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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>consider </b><font color="Maroon">p1</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>, <font color="Maroon">p2</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="E2:13"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</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>, <font color="Maroon">y</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="E4:13"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="topreal2.html#T4">TOPREAL2:4</a></i>;<br/>
<a NAME="E5:13"/>
(  <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="jordan1k.html#T22">JORDAN1K:22</a>, </i>;<br/>
<a NAME="E6:13"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#T102">TOPREAL6:102</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="jgraph_2.html#K2">AffineMap</a> 1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> as  <a href="tops_2.html#V3">being_homeomorphism</a> <a href="struct_0.html#NM6">Function</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_7.html#T50">JGRAPH_7:50</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">q1</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>;<br/>
<a NAME="E8:13"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E66</font></i>: <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_2.html#D2">JGRAPH_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>

;<br/>
<b>set </b><font color="Maroon">arg</font> =  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2"/>
(  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1"><b>suppose </b></a><a NAME="E1:13_2_1"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:13_2_1"/><b>then </b>
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex2.html#T34">COMPLEX2:34</a></i>;<br/><a NAME="E3:13_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <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="E3:13_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, </i>;<br/><b>hence </b><a NAME="E4:13_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><a NAME="E5:13_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E6:13_2_1"/><b>then </b>
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</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#T141">XREAL_1:141</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> =  <a href="jgraph_2.html#K2">AffineMap</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,0 as  <a href="tops_2.html#V3">being_homeomorphism</a> <a href="struct_0.html#NM6">Function</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_7.html#T50">JGRAPH_7:50</a></i>;<br/><b>set </b><font color="Maroon">E</font> = <font color="Maroon">j</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span>;<br/><b>set </b><font color="Maroon">r1</font> = <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>;<br/><b>set </b><font color="Maroon">r2</font> = <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span>;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_2.html#D2">JGRAPH_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><i><font color="Green">E82</font></i>: <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_2.html#D2">JGRAPH_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, </i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i/>
;<br/><a NAME="E10:13_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>,<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">j</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">f</font> = <font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>;<br/><a NAME="E11:13_2_1"/>
<font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> is    <a href="topgrp_1.html#M3">Homeomorphism</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_2_1_4"/>
<font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D4">TOPGRP_1:def 4</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as    <a href="topgrp_1.html#M3">Homeomorphism</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E13:13_2_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> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_2"><b>suppose </b></a><a NAME="E1:13_2_2"/>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_2_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> &amp;  <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i><a class="ref" href="comptrig.html#T52">COMPTRIG:52</a></i>;<br/><a NAME="E3:13_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E4:13_2_2"/><i><font color="Green">E85</font></i>: 
 <a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">h0</font> =  <a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="jordan24.html#V1" target="_self">isometric</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E6:13_2_2"/>
<font color="Maroon">h0</font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 ;<br/><b>then reconsider </b><font color="Maroon">h</font> =  <a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="tops_2.html#V3">being_homeomorphism</a> <a href="struct_0.html#NM6">Function</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon">F</font> = <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span>;<br/><b>set </b><font color="Maroon">s1</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>;<br/><b>set </b><font color="Maroon">s2</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">z</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/><a NAME="E9:13_2_2"/><i><font color="Green">E86</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex2.html#T66">COMPLEX2:66</a></i>;<br/><i><font color="Green">E87</font></i>: <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>reconsider </b><font color="Maroon">qq</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/><a NAME="E12:13_2_2"/><i><font color="Green">E88</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">qq</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T69">COMPLEX2:69</a></i>;<br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
;<br/><a NAME="E14:13_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:13_2_2"/><b>then </b><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_2_3"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_2_2_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <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="E3:13_2_2_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, </i>;<br/><a NAME="E4:13_2_2_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <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="E5:13_2_2_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, </i>;<br/><b>hence </b><a NAME="E6:13_2_2_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><a NAME="E17:13_2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E18:13_2_2"/><b>then </b>
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</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#T141">XREAL_1:141</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> =  <a href="jgraph_2.html#K2">AffineMap</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,0 as  <a href="tops_2.html#V3">being_homeomorphism</a> <a href="struct_0.html#NM6">Function</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_7.html#T50">JGRAPH_7:50</a></i>;<br/><b>set </b><font color="Maroon">E</font> = <font color="Maroon">j</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">r1</font> = <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">r2</font> = <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">f</font> = <font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span>;<br/><a NAME="E20:13_2_2"/>
<font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> is    <a href="topgrp_1.html#M3">Homeomorphism</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_2_4"/>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>hence </b><a NAME="E2:13_2_2_4"/>
<font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> is <a href="tops_2.html#V3">being_homeomorphism</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D4">TOPGRP_1:def 4</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> as    <a href="topgrp_1.html#M3">Homeomorphism</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><i><font color="Green">E98</font></i>: <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_2.html#D2">JGRAPH_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_2.html#D2">JGRAPH_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, </i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i/>
;<br/><a NAME="E24:13_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/><a NAME="E25:13_2_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/><a NAME="E26:13_2_2"/>
<span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1">Arg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E27:13_2_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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