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

<b>let </b><font color="Maroon">sn</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">K1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> ;<br/>





<b>assume </b><a NAME="E1:107"/><i><font color="Green">E34</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">sn</font> &amp; <font color="Maroon">sn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/>( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sn</font> &amp; <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) ) )
 ;<br/>

<b>set </b><font color="Maroon">a</font> = <font color="Maroon">sn</font>;<br/>
<b>set </b><font color="Maroon">b</font> = 1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font>;<br/>
<a NAME="E2:107"/><i><font color="Green">E36</font></i>: 
1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="jgraph_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default">2 <a href="jgraph_4.html#K1" target="_self">NormF</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="jgraph_4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:107"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E6:107"/><b>then </b>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>  holds <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g3</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E8:107"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">r2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">sn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon">g3</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_4.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E9:107"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:107"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:107"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:107_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:107_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_4.html#T12" target="_self">Th12</a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> <b>by </b><i/>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:107_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="toprns_1.html#T25">TOPRNS_1:25</a></i>;<br/><b>hence </b><a NAME="E3:107_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:107_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E7:107_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/>
<a NAME="E8:107_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </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"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
;<br/>
<a NAME="E9:107_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
;<br/>
<a NAME="E10:107_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K21">`1</a> </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="E11:107_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K21">`1</a> </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#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E12:107_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">r</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T14" target="_self">Th14</a>, , <a class="ref" href="xreal_1.html#T95">XREAL_1:95</a></i>;<br/>
<a NAME="E13:107_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_4.html#T14" target="_self">Th14</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E14:107_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T13" target="_self">Th13</a>, <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/>
<a NAME="E15:107_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E16:107_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span>
 
;<br/>
<a NAME="E17:107_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T10" target="_self">Th10</a>, <a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/>
<a NAME="E18:107_1"/>
<font color="Maroon">sn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:107_1"/><b>then </b>
<font color="Maroon">sn</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E20:107_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E21:107_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">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="jgraph_4.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/>
<a NAME="E22:107_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E23:107_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E24:107_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>;<br/>
<a NAME="E25:107_1"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E26:107_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E27:107_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">sn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:107_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E29:107_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="jgraph_4.html#K1" target="_self">NormF</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E30:107_1"/><i><font color="Green">E61</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E31:107_1"/>
<span class="p1">(<span class="default">2 <a href="jgraph_4.html#K1" target="_self">NormF</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">r</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E32:107_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="jgraph_4.html#T15" target="_self">Th15</a>, , <a class="ref" href="jgraph_4.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:107"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="jgraph_4.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
