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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>;<br/>
<a NAME="E2:28"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon">r3</font> = <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>;<br/>
<a NAME="E2:28_1"/>
for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 
;<br/>
<b>hence </b><a NAME="E3:28_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:28"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="jgraph_2.html#T6" target="_self">Th6</a>);<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:28"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">g0</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:28"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:28"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:28_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="frechet.html#T8">FRECHET:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:28_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E7:28_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E8:28_2"/><b>then </b><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E9:28_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:28_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1"><b>case </b></a><a NAME="E1:28_2_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span>;<br/><a NAME="E3:28_2_1_1_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:28_2_1_1_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E5:28_2_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>

;<br/><a NAME="E7:28_2_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E9:28_2_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E10:28_2_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:28_2_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:28"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E12:28_2_1_1_1"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E14:28_2_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E2:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jgraph_2.html#E1:28_2"><i><font color="Green">E44</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font>;<br/><a NAME="E15:28_2_1_1_1"/>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_2_1_1_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="jgraph_2.html#E6:28_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E5:28_2_1_1_1_2"/>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E6:28_2_1_1_1_2"/><b>then </b>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E6:28_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:28_2_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a>, <a class="txt" href="jgraph_2.html#E6:28_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E11:28_2_1_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> &amp; <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E4:28_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jgraph_2.html#E7:28_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E12:28_2_1_1_1_2"/>
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:28_2_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T110">REAL_2:110</a></i>;<br/>
<a NAME="E14:28_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E15:28_2_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E9:28_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">aa1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E17:28_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E9:28_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E18:28_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">aa1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E10:28_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E19:28_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E20:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E21:28_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E5:28"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E22:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">rx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E8:28_2"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<a NAME="E23:28_2_1_1_1_2"/>
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_1_1_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:28_2_1_1_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E9:28_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E3:28_2_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E4:28_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><a NAME="E5:28_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:28_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:28_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E7:28_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E8:28_2_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E5:28"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E9:28_2_1_1_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:28_2_1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:28_2_1_1_1_2_1_1_2"/>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:28_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E4:28_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <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"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E5:28_2_1_1_1_2_1_1_2"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/><a NAME="E6:28_2_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/><a NAME="E7:28_2_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <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"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:28_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E8:28_2_1_1_1_2_1_1_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_2.html#E5:28"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_2.html#E2:28_2_1_1_1"><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="E26:28_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E8:28_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_2.html#E1:28_2_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:28_2_1_1_1"/><b>then </b>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E17:28_2_1_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E4:28_2"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2"><b>case </b></a><a NAME="E1:28_2_1_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:28_2_1_1_2"/><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E4:28_2_1_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, , <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span>;<br/><a NAME="E5:28_2_1_1_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:28_2_1_1_2"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E7:28_2_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, , , <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>

;<br/><a NAME="E9:28_2_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E5:28"><i><font color="Green">E42</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E11:28_2_1_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:28"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E12:28_2_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E13:28_2_1_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E4:28_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E14:28_2_1_1_2"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:28_2_1_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E2:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jgraph_2.html#E6:28_2"><i><font color="Green">E46</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font>;<br/><a NAME="E17:28_2_1_1_2"/>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_2_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_2_1_1_2_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="jgraph_2.html#E8:28_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E5:28_2_1_1_2_2"/>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E6:28_2_1_1_2_2"/><b>then </b>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:28_2_1_1_2_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E8:28_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:28_2_1_1_2_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a>, <a class="txt" href="jgraph_2.html#E8:28_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E11:28_2_1_1_2_2"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> &amp; <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:28_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_2.html#E9:28_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E12:28_2_1_1_2_2"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:28_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:28_2_1_1_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E15:28_2_1_1_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">aa1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:28_2_1_1_2_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">aa1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E17:28_2_1_1_2_2"/>
<font color="Maroon">aa1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E18:28_2_1_1_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E19:28_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E20:28_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">aa1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E21:28_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E22:28_2_1_1_2_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E23:28_2_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_2.html#E1:28_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E24:28_2_1_1_2_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">rx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E10:28_2"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E25:28_2_1_1_2_2"/>
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E26:28_2_1_1_2_2"/><b>then </b><i><font color="Green">E169</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1_1_2_2_1_1"/>
( 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:28_2_1_1_2_2_1_1_1"/>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:28_2_1_1_2_2_1_1_1"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:28_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E5:28_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">aa1</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E6:28_2_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E7:28_2_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E8:28_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><a NAME="E9:28_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:28_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E11:28_2_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">aa1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E12:28_2_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E1:28_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E13:28_2_1_1_2_2_1_1_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:28_2_1_1_2_2_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:28_2_1_1_2_2_1_1_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:28_2_1_1_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E4:28_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E5:28_2_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <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"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:28_2_1_1_2_2_1_1_2"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><a NAME="E7:28_2_1_1_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:28"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><a NAME="E8:28_2_1_1_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <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"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</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"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E9:28_2_1_1_2_2_1_1_2"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">aa1</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_2.html#E1:28_2"><i><font color="Green">E44</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="E28:28_2_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E10:28_2"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:28_2_1_1_2"/><b>then </b>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E19:28_2_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:28_2"><i><font color="Green">E47</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="E12:28_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:28"/><b>then </b>
<font color="Maroon">g0</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:28"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>


</div>
