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<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</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>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan_a.html#D2" target="_self">JORDAN_A:def 2</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>5</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="E2:11_1_1"/><i><font color="Green">E7</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:11_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</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"><span class="p4">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</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/>
<a NAME="E4:11_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:11_1_1"/><i><font color="Green">E9</font></i>: 
( <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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E8:11_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</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>6</sub></font>,<font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</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#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan_a.html#E7:11_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E11:11_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</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="jordan_a.html#E2:11_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E12:11_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E11:11_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard6.html#T94">GOBOARD6:94</a></i>;<br/>
<a NAME="E13:11_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E11:11_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard6.html#T94">GOBOARD6:94</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="jordan_a.html#E8:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan_a.html#E12:11_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_1.html#T48">BORSUK_1:48</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<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="E15:11_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>15</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>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E18:11_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E19:11_1_1"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E15:11_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E20:11_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E21:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:11_1_1"/><i><font color="Green">E16</font></i>: 
( <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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E24:11_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>15</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>16</sub></font>,<font color="Maroon">c<sub>17</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>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan_a.html#E23:11_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</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>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E28:11_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan_a.html#E18:11_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan_a.html#E24:11_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:11_1_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E30:11_1_1"/><b>then </b><i><font color="Green">E18</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>22</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</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> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E31:11_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E32:11_1_1"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan_a.html#E30:11_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>;<br/>
<a NAME="E33:11_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E34:11_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> holds <span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> be   <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/>

<b>thus </b><span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>

<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="jordan_a.html#D1" target="_self">Def1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E35:11_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E8:11_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E36:11_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E19:11_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan_a.html#E24:11_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan_a.html#E34:11_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
;<br/>
<a NAME="E38:11_1_1"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T31">TOPRNS_1:31</a></i>;<br/>
<a NAME="E39:11_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E35:11_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan_a.html#E36:11_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toprns_1.html#T33">TOPRNS_1:33</a></i>;<br/>
<a NAME="E40:11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><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="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E35:11_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan_a.html#E36:11_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toprns_1.html#T33">TOPRNS_1:33</a></i>;<br/>
<a NAME="E41:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E42:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E37:11_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan_a.html#E38:11_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E43:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E33:11_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E44:11_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="real_2.html#T165">REAL_2:165</a></i>;<br/>
<a NAME="E45:11_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E37:11_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan_a.html#E38:11_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan_a.html#E39:11_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E46:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E33:11_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E47:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" target="_self">Eucl_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E48:11_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan_a.html#E44:11_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div>
