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<b>set </b><font color="Maroon" title="c2">f</font> =  <a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan_a.html#D2" target="_self" title="JORDAN_A:def.2">JORDAN_A:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font>  ex <font color="Olive" title="b2">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font> st <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font></span><br/><b>let </b><font color="Maroon" title="c4">N</font> be    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font> st <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">N</font></span><br/>



<b>consider </b><font color="Maroon" title="c5">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>and </b><br/><a NAME="E3:10_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r</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" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7" title="RCOMP_1:def.7">RCOMP_1:def 7</a></i>;<br/>
<a NAME="E4:10_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
;<br/>
<a NAME="E5:10_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:10_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:10_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c9">p2</font> = <font color="Maroon" title="c7">p2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan_a.html#E7:10_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p1'</font> = <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c11">p2'</font> = <font color="Maroon" title="c9">p2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E11:10_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan_a.html#E2:10_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E12:10_1_1"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">p1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">p1</font>
 
<b>by </b><i><a class="ref" href="goboard6.html#T94" title="GOBOARD6:th.94">GOBOARD6:94</a></i>;<br/>
<a NAME="E13:10_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">p2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">p2</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E11:10_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="goboard6.html#T94" title="GOBOARD6:th.94">GOBOARD6:94</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">V</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">p1'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">p2'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font> <b>by </b><i><a class="txt" href="jordan_a.html#E8:10_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jordan_a.html#E12:10_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="borsuk_1.html#T48" title="BORSUK_1:th.48">BORSUK_1:48</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">V</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">N</font></span><br/>

<b>let </b><font color="Maroon" title="c13">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">V</font> or <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )</span><br/>

<b>assume </b><a NAME="E15:10_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c15">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E18:10_1_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V</font>
 <b>and </b><br/><a NAME="E19:10_1_1"/><i><font color="Green" title="E15">A9</font></i>: 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E15:10_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E20:10_1_1"/>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
;<br/>
<a NAME="E21:10_1_1"/><b>then </b>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">q1</font>, <font color="Maroon" title="c17">q2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:10_1_1"/><i><font color="Green" title="E16">A10</font></i>: 
( <font color="Maroon" title="c16">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c17">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E24:10_1_1"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">q1</font>,<font color="Maroon" title="c17">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">q1</font> = <font color="Maroon" title="c16">q1</font>, <font color="Maroon" title="c19">q2</font> = <font color="Maroon" title="c17">q2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan_a.html#E23:10_1_1"><i><font color="Green" title="E16">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">q1'</font> = <font color="Maroon" title="c18">q1</font>, <font color="Maroon" title="c21">q2'</font> = <font color="Maroon" title="c19">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">q1''</font> = <font color="Maroon" title="c20">q1'</font>, <font color="Maroon" title="c23">q2''</font> = <font color="Maroon" title="c21">q2'</font>, <font color="Maroon" title="c24">p1''</font> = <font color="Maroon" title="c10">p1'</font>, <font color="Maroon" title="c25">p2''</font> = <font color="Maroon" title="c11">p2'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E28:10_1_1"/>
( <font color="Maroon" title="c20">q1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">p1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; <font color="Maroon" title="c21">q2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">p2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan_a.html#E18:10_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="jordan_a.html#E24:10_1_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:10_1_1"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">q1'</font>,<font color="Maroon" title="c10">p1'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c21">q2'</font>,<font color="Maroon" title="c11">p2'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E30:10_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">q1''</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c24">p1''</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">q2''</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">p2''</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20" title="SPPOL_1:th.20">SPPOL_1:20</a></i>;<br/>
<a NAME="E31:10_1_1"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E32:10_1_1"/><i><font color="Green" title="E19">A13</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b2">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">p</font>, <font color="Maroon" title="c27">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c26">p</font>,<font color="Maroon" title="c27">q</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c27">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c26">p</font>,<font color="Maroon" title="c27">q</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">p</font>,<font color="Maroon" title="c27">q</font>

<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c27">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan_a.html#D1" target="_self" title="JORDAN_A:def.1">Def1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E33:10_1_1"/><b>then </b><i><font color="Green" title="E20">A14</font></i>: 
<span class="p1">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E8:10_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<a NAME="E34:10_1_1"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E19:10_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="jordan_a.html#E24:10_1_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="jordan_a.html#E32:10_1_1"><i><font color="Green" title="E19">A13</font></i></a></i>;<br/>
<i><font color="Green" title="E22">A16</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c8">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
;<br/>
<a NAME="E36:10_1_1"/><i><font color="Green" title="E23">A17</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T31" title="TOPRNS_1:th.31">TOPRNS_1:31</a></i>;<br/>
<a NAME="E37:10_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E33:10_1_1"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="jordan_a.html#E34:10_1_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="toprns_1.html#T33" title="TOPRNS_1:th.33">TOPRNS_1:33</a></i>;<br/>
<a NAME="E38:10_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E33:10_1_1"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="jordan_a.html#E34:10_1_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="toprns_1.html#T33" title="TOPRNS_1:th.33">TOPRNS_1:33</a></i>;<br/>
<a NAME="E39:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E40:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E35:10_1_1"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="jordan_a.html#E36:10_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E31:10_1_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E42:10_1_1"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E43:10_1_1"/>
<font color="Maroon" title="c14">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E35:10_1_1"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="jordan_a.html#E36:10_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="jordan_a.html#E37:10_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E44:10_1_1"/><b>then </b>
<font color="Maroon" title="c14">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E31:10_1_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E45:10_1_1"/><b>then </b>
<font color="Maroon" title="c14">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan_a.html#K1" title="JORDAN_A:func.1">Eucl_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E46:10_1_1"/>
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:10_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="jordan_a.html#E42:10_1_1"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
