<?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">K0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">B0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>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">K0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:34"/><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; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K0</font> &amp; <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &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 ) </span>}</span>  &amp; <font color="Maroon">K0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <span class="p1">(<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> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sn</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 ;<br/>

<b>set </b><font color="Maroon">cn</font> =  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">p0</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E2:34"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">sn</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E3:34"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <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"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:34"/>
<font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="jgraph_2.html#T8">JGRAPH_2:8</a></i>;<br/>
<a NAME="E5:34"/><b>then </b><i><font color="Green">E39</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </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/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E2:34_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jgraph_2.html#T11">JGRAPH_2:11</a></i>;<br/><b>hence </b><a NAME="E3:34_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_4.html#E8:34"><i><font color="Green">E41</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:34"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </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="txt" href="jgraph_4.html#E8:34"><i><font color="Green">E41</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E6:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_4.html#E5:34"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jgraph_4.html#E6:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

;<br/>
<a NAME="E10:34"/>
<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"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E8:34"><i><font color="Green">E41</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E11:34"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">sn</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E14:34"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>;<br/>
<a NAME="E12:34"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">sn</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">sn</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jgraph_4.html#E9:34"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_4.html#E12:34"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K1</font> = <font color="Maroon">K0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E14:34"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">K0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><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="E1:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p8</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><b> such that </b><br/><a NAME="E3:34_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p8</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">p8</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="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:34"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E16:34"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><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="E1:34_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:34_4"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:34_4"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:34_4"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:34_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E7:34_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E62</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E18:34"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E19:34"/>
 <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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="E20:34"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g2</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> as   <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="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E22:34"/><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E23:34"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><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="E1:34_6"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:34_6"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:34_6"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:34_6"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:34_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E7:34_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E69</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E25:34"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K13">proj1</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E26:34"/>
 <a href="relset_1.html#K5">rng</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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="E27:34"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> as   <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="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E29:34"/><i><font color="Green">E71</font></i>: 
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#R1">&lt;=</a> 0 &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 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_8"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">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>
 ;<br/>

<a NAME="E2:34_8"/>
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> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E4:34_8"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p3</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">p3</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="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E5:34_8"/>
( <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">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><a class="txt" href="jgraph_4.html#E15:34"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:34"/>
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">g2</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4"><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="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_9"><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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_9"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">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>
 ;<br/>

<i><font color="Green">E77</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E3:34_9"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E5:34_9"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p3</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">p3</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="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E6:34_9"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jgraph_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:34_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then </b><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E15:34"><i><font color="Green">E55</font></i></a>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <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/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></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"><span class="p4">(<span class="default"><font color="Maroon">p</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">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="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<b>hence </b><a NAME="E9:34_9"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></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"><span class="p4">(<span class="default"><font color="Maroon">p</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">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="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">f2</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="E32:34"/><i><font color="Green">E83</font></i>: 
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">f2</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4"><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="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E33:34"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, , , </i>;<br/>
<a NAME="E34:34"/>
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">g1</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="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<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="p2"><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="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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_10"><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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_10"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">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>
 ;<br/>

<i><font color="Green">E87</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E3:34_10"/><i><font color="Green">E89</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E5:34_10"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p3</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">p3</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="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E6:34_10"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jgraph_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:34_10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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">p</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">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="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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><font color="Maroon">p</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">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="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">sn</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">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="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</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">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="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><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<b>hence </b><a NAME="E9:34_10"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">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="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</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">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="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>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">f1</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="E36:34"/><i><font color="Green">E92</font></i>: 
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">f1</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="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<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="p2"><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="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>
 ;<br/>
<a NAME="E37:34"/>
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#R1">&lt;=</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 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_11"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">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>
 ;<br/>

<a NAME="E2:34_11"/>
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> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E4:34_11"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p3</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">p3</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="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E5:34_11"/>
( <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">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="Maroon">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/>


</div><b>end;</b></div>
<a NAME="E38:34"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, , </i>;<br/>
<a NAME="E39:34"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:34_12"/><i><font color="Green">E157</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/>

<b>set </b><font color="Maroon">p99</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E2:34_12"/><i><font color="Green">E159</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E4:34_12"/><i><font color="Green">E160</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p3</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">p3</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="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E5:34_12"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></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/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">K0</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K3" target="_self">-FanMorphW</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><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"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></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> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <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"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span></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></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jgraph_4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, , , </i>
;<br/>
<b>hence </b><a NAME="E7:34_12"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_4.html#E3:34"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E40:34"/>
<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#T7" target="_self">Th7</a>, <a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a>, <a class="txt" href="jgraph_4.html#E15:34"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="jgraph_2.html#T45">JGRAPH_2:45</a></i>;<br/>


</div>
