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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:101_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:101_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 ;<br/>

<b>thus </b><a NAME="E3:101_1_1"/>
<font color="Maroon">f</font> is <a href="seq_2.html#V1">bounded_above</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">s</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a>,<a class="ref" href="pscomp_1.html#D12" target="_self">PSCOMP_1:def 12</a><br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span> ;<br/>
<a NAME="E3:101_1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span>  is   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<a NAME="E4:101_1_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="limfunc1.html#K3">right_closed_halfline</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E7:101_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">rchx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">rchx</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E4:101_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">rchx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:101_1_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">rchx</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:101_1_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">rchx</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><a NAME="E8:101_1_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">rchx</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E10:101_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E11:101_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r3</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:101_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E14:101_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><a NAME="E15:101_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E16:101_1_1_1_1"/>
<font color="Maroon">rchx</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r3</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/><a NAME="E17:101_1_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rchx</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E18:101_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:101_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> is <a href="setfam_1.html#V1">without_zero</a>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E10:101_1_1_1"/>
<font color="Maroon">F</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:101_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E4:101_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:101_1_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:101_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E8:101_1_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E9:101_1_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:101_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E12:101_1_1_1_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E13:101_1_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:101_1_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E16:101_1_1_1_2"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E17:101_1_1_1_2"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> )
 
;<br/>
<a NAME="E18:101_1_1_1_2"/><b>then </b><i><font color="Green">E193</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="limfunc1.html#T9">LIMFUNC1:9</a></i>;<br/>
<a NAME="E19:101_1_1_1_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, , , </i>;<br/>
<a NAME="E20:101_1_1_1_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E21:101_1_1_1_2"/>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:101_1_1_1"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">F</font> is <a href="compts_1.html#V1">centered</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:101_1_1_1"/>
<font color="Maroon">R</font> is <a href="pscomp_1.html#V4" target="_self">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D6" target="_self">PSCOMP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:101_1_1_1_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:101_1_1_1_3"/><b>then </b>
 ex <font color="Olive">r3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">r3</font> &amp; <font color="Olive">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:101_1_1_1_3"/>
<font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="fcont_3.html#T5">FCONT_3:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:101_1_1_1"/><b>then </b>
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:101_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="compts_1.html#T13">COMPTS_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:101_1_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E18:101_1_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">eR</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">R</font>;<br/>
<a NAME="E20:101_1_1_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">eR</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E21:101_1_1_1"/>
<font color="Maroon">eR</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon">er</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E23:101_1_1_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">eR</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">er</font> &amp; <font color="Maroon">er</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 ;<br/>
<a NAME="E24:101_1_1_1"/>
 <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">er</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r3</font> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">er</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E26:101_1_1_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">er</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E27:101_1_1_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">fx'</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E29:101_1_1_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">fx'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T3">XREAL_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fx'</font> = <font color="Maroon">fx'</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E31:101_1_1_1"/>
<font color="Maroon">fx'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:101_1_1_1"/><b>then </b>
 <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">fx'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E33:101_1_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">fx'</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E34:101_1_1_1"/>
 <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">fx'</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r3</font> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">fx'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/>
<a NAME="E35:101_1_1_1"/><b>then </b>
 ex <font color="Olive">r3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r3</font> &amp; <font color="Olive">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">fx'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E36:101_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a></i>;<br/>


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

<b>assume </b><a NAME="E4:101_1_1"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   ex <font color="Olive">r3</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Olive">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a>,<a class="ref" href="pscomp_1.html#D11" target="_self">PSCOMP_1:def 11</a><br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span> ;<br/>
<a NAME="E6:101_1_1"/><i><font color="Green">E206</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span>  is   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<a NAME="E7:101_1_1"/><i><font color="Green">E207</font></i>: 
 <a href="limfunc1.html#K2">left_closed_halfline</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r3</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E10:101_1_1"/><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">rchx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_1_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">rchx</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E4:101_1_1_2"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">rchx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:101_1_1_2"/><i><font color="Green">E211</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">rchx</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:101_1_1_2"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">rchx</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><a NAME="E8:101_1_1_2"/><i><font color="Green">E213</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">rchx</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E10:101_1_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E11:101_1_1_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:101_1_1_2"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E14:101_1_1_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><a NAME="E15:101_1_1_2"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">r1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E16:101_1_1_2"/>
<font color="Maroon">rchx</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r3</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="limfunc1.html#D1">LIMFUNC1:def 1</a></i>;<br/><a NAME="E17:101_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rchx</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E18:101_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:101_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">F</font> is <a href="setfam_1.html#V1">without_zero</a>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E13:101_1_1"/>
<font color="Maroon">F</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:101_1_1_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_1_1_3"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E4:101_1_1_3"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:101_1_1_3"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:101_1_1_3"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E8:101_1_1_3"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E9:101_1_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:101_1_1_3"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E12:101_1_1_3"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E13:101_1_1_3"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" target="_self">"</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:101_1_1_3"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E16:101_1_1_3"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E17:101_1_1_3"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> )
 
;<br/>
<a NAME="E18:101_1_1_3"/><b>then </b><i><font color="Green">E228</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="limfunc1.html#T14">LIMFUNC1:14</a></i>;<br/>
<a NAME="E19:101_1_1_3"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, , , </i>;<br/>
<a NAME="E20:101_1_1_3"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E21:101_1_1_3"/>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:101_1_1"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">F</font> is <a href="compts_1.html#V1">centered</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:101_1_1"/>
<font color="Maroon">R</font> is <a href="pscomp_1.html#V4" target="_self">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D6" target="_self">PSCOMP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:101_1_1_4"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:101_1_1_4"/><b>then </b>
 ex <font color="Olive">r3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">r3</font> &amp; <font color="Olive">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:101_1_1_4"/>
<font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="fcont_3.html#T6">FCONT_3:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:101_1_1"/><b>then </b>
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:101_1_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="compts_1.html#T13">COMPTS_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:101_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E21:101_1_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">eR</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">R</font>;<br/>
<a NAME="E23:101_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">eR</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E24:101_1_1"/>
<font color="Maroon">eR</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon">er</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E26:101_1_1"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">eR</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">er</font> &amp; <font color="Maroon">er</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 ;<br/>
<a NAME="E27:101_1_1"/>
 <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">er</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r3</font> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">er</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#D1">LIMFUNC1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E29:101_1_1"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">er</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E30:101_1_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">fx'</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E32:101_1_1"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">fx'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T4">XREAL_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fx'</font> = <font color="Maroon">fx'</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E34:101_1_1"/>
<font color="Maroon">fx'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:101_1_1"/><b>then </b>
 <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">fx'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
;<br/>
<a NAME="E36:101_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">fx'</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E37:101_1_1"/>
 <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">fx'</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r3</font> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fx'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#D1">LIMFUNC1:def 1</a></i>;<br/>
<a NAME="E38:101_1_1"/><b>then </b>
 ex <font color="Olive">r3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r3</font> &amp; <font color="Olive">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fx'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E39:101_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a></i>;<br/>


</div>
