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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:98_1_1"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D27" target="_self" title="PSCOMP_1:def.27">PSCOMP_1:def 27</a><br/>

<b>assume </b><a NAME="E2:98_1_1"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/>

<b>thus </b><a NAME="E3:98_1_1"/>
<font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1_1_1"/><i><font color="Green" title="E60">A3</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>   ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">s</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a>,<a class="ref" href="pscomp_1.html#D12" target="_self" title="PSCOMP_1:def.12">PSCOMP_1:def 12</a><br/>

<b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b><font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>set </b><font color="Maroon" title="c4">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span> ;<br/>
<a NAME="E3:98_1_1_1"/><i><font color="Green" title="E61">A4</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<a NAME="E4:98_1_1_1"/><i><font color="Green" title="E62">A5</font></i>: 
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="pscomp_1.html#E3:98_1_1_1"><i><font color="Green" title="E61">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">R</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E7:98_1_1_1"/><i><font color="Green" title="E63">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_1_1_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">rchx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:98_1_1_1_1"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E4:98_1_1_1_1"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E5:98_1_1_1_1"/><i><font color="Green" title="E66">A9</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:98_1_1_1_1"/><i><font color="Green" title="E67">A10</font></i>: 
( <font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c8">r3</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1_1_1"><i><font color="Green" title="E65">A8</font></i></a></i>;<br/><a NAME="E8:98_1_1_1_1"/><i><font color="Green" title="E68">A11</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:98_1_1_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="pscomp_1.html#E5:98_1_1_1_1"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self" title="PSCOMP_1:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">r1</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="E10:98_1_1_1_1"/><i><font color="Green" title="E69">A12</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E11:98_1_1_1_1"/><i><font color="Green" title="E70">A13</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:98_1_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:98_1_1_1_1"/><i><font color="Green" title="E71">A14</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>and </b><br/><a NAME="E14:98_1_1_1_1"/><i><font color="Green" title="E72">A15</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E10:98_1_1_1_1"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><a NAME="E15:98_1_1_1_1"/><i><font color="Green" title="E73">A16</font></i>: 
<font color="Maroon" title="c9">r1</font> is   <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="E16:98_1_1_1_1"/>
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">r3</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1_1_1"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="limfunc1.html#T154" title="LIMFUNC1:th.154">LIMFUNC1:154</a></i>;<br/><a NAME="E17:98_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E11:98_1_1_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="pscomp_1.html#E15:98_1_1_1_1"><i><font color="Green" title="E73">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E18:98_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:98_1_1_1_1"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="pscomp_1.html#E13:98_1_1_1_1"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="pscomp_1.html#E14:98_1_1_1_1"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:98_1_1_1"/><b>then </b><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E10:98_1_1_1"/>
<font color="Maroon" title="c6">F</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font>, <font color="Maroon" title="c8">Y</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="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:98_1_1_1_2"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:98_1_1_1_2"/><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E4:98_1_1_1_2"/><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E5:98_1_1_1_2"/><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:98_1_1_1_2"/><i><font color="Green" title="E68">A21</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c10">r1</font>
 <b>and </b><br/><a NAME="E8:98_1_1_1_2"/>
<font color="Maroon" title="c10">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1_1_2"><i><font color="Green" title="E66">A19</font></i></a></i>;<br/>
<b>assume </b><a NAME="E9:98_1_1_1_2"/>
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:98_1_1_1_2"/><i><font color="Green" title="E69">A22</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E12:98_1_1_1_2"/><i><font color="Green" title="E70">A23</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E13:98_1_1_1_2"/><i><font color="Green" title="E71">A24</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E15:98_1_1_1_2"/><i><font color="Green" title="E72">A25</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c12">r2</font>
 <b>and </b><br/><a NAME="E16:98_1_1_1_2"/>
<font color="Maroon" title="c12">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E12:98_1_1_1_2"><i><font color="Green" title="E70">A23</font></i></a></i>;<br/>
<a NAME="E17:98_1_1_1_2"/><i><font color="Green" title="E73">A26</font></i>: 
( <font color="Maroon" title="c10">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">r2</font> or <font color="Maroon" title="c12">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">r1</font> )
 
;<br/>
<a NAME="E18:98_1_1_1_2"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">B</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:98_1_1_1_2"><i><font color="Green" title="E65">A18</font></i></a>, <a class="txt" href="pscomp_1.html#E5:98_1_1_1_2"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="pscomp_1.html#E11:98_1_1_1_2"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="pscomp_1.html#E13:98_1_1_1_2"><i><font color="Green" title="E71">A24</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self" title="PSCOMP_1:def.2">Def2</a></i>;<br/>
<a NAME="E19:98_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> or <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1_1_2"><i><font color="Green" title="E68">A21</font></i></a>, <a class="txt" href="pscomp_1.html#E15:98_1_1_1_2"><i><font color="Green" title="E72">A25</font></i></a>, <a class="txt" href="pscomp_1.html#E17:98_1_1_1_2"><i><font color="Green" title="E73">A26</font></i></a>, <a class="ref" href="limfunc1.html#T9" title="LIMFUNC1:th.9">LIMFUNC1:9</a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E20:98_1_1_1_2"/>
<font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:98_1_1_1"/><b>then </b><i><font color="Green" title="E65">A27</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="pscomp_1.html#E9:98_1_1_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="pscomp_1.html#T1" target="_self" title="PSCOMP_1:th.1">Th1</a></i>;<br/>
<a NAME="E12:98_1_1_1"/>
<font color="Maroon" title="c5">R</font> is <a href="pscomp_1.html#V4" title="PSCOMP_1:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font> be   <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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D6" target="_self" title="PSCOMP_1:def.6">PSCOMP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:98_1_1_1_3"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/>

<a NAME="E2:98_1_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">r3</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:98_1_1_1_3"/>
<font color="Maroon" title="c7">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="fcont_3.html#T5" title="FCONT_3:th.5">FCONT_3:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:98_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:98_1_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T59" target="_self" title="PSCOMP_1:th.59">Th59</a></i>;<br/>
<a NAME="E14:98_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:98_1_1"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E11:98_1_1_1"><i><font color="Green" title="E65">A27</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:98_1_1_1"/><i><font color="Green" title="E66">A28</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="pscomp_1.html#E16:98_1_1_1"><i><font color="Green" title="E66">A28</font></i></a></i>;<br/>
<a NAME="E18:98_1_1_1"/><i><font color="Green" title="E67">A29</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E16:98_1_1_1"><i><font color="Green" title="E66">A28</font></i></a>, <a class="ref" href="pscomp_1.html#T2" target="_self" title="PSCOMP_1:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">eR</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">R</font>;<br/>
<a NAME="E20:98_1_1_1"/><i><font color="Green" title="E68">A30</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">eR</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E18:98_1_1_1"><i><font color="Green" title="E67">A29</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E21:98_1_1_1"/>
<font color="Maroon" title="c9">eR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">er</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E23:98_1_1_1"/><i><font color="Green" title="E69">A31</font></i>: 
( <font color="Maroon" title="c9">eR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c10">er</font> &amp; <font color="Maroon" title="c10">er</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 ;<br/>
<a NAME="E24:98_1_1_1"/>
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c10">er</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r3</font> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">er</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T154" title="LIMFUNC1:th.154">LIMFUNC1:154</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E26:98_1_1_1"/><i><font color="Green" title="E70">A32</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> &amp; <font color="Maroon" title="c11">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">er</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E20:98_1_1_1"><i><font color="Green" title="E68">A30</font></i></a>, <a class="txt" href="pscomp_1.html#E23:98_1_1_1"><i><font color="Green" title="E69">A31</font></i></a></i>;<br/>
<a NAME="E27:98_1_1_1"/><i><font color="Green" title="E71">A33</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E23:98_1_1_1"><i><font color="Green" title="E69">A31</font></i></a>, <a class="txt" href="pscomp_1.html#E26:98_1_1_1"><i><font color="Green" title="E70">A32</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="c12">fx'</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="E29:98_1_1_1"/><i><font color="Green" title="E72">A34</font></i>: 
<font color="Maroon" title="c12">fx'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T3" title="XREAL_1:th.3">XREAL_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">fx'</font> = <font color="Maroon" title="c12">fx'</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="E31:98_1_1_1"/>
<font color="Maroon" title="c13">fx'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E27:98_1_1_1"><i><font color="Green" title="E71">A33</font></i></a>, <a class="txt" href="pscomp_1.html#E29:98_1_1_1"><i><font color="Green" title="E72">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:98_1_1_1"/><b>then </b>
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c13">fx'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
;<br/>
<a NAME="E33:98_1_1_1"/><b>then </b><i><font color="Green" title="E73">A35</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c13">fx'</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E18:98_1_1_1"><i><font color="Green" title="E67">A29</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E34:98_1_1_1"/>
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c13">fx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r3</font> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">fx'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T154" title="LIMFUNC1:th.154">LIMFUNC1:154</a></i>;<br/>
<a NAME="E35:98_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r3</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">fx'</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E33:98_1_1_1"><i><font color="Green" title="E73">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E36:98_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E29:98_1_1_1"><i><font color="Green" title="E72">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:98_1_1"/><i><font color="Green" title="E60">A36</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>   ex <font color="Olive" title="b2">r3</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 <br/>( <font color="Olive" title="b2">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a>,<a class="ref" href="pscomp_1.html#D11" target="_self" title="PSCOMP_1:def.11">PSCOMP_1:def 11</a><br/>

<b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b><font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>set </b><font color="Maroon" title="c4">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span> ;<br/>
<a NAME="E6:98_1_1"/><i><font color="Green" title="E61">A37</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<a NAME="E7:98_1_1"/><i><font color="Green" title="E62">A38</font></i>: 
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r3</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="pscomp_1.html#E6:98_1_1"><i><font color="Green" title="E61">A37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">R</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E10:98_1_1"/><i><font color="Green" title="E63">A39</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">rchx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:98_1_1_2"/><i><font color="Green" title="E64">A40</font></i>: 
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E4:98_1_1_2"/><i><font color="Green" title="E65">A41</font></i>: 
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E5:98_1_1_2"/><i><font color="Green" title="E66">A42</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:98_1_1_2"/><i><font color="Green" title="E67">A43</font></i>: 
( <font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c8">r3</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1_2"><i><font color="Green" title="E65">A41</font></i></a></i>;<br/><a NAME="E8:98_1_1_2"/><i><font color="Green" title="E68">A44</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:98_1_1_2"><i><font color="Green" title="E64">A40</font></i></a>, <a class="txt" href="pscomp_1.html#E5:98_1_1_2"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self" title="PSCOMP_1:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">r1</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="E10:98_1_1_2"/><i><font color="Green" title="E69">A45</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E11:98_1_1_2"/><i><font color="Green" title="E70">A46</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r3</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1"><i><font color="Green" title="E60">A36</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:98_1_1_2"/><i><font color="Green" title="E71">A47</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>and </b><br/><a NAME="E14:98_1_1_2"/><i><font color="Green" title="E72">A48</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E10:98_1_1_2"><i><font color="Green" title="E69">A45</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><a NAME="E15:98_1_1_2"/><i><font color="Green" title="E73">A49</font></i>: 
<font color="Maroon" title="c9">r1</font> is   <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="E16:98_1_1_2"/>
<font color="Maroon" title="c7">rchx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r3</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1_2"><i><font color="Green" title="E67">A43</font></i></a>, <a class="ref" href="limfunc1.html#T153" title="LIMFUNC1:th.153">LIMFUNC1:153</a></i>;<br/><a NAME="E17:98_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rchx</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E11:98_1_1_2"><i><font color="Green" title="E70">A46</font></i></a>, <a class="txt" href="pscomp_1.html#E15:98_1_1_2"><i><font color="Green" title="E73">A49</font></i></a></i>;<br/><b>hence </b><a NAME="E18:98_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:98_1_1_2"><i><font color="Green" title="E68">A44</font></i></a>, <a class="txt" href="pscomp_1.html#E13:98_1_1_2"><i><font color="Green" title="E71">A47</font></i></a>, <a class="txt" href="pscomp_1.html#E14:98_1_1_2"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:98_1_1"/><b>then </b><i><font color="Green" title="E64">A50</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E13:98_1_1"/>
<font color="Maroon" title="c6">F</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font>, <font color="Maroon" title="c8">Y</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="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:98_1_1_3"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:98_1_1_3"/><i><font color="Green" title="E65">A51</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E4:98_1_1_3"/><i><font color="Green" title="E66">A52</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E5:98_1_1_3"/><i><font color="Green" title="E67">A53</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:98_1_1_3"/><i><font color="Green" title="E68">A54</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c10">r1</font>
 <b>and </b><br/><a NAME="E8:98_1_1_3"/>
<font color="Maroon" title="c10">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:98_1_1_3"><i><font color="Green" title="E66">A52</font></i></a></i>;<br/>
<b>assume </b><a NAME="E9:98_1_1_3"/>
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:98_1_1_3"/><i><font color="Green" title="E69">A55</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E12:98_1_1_3"/><i><font color="Green" title="E70">A56</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>and </b><br/><a NAME="E13:98_1_1_3"/><i><font color="Green" title="E71">A57</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E15:98_1_1_3"/><i><font color="Green" title="E72">A58</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c12">r2</font>
 <b>and </b><br/><a NAME="E16:98_1_1_3"/>
<font color="Maroon" title="c12">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E12:98_1_1_3"><i><font color="Green" title="E70">A56</font></i></a></i>;<br/>
<a NAME="E17:98_1_1_3"/><i><font color="Green" title="E73">A59</font></i>: 
( <font color="Maroon" title="c10">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font> or <font color="Maroon" title="c12">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r1</font> )
 
;<br/>
<a NAME="E18:98_1_1_3"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">B</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:98_1_1_3"><i><font color="Green" title="E65">A51</font></i></a>, <a class="txt" href="pscomp_1.html#E5:98_1_1_3"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="pscomp_1.html#E11:98_1_1_3"><i><font color="Green" title="E69">A55</font></i></a>, <a class="txt" href="pscomp_1.html#E13:98_1_1_3"><i><font color="Green" title="E71">A57</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self" title="PSCOMP_1:def.2">Def2</a></i>;<br/>
<a NAME="E19:98_1_1_3"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> or <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E7:98_1_1_3"><i><font color="Green" title="E68">A54</font></i></a>, <a class="txt" href="pscomp_1.html#E15:98_1_1_3"><i><font color="Green" title="E72">A58</font></i></a>, <a class="txt" href="pscomp_1.html#E17:98_1_1_3"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="limfunc1.html#T14" title="LIMFUNC1:th.14">LIMFUNC1:14</a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E20:98_1_1_3"/>
<font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:98_1_1"/><b>then </b><i><font color="Green" title="E65">A60</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E10:98_1_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="pscomp_1.html#E12:98_1_1"><i><font color="Green" title="E64">A50</font></i></a>, <a class="ref" href="pscomp_1.html#T1" target="_self" title="PSCOMP_1:th.1">Th1</a></i>;<br/>
<a NAME="E15:98_1_1"/>
<font color="Maroon" title="c5">R</font> is <a href="pscomp_1.html#V4" title="PSCOMP_1:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font> be   <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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D6" target="_self" title="PSCOMP_1:def.6">PSCOMP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:98_1_1_4"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 ;<br/>

<a NAME="E2:98_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">r3</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:98_1_1_4"/>
<font color="Maroon" title="c7">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="fcont_3.html#T6" title="FCONT_3:th.6">FCONT_3:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:98_1_1"/><b>then </b>
<font color="Maroon" title="c6">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:98_1_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T59" target="_self" title="PSCOMP_1:th.59">Th59</a></i>;<br/>
<a NAME="E17:98_1_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:98_1_1"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E14:98_1_1"><i><font color="Green" title="E65">A60</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:98_1_1"/><i><font color="Green" title="E66">A61</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="pscomp_1.html#E19:98_1_1"><i><font color="Green" title="E66">A61</font></i></a></i>;<br/>
<a NAME="E21:98_1_1"/><i><font color="Green" title="E67">A62</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E19:98_1_1"><i><font color="Green" title="E66">A61</font></i></a>, <a class="ref" href="pscomp_1.html#T2" target="_self" title="PSCOMP_1:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">eR</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">R</font>;<br/>
<a NAME="E23:98_1_1"/><i><font color="Green" title="E68">A63</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">eR</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E21:98_1_1"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E24:98_1_1"/>
<font color="Maroon" title="c9">eR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">er</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E26:98_1_1"/><i><font color="Green" title="E69">A64</font></i>: 
( <font color="Maroon" title="c9">eR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c10">er</font> &amp; <font color="Maroon" title="c10">er</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 ;<br/>
<a NAME="E27:98_1_1"/>
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c10">er</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r3</font> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">er</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T153" title="LIMFUNC1:th.153">LIMFUNC1:153</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E29:98_1_1"/><i><font color="Green" title="E70">A65</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> &amp; <font color="Maroon" title="c11">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">er</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E23:98_1_1"><i><font color="Green" title="E68">A63</font></i></a>, <a class="txt" href="pscomp_1.html#E26:98_1_1"><i><font color="Green" title="E69">A64</font></i></a></i>;<br/>
<a NAME="E30:98_1_1"/><i><font color="Green" title="E71">A66</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E26:98_1_1"><i><font color="Green" title="E69">A64</font></i></a>, <a class="txt" href="pscomp_1.html#E29:98_1_1"><i><font color="Green" title="E70">A65</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="c12">fx'</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="E32:98_1_1"/><i><font color="Green" title="E72">A67</font></i>: 
<font color="Maroon" title="c12">fx'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T4" title="XREAL_1:th.4">XREAL_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">fx'</font> = <font color="Maroon" title="c12">fx'</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="E34:98_1_1"/>
<font color="Maroon" title="c13">fx'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E30:98_1_1"><i><font color="Green" title="E71">A66</font></i></a>, <a class="txt" href="pscomp_1.html#E32:98_1_1"><i><font color="Green" title="E72">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:98_1_1"/><b>then </b>
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c13">fx'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
;<br/>
<a NAME="E36:98_1_1"/><b>then </b><i><font color="Green" title="E73">A68</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c13">fx'</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E21:98_1_1"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E37:98_1_1"/>
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c13">fx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r3</font> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">fx'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T153" title="LIMFUNC1:th.153">LIMFUNC1:153</a></i>;<br/>
<a NAME="E38:98_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r3</font> &amp; <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">fx'</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E36:98_1_1"><i><font color="Green" title="E73">A68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E39:98_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E32:98_1_1"><i><font color="Green" title="E72">A67</font></i></a></i>;<br/>


</div>
