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

<b>reconsider </b><font color="Maroon" title="c19">A1</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; ( <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) );<br/>
<a NAME="E2:95_1_1"/><i><font color="Green" title="E60">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:95_1_1"/><i><font color="Green" title="E61">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>;<br/>

<a NAME="E1:95_1_1_1"/><i><font color="Green" title="E62">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D10" target="_self" title="TOPREALB:def.10">Def10</a></i>;<br/>
<a NAME="E2:95_1_1_1"/><b>then </b><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:95_1_1_1"/><i><font color="Green" title="E64">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">a</font>, <font color="Maroon" title="c22">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:95_1_1_1"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c21">a</font>,<font color="Maroon" title="c22">b</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="euclid.html#T55" title="EUCLID:th.55">EUCLID:55</a></i>;<br/>
<a NAME="E6:95_1_1_1"/><i><font color="Green" title="E66">A7</font></i>: 
not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:95_1_1_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">k</font> =  <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">x1</font> = <font color="Maroon" title="c20">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="toprealb.html#E3:95_1_1_1"><i><font color="Green" title="E64">A5</font></i></a></i>;<br/>
<a NAME="E8:95_1_1_1"/><i><font color="Green" title="E67">A8</font></i>: 
( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E9:95_1_1_1"/><b>then </b><i><font color="Green" title="E68">A9</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">a</font> &amp; <font color="Maroon" title="c21">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="toprealb.html#T27" target="_self" title="TOPREALB:th.27">Th27</a></i>;<br/>
<a NAME="E10:95_1_1_1"/><b>then </b><i><font color="Green" title="E69">A10</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E75"><i><font color="Green" title="E52">Lm24</font></i></a></i>;<br/>
<i><font color="Green" title="E70">A11</font></i>: 1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c24">x1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="toprealb.html#T12" target="_self" title="TOPREALB:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:95_1_1_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></i>
;<br/>
<div><i><font color="Green" title="E71">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_1_1_2"/><i><font color="Green" title="E72">A13</font></i>: 
 <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/><a NAME="E2:95_1_1_1_2"/><b>then </b><i><font color="Green" title="E73">A14</font></i>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E9:95_1_1_1"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="sin_cos6.html#T100" title="SIN_COS6:th.100">SIN_COS6:100</a></i>;<br/>1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="toprealb.html#E9:95_1_1_1"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="toprealb.html#E11:95_1_1_1"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="toprealb.html#E1:95_1_1_1_2"><i><font color="Green" title="E72">A13</font></i></a>, <a class="ref" href="sin_cos6.html#T100" title="SIN_COS6:th.100">SIN_COS6:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/><a NAME="E4:95_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E5:95_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="toprealb.html#E6:95_1_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="toprealb.html#E2:95_1_1_1_2"><i><font color="Green" title="E73">A14</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E72">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><a NAME="E2:95_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 2
 ;<br/><a NAME="E3:95_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></i>;<br/><b>hence </b><a NAME="E4:95_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E12:95_1_1_1"><i><font color="Green" title="E71">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T58" title="XCMPLX_1:th.58">XCMPLX_1:58</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E73">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:95_1_1_1_4"/><i><font color="Green" title="E74">A17</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:95_1_1_1_4"/><b>then </b><i><font color="Green" title="E75">A18</font></i>: 
( 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">y</font> &amp; <font color="Maroon" title="c25">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E10:95_1_1_1"><i><font color="Green" title="E69">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:95_1_1_1_4"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:95_1_1_1"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="toprealb.html#E1:95_1_1_1_4"><i><font color="Green" title="E74">A17</font></i></a></i>;<br/><a NAME="E4:95_1_1_1_4"/><b>then </b>
( 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">y</font> &amp; <font color="Maroon" title="c25">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1_1_4"><i><font color="Green" title="E75">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:95_1_1_1_4"/>
<font color="Maroon" title="c25">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1_1_5"/>
( <font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_5_1"><b>suppose </b></a><a NAME="E1:95_1_1_1_5_1"/><i><font color="Green" title="E74">A19</font></i>: 
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="toprealb.html#E6:95_1_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:95_1_1_1_5_1"/><b>then </b><i><font color="Green" title="E75">A20</font></i>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E11:95_1_1_1"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="toprealb.html#E1:95_1_1_1_5_1"><i><font color="Green" title="E74">A19</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">y</font> = 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E14:95_1_1_1"><i><font color="Green" title="E73">A16</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c25">y</font>
;<br/><b>take </b>
<font color="Maroon" title="c21">a</font>
;<br/><b>take </b>
<font color="Maroon" title="c22">b</font>
;<br/><b>thus </b><a NAME="E5:95_1_1_1_5_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c21">a</font>,<font color="Maroon" title="c22">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E6:95_1_1_1_5_1"/>
( ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="toprealb.html#E3:95_1_1_1_5_1"><i><font color="Green" title="E75">A20</font></i></a>, <a class="ref" href="sin_cos6.html#T97" title="SIN_COS6:th.97">SIN_COS6:97</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_5_2"><b>suppose </b></a><a NAME="E1:95_1_1_1_5_2"/><i><font color="Green" title="E74">A21</font></i>: 
<font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c25">y</font> = 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E14:95_1_1_1"><i><font color="Green" title="E73">A16</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c25">y</font>
;<br/><b>take </b>
<font color="Maroon" title="c21">a</font>
;<br/><b>take </b>
<font color="Maroon" title="c22">b</font>
;<br/><b>thus </b><a NAME="E3:95_1_1_1_5_2"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c21">a</font>,<font color="Maroon" title="c22">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="toprealb.html#E1:95_1_1_1_5_2"><i><font color="Green" title="E74">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_5_3"><b>suppose </b></a><a NAME="E1:95_1_1_1_5_3"/><i><font color="Green" title="E74">A22</font></i>: 
<font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c25">y</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span>;<br/><a NAME="E2:95_1_1_1_5_3"/><i><font color="Green" title="E75">A23</font></i>: 
( 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> &amp; 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E10:95_1_1_1"><i><font color="Green" title="E69">A10</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E3:95_1_1_1_5_3"/>
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="toprealb.html#E13:95_1_1_1"><i><font color="Green" title="E72">A15</font></i></a></i>;<br/><a NAME="E4:95_1_1_1_5_3"/><b>then </b>
( 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> &amp; 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1_1_5_3"><i><font color="Green" title="E75">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">y</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E2:95_1_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>take </b>
<font color="Maroon" title="c26">y</font>
;<br/><b>take </b>
<font color="Maroon" title="c21">a</font>
;<br/><b>take </b>
<font color="Maroon" title="c22">b</font>
;<br/><b>thus </b><a NAME="E6:95_1_1_1_5_3"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c21">a</font>,<font color="Maroon" title="c22">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c22">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c21">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="toprealb.html#E5:95_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="toprealb.html#E1:95_1_1_1_5_3"><i><font color="Green" title="E74">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:95_1_1"/>
 ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> st <br/>for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="toprealb.html#E3:95_1_1"><i><font color="Green" title="E61">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:95_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">3 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> st <br/>for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>  ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; ( <font color="Olive" title="b4">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> ) )
 ;<br/>


</div>
