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

<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E13">A2</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R5" title="JORDAN20:pred.5">is_OSin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:10_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><a NAME="E3:10_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font> &amp; <font color="Olive" title="b3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font> ) )
 <b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><b> such that </b><br/><a NAME="E5:10_1"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E6:10_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:10_1"/><i><font color="Green" title="E17">A6</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E8:10_1"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:10_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:10_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">xs</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:10_1"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c7">xs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">xs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">s2</font> = <font color="Maroon" title="c7">xs</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E13:10_1"/><i><font color="Green" title="E20">A9</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; <font color="Maroon" title="c8">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">pr1</font> =  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">pro1</font> = <font color="Maroon" title="c9">pr1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><a NAME="E16:10_1"/>
for <font color="Olive" title="b1">p7</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> holds  <font color="Maroon" title="c10">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p7</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p7</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/>

<a NAME="E1:10_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><a NAME="E2:10_1_1"/>
<font color="Maroon" title="c10">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p7</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:10_1"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c10">pro1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T39" title="JGRAPH_2:th.39">JGRAPH_2:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">pro1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/><a NAME="E19:10_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c11">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="jordan20.html#E17:10_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/><a NAME="E20:10_1"/><i><font color="Green" title="E23">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:10_1"/>
for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/><a NAME="E22:10_1"/><b>then </b><i><font color="Green" title="E24">A13</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <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> 2</span>)</span>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a></i>;<br/><a NAME="E23:10_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <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> 2</span>)</span>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</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><br/>

<b>assume </b><a NAME="E1:10_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <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> 2</span>)</span>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_2"/><i><font color="Green" title="E25">A14</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c13">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:10_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E3:10_1_2"><i><font color="Green" title="E25">A14</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">R</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <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> 2</span>)</span>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="jordan20.html#E22:10_1"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">s0</font> =  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c12">R</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E26:10_1"/>
<font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</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><br/>

<b>assume </b><a NAME="E1:10_1_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_3"/><i><font color="Green" title="E25">A15</font></i>: 
( <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">s</font> &amp; <font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 ;<br/>
<a NAME="E4:10_1_3"/>
<font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_3"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:10_1_3"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">R''</font> = <font color="Maroon" title="c12">R</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E28:10_1"/>
<font color="Maroon" title="c14">R''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R</font>
 ;<br/><a NAME="E29:10_1"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E22:10_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="borsuk_4.html#T51" title="BORSUK_4:th.51">BORSUK_4:51</a></i>;<br/><a NAME="E30:10_1"/><i><font color="Green" title="E26">A17</font></i>: 
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_4"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">s3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_4"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c16">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">s3</font> &amp; <font color="Maroon" title="c16">s3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s3</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 ;<br/>
<a NAME="E4:10_1_4"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">s2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_4"><i><font color="Green" title="E27">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_4"/>
<font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:10_1_4"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:10_1"/><b>then </b>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font>
 ;<br/><a NAME="E32:10_1"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T62" title="SEQ_4:th.62">SEQ_4:62</a></i>;<br/><a NAME="E33:10_1"/><b>then </b><i><font color="Green" title="E28">A20</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E34:10_1"/><b>then </b>
<font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E29:10_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E35:10_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E36:10_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">p9</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E38:10_1"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:10_1"><i><font color="Green" title="E26">A17</font></i></a></i>;<br/><a NAME="E39:10_1"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c12">R</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></i>;<br/><a NAME="E40:10_1"/><i><font color="Green" title="E30">A22</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E29:10_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="jordan20.html#E32:10_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></i>;<br/><a NAME="E41:10_1"/><i><font color="Green" title="E31">A23</font></i>: 
for <font color="Olive" title="b1">p8</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">p8</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_5"/><i><font color="Green" title="E32">A24</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c16">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:10_1_5"/><b>then </b><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c16">p8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:10_1_5"/><i><font color="Green" title="E34">A26</font></i>: 
( <font color="Maroon" title="c17">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c16">p8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x8</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:10_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">s8</font> = <font color="Maroon" title="c17">x8</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E6:10_1_5"/><i><font color="Green" title="E35">A27</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">s8</font> &amp; <font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p8</font> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:10_1_5"/><b>then </b><i><font color="Green" title="E36">A28</font></i>: 
<font color="Maroon" title="c13">s0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">s8</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_5"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E8:10_1_5"/><i><font color="Green" title="E37">A29</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_5"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_5_1"/><i><font color="Green" title="E38">A30</font></i>: 
<font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:10_1_5_1"/><b>then </b><i><font color="Green" title="E39">A31</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="jordan20.html#E8:10_1_5"><i><font color="Green" title="E37">A29</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">ee</font> = <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:10_1_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_5_1"><i><font color="Green" title="E38">A30</font></i></a></i>;<br/><a NAME="E5:10_1_5_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <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="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E6:10_1_5_1"/><b>then </b><i><font color="Green" title="E40">A32</font></i>: 
<font color="Maroon" title="c19">ee</font> <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="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">w</font> = <font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c20">w</font>,<font color="Maroon" title="c19">ee</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E9:10_1_5_1"/><i><font color="Green" title="E41">A33</font></i>: 
<font color="Maroon" title="c20">w</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="c20">w</font>,<font color="Maroon" title="c19">ee</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_5_1"><i><font color="Green" title="E40">A32</font></i></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/><i><font color="Green" title="E42">A34</font></i>: <font color="Maroon" title="c21">B</font> = 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s7</font> where <font color="Olive" title="b1">s7</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">ee</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s7</font> &amp; <font color="Olive" title="b1">s7</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">ee</font> ) </span>}</span> 
<b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_5_1"><i><font color="Green" title="E40">A32</font></i></a>, <a class="ref" href="jordan2b.html#T21" title="JORDAN2B:th.21">JORDAN2B:21</a></i>
<br/>.= 
<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"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">ee</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">ee</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#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>
;<br/><a NAME="E11:10_1_5_1"/><i><font color="Green" title="E43">A35</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:10_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a>, <a class="ref" href="uniform1.html#T3" title="UNIFORM1:th.3">UNIFORM1:3</a></i>;<br/><a NAME="E12:10_1_5_1"/><i><font color="Green" title="E44">A36</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">s8m</font> = <font color="Maroon" title="c18">s8</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14" title="TOPMETR:th.14">TOPMETR:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">s8n</font> = <font color="Maroon" title="c18">s8</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E15:10_1_5_1"/><i><font color="Green" title="E45">A37</font></i>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#D8" title="TOPMETR:def.8">TOPMETR:def 8</a></i>;<br/><font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s8</font> = 
<font color="Maroon" title="c10">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s8</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E20:10_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">p8</font>
<b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_5"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
;<br/><a NAME="E17:10_1_5_1"/><b>then </b>
<font color="Maroon" title="c18">s8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">B</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E20:10_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="jordan20.html#E9:10_1_5_1"><i><font color="Green" title="E41">A33</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">r0</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="E19:10_1_5_1"/><i><font color="Green" title="E46">A38</font></i>: 
( <font color="Maroon" title="c24">r0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c22">s8m</font>,<font color="Maroon" title="c24">r0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">B</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E11:10_1_5_1"><i><font color="Green" title="E43">A35</font></i></a>, <a class="txt" href="jordan20.html#E15:10_1_5_1"><i><font color="Green" title="E45">A37</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">r0</font> = <font color="Maroon" title="c24">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">r01</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s8</font></span>)</span>,<font color="Maroon" title="c25">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E22:10_1_5_1"/>
<font color="Maroon" title="c8">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s8</font> <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="jordan20.html#E2:10_1_5_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E23:10_1_5_1"/><b>then </b><i><font color="Green" title="E47">A39</font></i>: 
<font color="Maroon" title="c26">r01</font> <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="jordan20.html#E19:10_1_5_1"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="square_1.html#T113" title="SQUARE_1:th.113">SQUARE_1:113</a></i>;<br/><a NAME="E24:10_1_5_1"/><b>then </b>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c23">s8n</font>,<font color="Maroon" title="c26">r01</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"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</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="frechet.html#T7" title="FRECHET:th.7">FRECHET:7</a></i>;<br/><a NAME="E25:10_1_5_1"/><b>then </b><i><font color="Green" title="E48">A40</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c22">s8m</font>,<font color="Maroon" title="c26">r01</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"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:10_1_5_1"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="topmetr.html#T13" title="TOPMETR:th.13">TOPMETR:13</a></i>;<br/><a NAME="E26:10_1_5_1"/>
<font color="Maroon" title="c8">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">r01</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E27:10_1_5_1"/><b>then </b><i><font color="Green" title="E49">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s8</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">r01</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">s8</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E28:10_1_5_1"/>
<font color="Maroon" title="c26">r01</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">r0</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><a NAME="E29:10_1_5_1"/><b>then </b><i><font color="Green" title="E50">A42</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c22">s8m</font>,<font color="Maroon" title="c26">r01</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c22">s8m</font>,<font color="Maroon" title="c25">r0</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">Q</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E31:10_1_5_1"/><i><font color="Green" title="E51">A43</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E27:10_1_5_1"><i><font color="Green" title="E49">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E32:10_1_5_1"/><i><font color="Green" title="E52">A44</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">B</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:10_1_5_1"><i><font color="Green" title="E46">A38</font></i></a>, <a class="txt" href="jordan20.html#E25:10_1_5_1"><i><font color="Green" title="E48">A40</font></i></a>, <a class="txt" href="jordan20.html#E29:10_1_5_1"><i><font color="Green" title="E50">A42</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:10_1_5_1"/><i><font color="Green" title="E53">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:10_1_5_1"><i><font color="Green" title="E47">A39</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E34:10_1_5_1"/><b>then </b><i><font color="Green" title="E54">A46</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E35:10_1_5_1"/><i><font color="Green" title="E55">A47</font></i>: 
<font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="jordan20.html#E23:10_1_5_1"><i><font color="Green" title="E47">A39</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E36:10_1_5_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:10_1_5_1"><i><font color="Green" title="E47">A39</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E37:10_1_5_1"/><b>then </b>
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E38:10_1_5_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font>
 ;<br/><a NAME="E39:10_1_5_1"/><b>then </b>
( <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> &amp; <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:10_1_5_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E40:10_1_5_1"/><b>then </b><i><font color="Green" title="E56">A48</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</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#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E41:10_1_5_1"/><i><font color="Green" title="E57">A49</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="jordan20.html#E35:10_1_5_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E42:10_1_5_1"/>
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E31:10_1_5_1"><i><font color="Green" title="E51">A43</font></i></a>, <a class="txt" href="jordan20.html#E34:10_1_5_1"><i><font color="Green" title="E54">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E43:10_1_5_1"/><b>then </b>
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E41:10_1_5_1"><i><font color="Green" title="E57">A49</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E44:10_1_5_1"/><b>then </b>
<font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c26">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c26">r01</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E40:10_1_5_1"><i><font color="Green" title="E56">A48</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E45:10_1_5_1"/><b>then </b><i><font color="Green" title="E58">A50</font></i>: 
( <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">h</font> &amp; <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">B</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E32:10_1_5_1"><i><font color="Green" title="E52">A44</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">s70</font> = <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E47:10_1_5_1"/><i><font color="Green" title="E59">A51</font></i>: 
<font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s8</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r01</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E35:10_1_5_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E48:10_1_5_1"/><i><font color="Green" title="E60">A52</font></i>: 
( <font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c28">s70</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">s70</font> &amp; <font color="Maroon" title="c28">s70</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="jordan20.html#E27:10_1_5_1"><i><font color="Green" title="E49">A41</font></i></a>, <a class="txt" href="jordan20.html#E34:10_1_5_1"><i><font color="Green" title="E54">A46</font></i></a>, <a class="txt" href="jordan20.html#E35:10_1_5_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E49:10_1_5_1"/>
for <font color="Olive" title="b1">p7</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> 2</span>)</span>  st <font color="Olive" title="b1">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s70</font> holds <br/><font color="Olive" title="b1">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">p7</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_5_1_3"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c29">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s70</font>
 ;<br/>

<a NAME="E2:10_1_5_1_3"/>
<font color="Maroon" title="c28">s70</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E48:10_1_5_1"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:10_1_5_1_3"/><b>then </b>
<font color="Maroon" title="c28">s70</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="jordan20.html#E47:10_1_5_1"><i><font color="Green" title="E59">A51</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:10_1_5_1_3"/><b>then </b><i><font color="Green" title="E62">A54</font></i>: 
<font color="Maroon" title="c29">p7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_5_1_3"><i><font color="Green" title="E61">A53</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green" title="E63">A55</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s70</font> = 
<font color="Maroon" title="c10">pro1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s70</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E45:10_1_5_1"><i><font color="Green" title="E58">A50</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c9">pr1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p7</font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_5_1_3"><i><font color="Green" title="E61">A53</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_5_1_3"><i><font color="Green" title="E62">A54</font></i></a>, <a class="txt" href="jordan20.html#E5:10_1_5_1_3"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c29">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
;<br/>
<a NAME="E7:10_1_5_1_3"/><b>then </b><i><font color="Green" title="E64">A56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">ee</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c29">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">ee</font> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:10_1_5_1"><i><font color="Green" title="E42">A34</font></i></a>, <a class="txt" href="jordan20.html#E45:10_1_5_1"><i><font color="Green" title="E58">A50</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_5_1_3_3"/><i><font color="Green" title="E65">A57</font></i>: 
<font color="Maroon" title="c29">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_5_1_3_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_3_3_1_1_1"><b>case </b></a><a NAME="E1:10_1_5_1_3_3_1_1_1"/><i><font color="Green" title="E66">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:10_1_5_1_3_3_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1_5_1_3"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_5_1_3_3"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_1_5_1_3_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:10_1_5_1_3_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:10_1_5_1_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E67">A59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:10_1_5_1_3_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_5_1_3_3_1_1_1"><i><font color="Green" title="E66">A58</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E7:10_1_5_1_3_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1_5_1_3_3_1_1_1"><i><font color="Green" title="E67">A59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5_1_3_3_1_1_2"><b>case </b></a><a NAME="E1:10_1_5_1_3_3_1_1_2"/><i><font color="Green" title="E66">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:10_1_5_1_3_3_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <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="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1_5_1_3"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_5_1_3_3"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_1_5_1_3_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:10_1_5_1_3_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:10_1_5_1_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E67">A61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:10_1_5_1_3_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_5_1_3_3_1_1_2"><i><font color="Green" title="E66">A60</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E7:10_1_5_1_3_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1_5_1_3_3_1_1_2"><i><font color="Green" title="E67">A61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_5_1_3_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:10_1_5_1_3"/>
<font color="Maroon" title="c29">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c29">s7</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E51:10_1_5_1"/><i><font color="Green" title="E61">A62</font></i>: 
( <font color="Maroon" title="c18">s8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c29">s7</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">s7</font> &amp; <font color="Maroon" title="c29">s7</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">p7</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> 2</span>)</span>  st <font color="Olive" title="b1">p7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">s7</font> holds <br/><font color="Olive" title="b1">p7</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E48:10_1_5_1"><i><font color="Green" title="E60">A52</font></i></a></i>;<br/><a NAME="E52:10_1_5_1"/>
<font color="Maroon" title="c29">s7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E51:10_1_5_1"><i><font color="Green" title="E61">A62</font></i></a></i>;<br/><a NAME="E53:10_1_5_1"/><b>then </b>
<font color="Maroon" title="c29">s7</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E39:10_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E54:10_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1_5"><i><font color="Green" title="E36">A28</font></i></a>, <a class="txt" href="jordan20.html#E51:10_1_5_1"><i><font color="Green" title="E61">A62</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:10_1_5"/>
<font color="Maroon" title="c16">p8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">p4</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> 2</span>)</span><b> such that </b><br/><a NAME="E43:10_1"/><i><font color="Green" title="E32">A63</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c16">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">p9</font> &amp; ( for <font color="Olive" title="b1">p5</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> or for <font color="Olive" title="b1">p6</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan20.html#E40:10_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="jordan20.html#D5" target="_self" title="JORDAN20:def.5">Def5</a></i>;<br/><a NAME="E44:10_1"/><i><font color="Green" title="E33">A64</font></i>: 
<font color="Maroon" title="c15">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_6_1"/>
( for <font color="Olive" title="b1">p5</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> or for <font color="Olive" title="b1">p6</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_1"><b>case </b></a><a NAME="E1:10_1_6_1_1"/><i><font color="Green" title="E34">A65</font></i>: 
for <font color="Olive" title="b1">p5</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><div><i><font color="Green" title="E35">A66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_1_1"/>
 ex <font color="Olive" title="b1">p51</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> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c17">p51</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> 2</span>)</span><b> such that </b><br/><a NAME="E3:10_1_6_1_1_1"/><i><font color="Green" title="E36">A67</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E4:10_1_6_1_1_1"/><i><font color="Green" title="E37">A68</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E40:10_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_1_1"><i><font color="Green" title="E36">A67</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E5:10_1_6_1_1_1"/>
for <font color="Olive" title="b1">p5</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">p5</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_6_1_1_1_1"/><i><font color="Green" title="E38">A69</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:10_1_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E39">A70</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:10_1_6_1_1_1"><i><font color="Green" title="E36">A67</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/>
<a NAME="E3:10_1_6_1_1_1_1"/><i><font color="Green" title="E40">A71</font></i>: 
<font color="Maroon" title="c18">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_1_1_1"><i><font color="Green" title="E38">A69</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:10_1_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E41">A72</font></i>: 
( <font color="Maroon" title="c18">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p9</font> implies  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 
<b>by </b><i><a class="ref" href="jordan5c.html#T9" title="JORDAN5C:th.9">JORDAN5C:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_6_1_1_1_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E44:10_1"><i><font color="Green" title="E33">A64</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_1_1_1"><i><font color="Green" title="E40">A71</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_1_1_1"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:10_1_6_1_1_1_1_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_6_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_1"><i><font color="Green" title="E34">A65</font></i></a>, <a class="txt" href="jordan20.html#E2:10_1_6_1_1_1_1"><i><font color="Green" title="E39">A70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:10_1_6_1_1_1_1_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_6_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E41:10_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_6_1_1_1_1"><i><font color="Green" title="E38">A69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_6_1_1_1_1"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:10_1_6_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_1_1"><i><font color="Green" title="E36">A67</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_1_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="ref" href="jordan20.html#D1" target="_self" title="JORDAN20:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_1_2"/><i><font color="Green" title="E36">A73</font></i>: 
for <font color="Olive" title="b1">p51</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> 2</span>)</span>  holds <br/>(  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E2:10_1_6_1_1_2"/><i><font color="Green" title="E37">A74</font></i>: 
for <font color="Olive" title="b1">p51</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p51</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_6_1_1_2_1"/><i><font color="Green" title="E38">A75</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:10_1_6_1_1_2_1"/><b>then </b><i><font color="Green" title="E39">A76</font></i>: 
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_1_2"><i><font color="Green" title="E36">A73</font></i></a></i>;<br/>
<a NAME="E3:10_1_6_1_1_2_1"/>
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_1"><i><font color="Green" title="E34">A65</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_6_1_1_2_1"><i><font color="Green" title="E38">A75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_6_1_1_2_1"/>
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_1_2_1"><i><font color="Green" title="E39">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:10_1_6_1_1_2"/>
<font color="Maroon" title="c16">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><a NAME="E4:10_1_6_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">xs4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:10_1_6_1_1_2"/><i><font color="Green" title="E38">A77</font></i>: 
( <font color="Maroon" title="c17">xs4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c16">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xs4</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">s4</font> = <font color="Maroon" title="c17">xs4</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E8:10_1_6_1_1_2"/><i><font color="Green" title="E39">A78</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">s4</font> &amp; <font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_6_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:10_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E40">A79</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E29:10_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_1_2_2"/>
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">s0</font>
 ;<br/><a NAME="E2:10_1_6_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font> <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="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E3:10_1_6_1_1_2_2"/><b>then </b><i><font color="Green" title="E41">A80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E4:10_1_6_1_1_2_2"/><b>then </b><i><font color="Green" title="E42">A81</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">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="E6:10_1_6_1_1_2_2"/><i><font color="Green" title="E43">A82</font></i>: 
( <font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font> &amp; <font color="Maroon" title="c13">s0</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"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E39:10_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_1_2_2"><i><font color="Green" title="E41">A80</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E7:10_1_6_1_1_2_2"/><i><font color="Green" title="E44">A83</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E39:10_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E8:10_1_6_1_1_2_2"/><i><font color="Green" title="E45">A84</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:10_1_6_1_1_2_2"><i><font color="Green" title="E42">A81</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">s7</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:10_1_6_1_1_2_2"/><i><font color="Green" title="E46">A85</font></i>: 
( <font color="Maroon" title="c20">s7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">s7</font> &amp; <font color="Maroon" title="c20">s7</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s7</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_6_1_1_2_2"><i><font color="Green" title="E43">A82</font></i></a></i>;<br/><a NAME="E11:10_1_6_1_1_2_2"/><i><font color="Green" title="E47">A86</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E10:10_1_6_1_1_2_2"><i><font color="Green" title="E46">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:10_1_6_1_1_2_2"/><b>then </b><i><font color="Green" title="E48">A87</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:10_1_6_1_1_2"><i><font color="Green" title="E39">A78</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_1_2_2"><i><font color="Green" title="E42">A81</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">rss</font> = <font color="Maroon" title="c19">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E14:10_1_6_1_1_2_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:10_1_6_1_1_2_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:10_1_6_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">pss</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E17:10_1_6_1_1_2_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c22">pss</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c22">pss</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="txt" href="jordan20.html#E8:10_1_6_1_1_2"><i><font color="Green" title="E39">A78</font></i></a>, <a class="txt" href="jordan20.html#E7:10_1_6_1_1_2_2"><i><font color="Green" title="E44">A83</font></i></a>, <a class="txt" href="jordan20.html#E8:10_1_6_1_1_2_2"><i><font color="Green" title="E45">A84</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1_6_1_1_2_2"><i><font color="Green" title="E47">A86</font></i></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></i>;<br/><a NAME="E18:10_1_6_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c22">pss</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_1_2"><i><font color="Green" title="E37">A74</font></i></a></i>;<br/><b>hence </b><a NAME="E19:10_1_6_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E10:10_1_6_1_1_2_2"><i><font color="Green" title="E46">A85</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:10_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="txt" href="jordan20.html#E9:10_1_6_1_1_2"><i><font color="Green" title="E40">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:10_1_6_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_1"><i><font color="Green" title="E35">A66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_2"><b>case </b></a><a NAME="E1:10_1_6_1_2"/><i><font color="Green" title="E34">A88</font></i>: 
for <font color="Olive" title="b1">p6</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p6</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p6</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><div><i><font color="Green" title="E35">A89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_2_1"/>
 ex <font color="Olive" title="b1">p51</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> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c17">p51</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> 2</span>)</span><b> such that </b><br/><a NAME="E3:10_1_6_1_2_1"/><i><font color="Green" title="E36">A90</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E4:10_1_6_1_2_1"/><i><font color="Green" title="E37">A91</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E40:10_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_2_1"><i><font color="Green" title="E36">A90</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E5:10_1_6_1_2_1"/>
for <font color="Olive" title="b1">p5</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">p5</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_6_1_2_1_1"/><i><font color="Green" title="E38">A92</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:10_1_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E39">A93</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:10_1_6_1_2_1"><i><font color="Green" title="E36">A90</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/>
<a NAME="E3:10_1_6_1_2_1_1"/><i><font color="Green" title="E40">A94</font></i>: 
<font color="Maroon" title="c18">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_2_1_1"><i><font color="Green" title="E38">A92</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:10_1_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E41">A95</font></i>: 
( <font color="Maroon" title="c18">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p9</font> implies  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 
<b>by </b><i><a class="ref" href="jordan5c.html#T9" title="JORDAN5C:th.9">JORDAN5C:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_6_1_2_1_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E44:10_1"><i><font color="Green" title="E33">A64</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_2_1_1"><i><font color="Green" title="E40">A94</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_2_1_1"><i><font color="Green" title="E41">A95</font></i></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:10_1_6_1_2_1_1_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_6_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_2"><i><font color="Green" title="E34">A88</font></i></a>, <a class="txt" href="jordan20.html#E2:10_1_6_1_2_1_1"><i><font color="Green" title="E39">A93</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_6_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:10_1_6_1_2_1_1_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c18">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_6_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E41:10_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_6_1_2_1_1"><i><font color="Green" title="E38">A92</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_6_1_2_1_1"/>
<font color="Maroon" title="c18">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:10_1_6_1_2_1"/>
<font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_2_1"><i><font color="Green" title="E36">A90</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_2_1"><i><font color="Green" title="E37">A91</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self" title="JORDAN20:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_2_2"/><i><font color="Green" title="E36">A96</font></i>: 
for <font color="Olive" title="b1">p51</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> 2</span>)</span>  holds <br/>(  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E2:10_1_6_1_2_2"/><i><font color="Green" title="E37">A97</font></i>: 
for <font color="Olive" title="b1">p51</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> 2</span>)</span>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p51</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10_1_6_1_2_2_1"/><i><font color="Green" title="E38">A98</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:10_1_6_1_2_2_1"/><b>then </b><i><font color="Green" title="E39">A99</font></i>: 
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_2_2"><i><font color="Green" title="E36">A96</font></i></a></i>;<br/>
<a NAME="E3:10_1_6_1_2_2_1"/>
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:10_1_6_1_2"><i><font color="Green" title="E34">A88</font></i></a>, <a class="txt" href="jordan20.html#E1:10_1_6_1_2_2_1"><i><font color="Green" title="E38">A98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_6_1_2_2_1"/>
<font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_2_2_1"><i><font color="Green" title="E39">A99</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:10_1_6_1_2_2"/>
<font color="Maroon" title="c16">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><a NAME="E4:10_1_6_1_2_2"/><b>then </b>
<font color="Maroon" title="c16">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">xs4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:10_1_6_1_2_2"/><i><font color="Green" title="E38">A100</font></i>: 
( <font color="Maroon" title="c17">xs4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c16">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xs4</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">s4</font> = <font color="Maroon" title="c17">xs4</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2"><i><font color="Green" title="E38">A100</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E8:10_1_6_1_2_2"/><i><font color="Green" title="E39">A101</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">s4</font> &amp; <font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_6_1_2_2"><i><font color="Green" title="E38">A100</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:10_1_6_1_2_2"/><b>then </b><i><font color="Green" title="E40">A102</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E29:10_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2"><i><font color="Green" title="E38">A100</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_6_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_6_1_2_2_2"/>
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">s0</font>
 ;<br/><a NAME="E2:10_1_6_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font> <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="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E3:10_1_6_1_2_2_2"/><b>then </b><i><font color="Green" title="E41">A103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E4:10_1_6_1_2_2_2"/><b>then </b><i><font color="Green" title="E42">A104</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">s4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">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="E6:10_1_6_1_2_2_2"/><i><font color="Green" title="E43">A105</font></i>: 
( <font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R</font> &amp; <font color="Maroon" title="c13">s0</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"><font color="Maroon" title="c13">s0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E39:10_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:10_1_6_1_2_2_2"><i><font color="Green" title="E41">A103</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E7:10_1_6_1_2_2_2"/><i><font color="Green" title="E44">A106</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">s0</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E39:10_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2_2"><i><font color="Green" title="E43">A105</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E8:10_1_6_1_2_2_2"/><i><font color="Green" title="E45">A107</font></i>: 
<font color="Maroon" title="c18">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:10_1_6_1_2_2_2"><i><font color="Green" title="E42">A104</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2_2"><i><font color="Green" title="E43">A105</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">s7</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:10_1_6_1_2_2_2"/><i><font color="Green" title="E46">A108</font></i>: 
( <font color="Maroon" title="c20">s7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">s7</font> &amp; <font color="Maroon" title="c20">s7</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">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> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s7</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:10_1_6_1_2_2_2"><i><font color="Green" title="E43">A105</font></i></a></i>;<br/><a NAME="E11:10_1_6_1_2_2_2"/><i><font color="Green" title="E47">A109</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E13:10_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan20.html#E10:10_1_6_1_2_2_2"><i><font color="Green" title="E46">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:10_1_6_1_2_2_2"/><b>then </b><i><font color="Green" title="E48">A110</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:10_1_6_1_2_2"><i><font color="Green" title="E39">A101</font></i></a>, <a class="txt" href="jordan20.html#E4:10_1_6_1_2_2_2"><i><font color="Green" title="E42">A104</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2_2"><i><font color="Green" title="E43">A105</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">rss</font> = <font color="Maroon" title="c19">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E14:10_1_6_1_2_2_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:10_1_6_1_2_2_2"><i><font color="Green" title="E48">A110</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:10_1_6_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:10_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">pss</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">rss</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E17:10_1_6_1_2_2_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p4</font>,<font color="Maroon" title="c22">pss</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c22">pss</font>,<font color="Maroon" title="c15">p9</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:10_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan20.html#E33:10_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2"><i><font color="Green" title="E38">A100</font></i></a>, <a class="txt" href="jordan20.html#E8:10_1_6_1_2_2"><i><font color="Green" title="E39">A101</font></i></a>, <a class="txt" href="jordan20.html#E7:10_1_6_1_2_2_2"><i><font color="Green" title="E44">A106</font></i></a>, <a class="txt" href="jordan20.html#E8:10_1_6_1_2_2_2"><i><font color="Green" title="E45">A107</font></i></a>, <a class="txt" href="jordan20.html#E11:10_1_6_1_2_2_2"><i><font color="Green" title="E47">A109</font></i></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></i>;<br/><a NAME="E18:10_1_6_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c22">pss</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_2_2"><i><font color="Green" title="E37">A97</font></i></a></i>;<br/><b>hence </b><a NAME="E19:10_1_6_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E10:10_1_6_1_2_2_2"><i><font color="Green" title="E46">A108</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:10_1_6_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E43:10_1"><i><font color="Green" title="E32">A63</font></i></a>, <a class="txt" href="jordan20.html#E6:10_1_6_1_2_2"><i><font color="Green" title="E38">A100</font></i></a>, <a class="txt" href="jordan20.html#E9:10_1_6_1_2_2"><i><font color="Green" title="E40">A102</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:10_1_6_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:10_1_6_1_2"><i><font color="Green" title="E35">A89</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E46:10_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R5" title="JORDAN20:pred.5">is_OSin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ;<br/>


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