<?xml version="1.0"?>
<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>let </b><font color="Maroon">e</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">f</font> = 1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>assume </b><a NAME="E1:49"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>

<b>let </b><font color="Maroon">a</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="E2:49"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:49"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E5:49"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E6:49"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:49"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E8:49"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E9:49"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E10:49"/><b>then </b><i><font color="Green">E70</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E11:49"/>
( <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><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="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<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="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>
<a NAME="E12:49"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:49"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1
 <b>and </b><br/><a NAME="E15:49"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:49"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2
 <b>and </b><br/><a NAME="E18:49"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E19:49"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>;<br/>
<a NAME="E20:49"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E21:49"/><b>then </b>
0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E22:49"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E23:49"/><b>then </b><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:49_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:49_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="real_2.html#T126">REAL_2:126</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:49"/><i><font color="Green">E86</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:49_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:49"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E28:49"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E31:49"/><i><font color="Green">E88</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E32:49"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  )
 
<b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/>
<a NAME="E33:49"/><b>then </b><i><font color="Green">E89</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E34:49"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E35:49"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  )
 
<b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>
<a NAME="E36:49"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#T69">SQUARE_1:69</a></i>;<br/>
<a NAME="E37:49"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E38:49"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>;<br/>
<a NAME="E39:49"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E40:49"/><b>then </b>
 <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"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/>
<a NAME="E41:49"/><b>then </b><i><font color="Green">E90</font></i>: 
 <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"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<i><font color="Green">E91</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T12">TOPREAL3:12</a></i>
;<br/>
<a NAME="E43:49"/>
( <font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E44:49"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">e</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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