"/>
<font color="Maroon">c<sub>28</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>26</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T139">FUNCT_1:139</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E4:16_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E42:16_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
;<br/>
<a NAME="E8:16_1_3"/>
<font color="Maroon">c<sub>29</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topreal2.html#E3:16_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E9:16_1_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>29</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E9:16_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_3"/>
<font color="Maroon">c<sub>26</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topreal2.html#E38:16_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:16_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span><b> such that </b><br/><a NAME="E46:16_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E47:16_1"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E48:16_1"/><i><font color="Green">E63</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  non <a href="xboole_0.html#V1">empty</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/>
<a NAME="E50:16_1"/>
( <font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E46:16_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E51:16_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E48:16_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="compts_1.html#T23">COMPTS_1:23</a></i>;<br/>
<a NAME="E52:16_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font> is <a href="compts_1.html#V3">being_T2</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E53:16_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E32:16_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topreal2.html#E39:16_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal2.html#E40:16_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topreal2.html#E44:16_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topreal2.html#E51:16_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<a NAME="E54:16_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E55:16_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<i><font color="Green">E67</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E55:16_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>
;<br/>
<a NAME="E57:16_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E58:16_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E58:16_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>
;<br/>
<b>hence </b><a NAME="E60:16_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topreal2.html#E53:16_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal2.html#E56:16_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal2.html#T3" target="_self">Th3</a></i>;<br/>

<i><font color="Green">E69</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>

;<br/>
<i><font color="Green">E70</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<a href="topreal1.html#K2">R^2-unit_square</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E63:16_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E61:16_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E65:16_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E66:16_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E67:16_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>
 
;<br/>
<a NAME="E68:16_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T4">TOPMETR:4</a></i>;<br/>
<a NAME="E69:16_1"/><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E61:16_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E70:16_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>30</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E71:16_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_8"/>
<font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <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> such that </b><br/><a NAME="E3:16_1_8"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>32</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:16_1_8"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E5:16_1_8"/>
<font color="Maroon">c<sub>32</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/>
<i><font color="Green">E76</font></i>: <font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>31</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T139">FUNCT_1:139</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E4:16_1_8"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="topreal2.html#E42:16_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
;<br/>
<a NAME="E8:16_1_8"/>
<font color="Maroon">c<sub>33</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topreal2.html#E3:16_1_8"><i><font color="Green">E74</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E9:16_1_8"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>33</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E9:16_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_8"/>
<font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topreal2.html#E68:16_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1_8"><i><font color="Green">E76</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E72:16_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span><b> such that </b><br/><a NAME="E74:16_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E75:16_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E76:16_1"/>
( <font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E74:16_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E77:16_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>29</sub></font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E48:16_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="compts_1.html#T23">COMPTS_1:23</a></i>;<br/>
<a NAME="E78:16_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font> is <a href="compts_1.html#V3">being_T2</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E79:16_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E62:16_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topreal2.html#E69:16_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topreal2.html#E70:16_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topreal2.html#E72:16_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topreal2.html#E77:16_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<a NAME="E80:16_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E81:16_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E82:16_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<a NAME="E83:16_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E81:16_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>
<a NAME="E84:16_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E85:16_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E86:16_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<a NAME="E87:16_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E85:16_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E88:16_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topreal2.html#E79:16_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topreal2.html#E83:16_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreal2.html#T3" target="_self">Th3</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topreal2.html#E15:16_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T121">FUNCT_1:121</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topreal2.html#E82:16_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topreal2.html#E86:16_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <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> such that </b><a NAME="E2:16"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E3:16"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being non <a href="xboole_0.html#V1">empty</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> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<a NAME="E4:16"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E2:16"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">c<sub>10</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>by </b><i><a class="txt" href="topreal2.html#E2:16"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topreal2.html#E32"><i><font color="Green">Lemma32</font></i></a></i>;<br/>


</div>
