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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>
<a NAME="E2:18"/><i><font color="Green">E16</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E17</font></i>: 
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E4:18"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:18_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E2:18_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E5:18_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:18_1"/><i><font color="Green">E21</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E11:18_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E12:18_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E13:18_1"/><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T12">EUCLID:12</a></i>;<br/>
<i><font color="Green">E25</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18"><i><font color="Green">E15</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2"/>
( 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1"><b>suppose </b></a><a NAME="E1:18_1_2_1"/><i><font color="Green">E26</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="topalg_1.html#E12:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/><a NAME="E4:18_1_2_1"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E5:18_1_2_1"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E13:18_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topalg_1.html#E4:18_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E6:18_1_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:18_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E8:18_1_2_1"/><b>then </b><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E10:18_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><a NAME="E11:18_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:18_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/><a NAME="E13:18_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E14:18_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_1.html#E10:18_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E15:18_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><a NAME="E16:18_1_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E17:18_1_2_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/><a NAME="E19:18_1_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_1.html#E16:18_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_1.html#E17:18_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 = 
<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"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_1.html#E18:18_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<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"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
;<br/><a NAME="E21:18_1_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E22:18_1_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E23:18_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/><a NAME="E24:18_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_1.html#E20:18_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T126">REAL_2:126</a></i>;<br/><a NAME="E25:18_1_2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E26:18_1_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E19:18_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1_3_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_3_1_1"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_3_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_2_1_3_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:18_1_2_1_3_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1_3_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E4:18_1_2_1_3_1_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_1.html#E25:18_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_1.html#E1:18_1_2_1_3_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_2_1_3_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1_3_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_3_1_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_1.html#E25:18_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:18_1_2_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1_3_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_1_3_1_1_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E26:18_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_4.html#T70">BORSUK_4:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_3_1_1_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>;<br/><a NAME="E3:18_1_2_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="pre_topc.html#D2">PRE_TOPC:def 2</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_1_3_1_1_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T52">TOPS_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:18_1_2_1_3_1_1"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_3_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_3_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_3_1_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_2_1_3_1_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:18_1_2_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E4:18_1_2_1_3_1_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_1_3_1_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E5:18_1_2_1_3_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1_3_1_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><b>hence </b><a NAME="E6:18_1_2_1_3_1_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_1_3_1_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>19</sub></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="E2:18_1_2_1_3_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:18_1_2_1_3_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_1_3_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E5:18_1_2_1_3_1_2_1"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_1_3_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E6:18_1_2_1_3_1_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1_3_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topalg_1.html#E4:18_1_2_1_3_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E7:18_1_2_1_3_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="topalg_1.html#E25:18_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_1.html#E4:18_1_2_1_3_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:18_1_2_1_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_1_3_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E9:18_1_2_1_3_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1_3_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_2_1_3_1_2"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E26:18_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topalg_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E28:18_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E11:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><b>hence </b><a NAME="E29:18_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E27:18_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/><a NAME="E30:18_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_2_1_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2_1_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:18_1_2_1_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E5:18_1_2_1_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_1.html#E3:18_1_2_1_4"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>23</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E8:18_1_2_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_1_4"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18"><i><font color="Green">E15</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E12:18_1_2_1_4"/><i><font color="Green">E46</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E13:18_1_2_1_4"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T22">EUCLID:22</a></i>;<br/>
<a NAME="E14:18_1_2_1_4"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_1.html#E8:18_1_2_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:18_1_2_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E16:18_1_2_1_4"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E17:18_1_2_1_4"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topalg_1.html#E15:18_1_2_1_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E18:18_1_2_1_4"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xcmplx_1.html#T98">XCMPLX_1:98</a></i>;<br/>
<a NAME="E19:18_1_2_1_4"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>23</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E20:18_1_2_1_4"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="topalg_1.html#E10:18_1_2_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<b>then </b><i><font color="Green">E53</font></i>: <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topalg_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E19:18_1_2_1_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>
;<br/>
<a NAME="E22:18_1_2_1_4"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T12">EUCLID:12</a></i>;<br/>
<a NAME="E23:18_1_2_1_4"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_1.html#E8:18_1_2_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E24:18_1_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:18_1_2_1_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E26:18_1_2_1_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E19:18_1_2_1_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/>
<a NAME="E27:18_1_2_1_4"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E22:18_1_2_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E28:18_1_2_1_4"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T21">EUCLID:21</a></i>;<br/>
<a NAME="E29:18_1_2_1_4"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T18">EUCLID:18</a></i>;<br/>
<a NAME="E30:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E15:18_1_2_1_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topalg_1.html#E16:18_1_2_1_4"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_1.html#E28:18_1_2_1_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E31:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E32:18_1_2_1_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E6:18_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T99">XCMPLX_1:99</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
;<br/>
<a NAME="E34:18_1_2_1_4"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="topalg_1.html#E27:18_1_2_1_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_1.html#E32:18_1_2_1_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E35:18_1_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T48">XCMPLX_1:48</a></i>;<br/>
<a NAME="E36:18_1_2_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topalg_1.html#E17:18_1_2_1_4"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_1.html#E18:18_1_2_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E37:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>;<br/>
<a NAME="E38:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topalg_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E39:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E40:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="topalg_1.html#E14:18_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E41:18_1_2_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E21:18_1_2_1_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_1.html#E34:18_1_2_1_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E42:18_1_2_1_4"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E12:18_1_2_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_1.html#E13:18_1_2_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E43:18_1_2_1_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E31:18_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2"><b>suppose </b></a><a NAME="E1:18_1_2_2"/>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_2_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:18_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><a NAME="E4:18_1_2_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E5:18_1_2_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,1</span><a href="rcomp_2.html#K2">.]</a></span>;<br/><a NAME="E7:18_1_2_2"/><i><font color="Green">E29</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_1.html#E13:18_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:18_1_2_2"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_1.html#E7:18_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E9:18_1_2_2"/>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E13:18_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:18_1_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="real_2.html#T117">REAL_2:117</a></i>;<br/><a NAME="E11:18_1_2_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:18_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_1.html#E10:18_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><a NAME="E13:18_1_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/><a NAME="E14:18_1_2_2"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_2_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:18_1_2_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E4:18_1_2_2_1"/>
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E5:18_1_2_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E11:18_1_2_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_2_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E3:18_1_2_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>,1</span><a href="rcomp_2.html#K2">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E16:18_1_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><a NAME="E17:18_1_2_2"/>
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E18:18_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>hence </b><a NAME="E19:18_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_1.html#E16:18_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E20:18_1_2_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E13:18_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topalg_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E21:18_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><b>hence </b><a NAME="E22:18_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E20:18_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/><a NAME="E23:18_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_2_2_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:18_1_2_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E5:18_1_2_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_1.html#E3:18_1_2_2_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>22</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E8:18_1_2_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_2_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:18"><i><font color="Green">E15</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>29</sub></font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E12:18_1_2_2_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T13">JORDAN2C:13</a></i>;<br/>
<a NAME="E13:18_1_2_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E10:18_1_2_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E14:18_1_2_2_2"/>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/>
<a NAME="E15:18_1_2_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/>
<i><font color="Green">E44</font></i>: <font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E14:18_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="topalg_1.html#E4:18_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_1.html#E12:18_1_2_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>
;<br/>
<i><font color="Green">E45</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>23</sub></font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E13:18_1_2_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_1.html#E16:18_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E15:18_1_2_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E18:18_1_2_2_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topalg_1.html#E8:18_1_2_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E19:18_1_2_2_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:18_1_2_2_2"/><i><font color="Green">E47</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E21:18_1_2_2_2"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T21">EUCLID:21</a></i>;<br/>
<a NAME="E22:18_1_2_2_2"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T18">EUCLID:18</a></i>;<br/>
<a NAME="E23:18_1_2_2_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E19:18_1_2_2_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_1.html#E20:18_1_2_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_1.html#E21:18_1_2_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E24:18_1_2_2_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_2_2_4_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_2_4_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_2_2_4_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_2_2_2_4_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:18_1_2_2_2_4_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E4:18_1_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topalg_1.html#E8:18_1_2_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:18_1_2_2_2_4_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E5:18_1_2_2_2_4_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:18_1_2_2_2_4_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:18_1_2_2_2_4_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E8:18_1_2_2_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_1.html#E2:18_1_2_2_2_4_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_2.html#T200">REAL_2:200</a></i>;<br/><a NAME="E9:18_1_2_2_2_4_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_1.html#T54">XCMPLX_1:54</a></i>;<br/><a NAME="E10:18_1_2_2_2_4_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E24:18_1_2_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:18_1_2_2_2_4_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_2_2_4_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E12:18_1_2_2_2_4_1_1"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:18_1_2_2_2_4_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_2_4_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_2_2_4_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_2_2_2_4_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan5a.html#T2">JORDAN5A:2</a></i>;<br/><a NAME="E3:18_1_2_2_2_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:18_1_2_2_2_4_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_2_2_4_1_2"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E7:18_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E26:18_1_2_2_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E5:18_1_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_1.html#E17:18_1_2_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E24:18_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:18_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


</div>
