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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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>assume </b><a NAME="E1:39"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 ;<br/>

<a NAME="E2:39"/><i><font color="Green">E41</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>

<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:39"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/>
<a NAME="E4:39"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:39"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E6:39"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</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="hidden.html#R1">=</a> 1 </span>}</span> 
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>] means <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1;<br/>
<a NAME="E9:39"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <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>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <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="hidden.html#R1">=</a> 1 </span>}</span>  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="jgraph_6.html#E8:39"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E11:39"/><i><font color="Green">E43</font></i>: 
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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E12:39"/><i><font color="Green">E44</font></i>: 
 <a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jgraph_2.html#T54">JGRAPH_2:54</a></i>;<br/>
<a NAME="E13:39"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E46</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</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>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E11:39"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jgraph_6.html#E13:39"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E15:39"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</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>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_4"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:39_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</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>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E14:39"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_6.html#E3:39_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</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>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></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="jgraph_6.html#E11:39"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rfunct_2.html#T5">RFUNCT_2:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jgraph_6.html#T29" target="_self">Th29</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>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</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>7</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>5</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_6.html#E14:39"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_6.html#E15:39"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E18:39"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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>7</sub></font></span>)</span>,<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="jgraph_6.html#E14:39"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</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>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K2">AffineMap</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></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="jgraph_6.html#E11:39"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rfunct_2.html#T5">RFUNCT_2:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jgraph_6.html#T29" target="_self">Th29</a></i>
;<br/>
<a NAME="E21:39"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E12:39"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E22:39"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E23:39"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="topreal2.html#V1">being_simple_closed_curve</a>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E24:39"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E25:39"/><b>then </b>
<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>7</sub></font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E26:39"/><b>then </b>
ex <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>7</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>5</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E20:39"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_6.html#E21:39"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_6.html#E22:39"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jgraph_1.html#T64">JGRAPH_1:64</a></i>;<br/>
<b>hence </b><a NAME="E27:39"/>
<font color="Maroon">c<sub>4</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E23:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jgraph_6.html#T30" target="_self">Th30</a></i>;<br/>


</div>
