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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="topreal6.html#T81">TOPREAL6:81</a></i>;<br/>
<a NAME="E2:15"/><i><font color="Green">E13</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> as  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> );<br/>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jct_misc.html#E4:15_1"><i><font color="Green">E15</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> =  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/><b>thus </b><a NAME="E7:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jct_misc.html#E5:15_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E6:15"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>(<a class="txt" href="jct_misc.html#E4:15"><i><font color="Green">E14</font></i></a>);<br/></i>
<a NAME="E7:15"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="rcomp_1.html#V3">open</a> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="rcomp_1.html#V3">open</a>
 ;<br/>

<a NAME="E2:15_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <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="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E1:15_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:15_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:15_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:15_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E6:15_2_1"/><i><font color="Green">E18</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E7:15_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:15_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:15_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:15"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> as   <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>17</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>, <font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<a NAME="E15:15_2_1"/>
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/>
<b>hence </b><a NAME="E16:15_2_1"/>
( <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tsp_1.html#D1">TSP_1:def 1</a></i>;<br/>

<a NAME="E17:15_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="jct_misc.html#E6:15_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E18:15_2_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="topreal6.html#T20">TOPREAL6:20</a></i>;<br/>
<a NAME="E19:15_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E20:15_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="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E21:15_2_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:15_2_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E18:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E23:15_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E9:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<a NAME="E24:15_2_1"/><i><font color="Green">E23</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:15_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</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="E1:15_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:15_2_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E4:15_2_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:15_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:15_2_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:15"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E8:15_2_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E3:15_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E6:15_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:15_2_1_1"/><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>11</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</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="E10:15_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E11:15_2_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E3:15_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jct_misc.html#E6:15_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:15_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</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>12</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</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="E13:15_2_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E14:15_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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>10</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E15:15_2_1_1"/><b>then </b><i><font color="Green">E30</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>23</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E10:15_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jct_misc.html#E13:15_2_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E16:15_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <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="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E17:15_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E15:15_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E18:15_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:15_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jct_misc.html#E10:15_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E4:15_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jct_misc.html#E7:15_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:15_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E4:15_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E27:15_2_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E24:15_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T163">FUNCT_1:163</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jct_misc.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="pscomp_1.html#T54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E9:15"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="pscomp_1.html#V2">with_min</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D15">PSCOMP_1:def 15</a></i>;<br/>
<a NAME="E10:15"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D4">PSCOMP_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E12:15"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E13:15"/><i><font color="Green">E17</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E14:15"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jct_misc.html#E12:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:15"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>and </b><br/><a NAME="E17:15"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E18:15"/>
( <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>10</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E16:15"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E20:15"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jct_misc.html#E17:15"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<a NAME="E21:15"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><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>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:15_3_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:15_3_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:15_3_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:15_3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:15"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E8:15_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jct_misc.html#E3:15_3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jct_misc.html#E6:15_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E9:15_3_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E10:15_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jct_misc.html#E4:15_3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jct_misc.html#E7:15_3_1"><i><font color="Green">E25</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>13</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="E2:15_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:15_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:15_3"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E6:15_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E5:15_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="jct_misc.html#E2:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:15_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:15_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:15"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E11:15_3"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:15_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:15_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E4:15_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jct_misc.html#E10:15_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E23:15"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E24:15"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E6:15"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E25:15"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E17:15"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jct_misc.html#E23:15"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E26:15"/>
 <a href="jct_misc.html#K1" target="_self">dist</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jct_misc.html#E13:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jct_misc.html#E21:15"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jct_misc.html#E24:15"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jct_misc.html#D2" target="_self">Def2</a></i>;<br/>


</div>
