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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">C0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> 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">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 </span>}</span>  &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C0</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b>( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E2:15"/>
<span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</font> 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="Maroon">S<sub>1</sub></font>[<font color="Olive">q1</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S1">JGRAPH_2:sch 1</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">KXP</font> = <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</font> 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">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b>( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E4:15"/>
<span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> 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="Maroon">S<sub>2</sub></font>[<font color="Olive">q2</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S1">JGRAPH_2:sch 1</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">KXN</font> = <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> 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">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b>( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E6:15"/>
<span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</font> 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="Maroon">S<sub>3</sub></font>[<font color="Olive">q3</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S1">JGRAPH_2:sch 1</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">KYP</font> = <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</font> 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">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b>( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E8:15"/>
<span class="p1">{<span class="default"> <font color="Olive">q4</font> where <font color="Olive">q4</font> 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="Maroon">S<sub>4</sub></font>[<font color="Olive">q4</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S1">JGRAPH_2:sch 1</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">KYN</font> = <span class="p1">{<span class="default"> <font color="Olive">q4</font> where <font color="Olive">q4</font> 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">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">q4</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">O</font> = 0 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E12:15"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>: <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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</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">0 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/>
<a NAME="E14:15"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:15"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KXN</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:15"/><i><font color="Green">E34</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/>
<a NAME="E18:15"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KXP</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:15"/><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</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="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><i><font color="Green">E47</font></i>: <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">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</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">0 <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="real_1.html#K1">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/>
<a NAME="E21:15"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E22:15"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KYN</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E23:15"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><i><font color="Green">E73</font></i>: <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">0,1</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">0 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/>
<a NAME="E25:15"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E26:15"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">KYP</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_5.html#T2" target="_self">Th2</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E27:15"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="txt" href="jgraph_5.html#E1"><i><font color="Green">Lemma26</font></i></a>, </i>;<br/>


</div>
