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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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>assume </b><a NAME="E1:47"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</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"><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>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</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> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:47"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>1</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#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T44" target="_self">Th44</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E5:47"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T42" target="_self">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>,<font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E8:47"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> as   <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> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E10:47"/><i><font color="Green">E49</font></i>: 
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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E2:47_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E3:47_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E4:47_1_1"/><i><font color="Green">E55</font></i>: 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:47_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:47_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:47_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:47_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:47_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:47_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:47_1_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E12:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:47_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_5.html#E10:47_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/><a NAME="E13:47_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E14:47_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:47_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E16:47_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E13:47_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:47_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jgraph_5.html#E15:47_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:47_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E18:47_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E19:47_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E15:47_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_5.html#E16:47_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E20:47_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E21:47_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E2:47_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E14:47_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jgraph_5.html#E20:47_1_1"><i><font color="Green">E62</font></i></a></i>
;<br/><a NAME="E23:47_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E24:47_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E25:47_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E26:47_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:47_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E28:47_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E29:47_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E30:47_1_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E31:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E26:47_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E29:47_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/><a NAME="E32:47_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E33:47_1_1"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E34:47_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E35:47_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E32:47_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:47_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E34:47_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E32:47_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E37:47_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E38:47_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E34:47_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E35:47_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E39:47_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E40:47_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E3:47_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E33:47_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E39:47_1_1"><i><font color="Green">E70</font></i></a></i>
;<br/><b>hence </b><a NAME="E42:47_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E4:47_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_5.html#E23:47_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E33:47_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E2:47_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E3:47_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E4:47_1_2"/><i><font color="Green">E56</font></i>: 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:47_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:47_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:47_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:47_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:47_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:47_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:47_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E12:47_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:47_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/><a NAME="E13:47_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E14:47_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:47_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E16:47_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E13:47_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:47_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jgraph_5.html#E15:47_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:47_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E18:47_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E19:47_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E15:47_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_5.html#E16:47_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E20:47_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E21:47_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E2:47_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E14:47_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jgraph_5.html#E20:47_1_2"><i><font color="Green">E62</font></i></a></i>
;<br/><a NAME="E23:47_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E24:47_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E25:47_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E26:47_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:47_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E28:47_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E29:47_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E30:47_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E31:47_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E26:47_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/><a NAME="E32:47_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E33:47_1_2"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E34:47_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E35:47_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E32:47_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:47_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E34:47_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E32:47_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E37:47_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E38:47_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E34:47_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E35:47_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E39:47_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E40:47_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E3:47_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E33:47_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E39:47_1_2"><i><font color="Green">E69</font></i></a></i>
;<br/><b>hence </b><a NAME="E42:47_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E4:47_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_5.html#E23:47_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E33:47_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_3_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:47_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:47_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1_3"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:47_1_3_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_3_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:47_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E1:47_1_3"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:47_1_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:47_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> iff <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:47_1"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:47"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:47"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E13:47"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E14:47"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:47"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E15:47"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E5:47"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E16:47"/>
ex <font color="Olive">b<sub>1</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> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/>( <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>5</sub></font> iff <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E8:47"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_5.html#E10:47"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_5.html#E13:47"><i><font color="Green">E51</font></i></a></i>;<br/>


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