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<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E49</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><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>;<br/>

<b>assume </b><a NAME="E2:8"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">a</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E4:8"/>
<font color="Maroon">g</font> is <a href="vectmetr.html#V3">isometric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:8_1"/>
 <a href="relset_1.html#K5">rng</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#K14">Euclid</a> 2</span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectmetr.html#D3">VECTMETR:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:8_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">o</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="E2:8_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">o</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>set </b><font color="Maroon">pz</font> = <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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <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>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <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/>
<b>set </b><font color="Maroon">arg</font> =  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( <font color="Maroon">pz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">pz</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">pz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">qz</font> =  <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E2:8_1_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex2.html#T66">COMPLEX2:66</a></i>;<br/><a NAME="E3:8_1_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E4:8_1_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font> &amp;  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font> <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>, , <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E5:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:8_1_1_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">pz</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T67">COMPLEX2:67</a></i>;<br/><a NAME="E7:8_1_1_1_1_1_1"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#D4">COMPLEX2:def 4</a></i>;<br/><a NAME="E8:8_1_1_1_1_1_1"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="complex2.html#T21">COMPLEX2:21</a></i>;<br/><a NAME="E9:8_1_1_1_1_1_1"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#D4">COMPLEX2:def 4</a></i>;<br/><a NAME="E10:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pz</font>
 <b>by </b><i><a class="ref" href="complex2.html#T19">COMPLEX2:19</a>, </i>;<br/><b>set </b><font color="Maroon">q</font> = <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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E11:8_1_1_1_1_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#K14">Euclid</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="E12:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</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> <font color="Maroon">pz</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">pz</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">pz</font></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">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><b>hence </b><a NAME="E14:8_1_1_1_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">qz</font> =  <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E2:8_1_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex2.html#T66">COMPLEX2:66</a></i>;<br/><a NAME="E3:8_1_1_1_1_1_2"/>
( 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> <font color="Maroon">a</font> &amp;  <a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a>, <a class="ref" href="comptrig.html#T52">COMPTRIG:52</a></i>;<br/><a NAME="E4:8_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</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> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/><a NAME="E5:8_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T51">XREAL_1:51</a></i>;<br/><a NAME="E6:8_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<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#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></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#T32">XREAL_1:32</a></i>;<br/><a NAME="E7:8_1_1_1_1_1_2"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">pz</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T67">COMPLEX2:67</a></i>;<br/><a NAME="E8:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#D4">COMPLEX2:def 4</a></i>;<br/><a NAME="E9:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T10">COMPLEX2:10</a></i>;<br/><a NAME="E10:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><span class="p5">(<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#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T9">COMPLEX2:9</a></i>;<br/><a NAME="E11:8_1_1_1_1_1_2"/><b>then </b>
 <a href="comptrig.html#K1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="complex2.html#T21">COMPLEX2:21</a></i>;<br/><a NAME="E12:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<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#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<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#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#D4">COMPLEX2:def 4</a></i>;<br/><a NAME="E13:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T10">COMPLEX2:10</a></i>;<br/><a NAME="E14:8_1_1_1_1_1_2"/><b>then </b>
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1">Arg</a> <font color="Maroon">pz</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="complex2.html#T9">COMPLEX2:9</a></i>;<br/><a NAME="E15:8_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="complex2.html#K2">Rotate</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pz</font>
 <b>by </b><i><a class="ref" href="complex2.html#T19">COMPLEX2:19</a>, </i>;<br/><b>set </b><font color="Maroon">q</font> = <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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E16:8_1_1_1_1_1_2"/>
 <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#K14">Euclid</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="E17:8_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></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"><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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></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> <font color="Maroon">pz</font>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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>,<font color="Maroon">a</font></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"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">a</font></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"><a href="complex1.html#K3">Re</a> <span class="p0">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">pz</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</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> <font color="Maroon">pz</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">pz</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">pz</font></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">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><b>hence </b><a NAME="E19:8_1_1_1_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/>
<font color="Maroon">pz</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <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>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E3:8_1_1_1_2"/>
 <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#K14">Euclid</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="E4:8_1_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<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/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></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>,<font color="Maroon">a</font></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>,<font color="Maroon">a</font></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"><font color="Maroon">z</font> <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>,<font color="Maroon">a</font></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>,<font color="Maroon">a</font></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"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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>,<font color="Maroon">a</font></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"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <a href="xcmplx_0.html#K1">&lt;i&gt;</a> </span>)</span></span>)</span>,<font color="Maroon">a</font></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"><font color="Maroon">z</font> <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>,<font color="Maroon">a</font></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> <font color="Maroon">z</font>,<font color="Maroon">a</font></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> <font color="Maroon">z</font>,<font color="Maroon">a</font></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> <font color="Maroon">z</font></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> <font color="Maroon">z</font>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T66">COMPLEX2:66</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> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T66">COMPLEX2:66</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a></i>
;<br/><b>hence </b><a NAME="E7:8_1_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">X</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>, <font color="Maroon">Y</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>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</font>, <font color="Maroon">y</font> = <font color="Maroon">Y</font>, <font color="Maroon">gx</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">X</font>, <font color="Maroon">gy</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Y</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/>
<a NAME="E3:8_1"/><i><font color="Green">E97</font></i>: 
( <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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span> &amp; <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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green">E98</font></i>: 
( <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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></span>)</span> &amp; <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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2">Rotate</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>, <font color="Maroon">yy</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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="E6:8_1"/><i><font color="Green">E101</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E7:8_1"/><i><font color="Green">E102</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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> <font color="Maroon">y</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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> <font color="Maroon">y</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E8:8_1"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></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"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>;<br/>
<a NAME="E9:8_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>hence </b> <a href="metric_1.html#K4">dist</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> = 
 <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">x</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">y</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">x</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">y</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="goboard6.html#T9">GOBOARD6:9</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>

<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"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T70">COMPLEX2:70</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"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T70">COMPLEX2:70</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"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">yy</font>,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T70">COMPLEX2:70</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"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">yy</font>,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">xx</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="complex2.html#K2">Rotate</a> <font color="Maroon">yy</font>,<font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T70">COMPLEX2:70</a></i>
<br/>.= 
 <a href="topreal6.html#K1">dist</a> <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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">y</font> <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"><font color="Maroon">y</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="topreal6.html#T101">TOPREAL6:101</a>, , </i>
<br/>.= 
 <a href="topreal6.html#K1">dist</a> <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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">gy</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">gx</font>,<font color="Maroon">gy</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon">f</font> is <a href="jordan24.html#V1" target="_self">isometric</a>
 <b>by </b><i/>;<br/>


</div>
