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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b2">G</font> being  <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>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">q</font></span>)</span> where <font color="Olive" title="b3">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> </span>}</span>  &amp; <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Olive" title="b2">G</font> )
 ;<br/>

<a NAME="E2:5"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T27" title="BORSUK_3:th.27">BORSUK_3:27</a></i>;<br/>
<a NAME="E3:5"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="jordan_a.html#T2" target="_self" title="JORDAN_A:th.2">Th2</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28" title="RCOMP_1:th.28">RCOMP_1:28</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E6:5"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E4:5_1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E6:5_1"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E8:5_1"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">p</font>,<font color="Olive" title="b1">q1</font></span>)</span> where <font color="Olive" title="b1">q1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> </span>}</span> 
 <b>and </b><br/><a NAME="E9:5_1"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="jordan_a.html#E4:5_1"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<a NAME="E10:5_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E6:5_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="jordan_a.html#E8:5_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<a NAME="E11:5_1"/>
<font color="Maroon" title="c9">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:5_1_1"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">q1</font>
 <b>and </b><br/><a NAME="E4:5_1_1"/><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E8:5_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<a NAME="E5:5_1_1"/><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">q1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:5_1_1"><i><font color="Green" title="E14">A11</font></i></a></i>;<br/>
<a NAME="E6:5_1_1"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">q1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:5_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="jordan_a.html#E4:5_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:5_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E5:5_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:5_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="jordan_a.html#E9:5_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="jordan_a.html#E10:5_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="jordan_a.html#T3" target="_self" title="JORDAN_A:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:5"/><b>then </b><i><font color="Green" title="E8">A14</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E5:5"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T3" title="RCOMP_1:th.3">RCOMP_1:3</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green" title="E9">A15</font></i>: 
for <font color="Olive" title="b1">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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> holds <br/> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">pq</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E3:5_2"/><i><font color="Green" title="E10">A16</font></i>: 
<font color="Maroon" title="c7">pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E4:5_2"/><i><font color="Green" title="E11">A17</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pq</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:5_2"/>
<font color="Maroon" title="c7">pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
;<br/>
<a NAME="E6:5_2"/><b>then </b>
<font color="Maroon" title="c7">pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:5_2"/><i><font color="Green" title="E12">A18</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E9:5_2"/><i><font color="Green" title="E13">A19</font></i>: 
<font color="Maroon" title="c7">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E10:5_2"/><i><font color="Green" title="E14">A20</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:5_2"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="jordan_a.html#E9:5_2"><i><font color="Green" title="E13">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan_a.html#E8:5_2"><i><font color="Green" title="E12">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E13:5_2"/><i><font color="Green" title="E15">A21</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">p</font>,<font color="Olive" title="b1">q1</font></span>)</span> where <font color="Olive" title="b1">q1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> </span>}</span> 
 <b>and </b><br/><a NAME="E14:5_2"/><i><font color="Green" title="E16">A22</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c12">G</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E15:5_2"/><i><font color="Green" title="E17">A23</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:5_2"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="jordan_a.html#E9:5_2"><i><font color="Green" title="E13">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:5_2"/>
<font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_2_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:5_2_1"/><i><font color="Green" title="E18">A24</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c14">q1</font>
 <b>and </b><br/><a NAME="E4:5_2_1"/><i><font color="Green" title="E19">A25</font></i>: 
<font color="Maroon" title="c14">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E13:5_2"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/>
<a NAME="E5:5_2_1"/><i><font color="Green" title="E20">A26</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c14">q1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:5_2_1"><i><font color="Green" title="E18">A24</font></i></a></i>;<br/>
<a NAME="E6:5_2_1"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c14">q1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E15:5_2"><i><font color="Green" title="E17">A23</font></i></a>, <a class="txt" href="jordan_a.html#E4:5_2_1"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:5_2_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E5:5_2_1"><i><font color="Green" title="E20">A26</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:5_2"/><b>then </b><i><font color="Green" title="E18">A27</font></i>: 
<font color="Maroon" title="c12">G</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E5:5"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T3" title="RCOMP_1:th.3">RCOMP_1:3</a></i>;<br/>
<a NAME="E18:5_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E4:5_2"><i><font color="Green" title="E11">A17</font></i></a>, <a class="txt" href="jordan_a.html#E9:5_2"><i><font color="Green" title="E13">A19</font></i></a></i>;<br/>
<a NAME="E19:5_2"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E10:5_2"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="jordan_a.html#E13:5_2"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/>
<a NAME="E20:5_2"/><b>then </b><i><font color="Green" title="E19">A28</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E14:5_2"><i><font color="Green" title="E16">A22</font></i></a>, <a class="txt" href="jordan_a.html#E17:5_2"><i><font color="Green" title="E18">A27</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E21:5_2"/>
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E15:5_2"><i><font color="Green" title="E17">A23</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E22:5_2"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E7:5"><i><font color="Green" title="E8">A14</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E23:5_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E20:5_2"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:5"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> 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="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:5_3"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:5_3"/><i><font color="Green" title="E10">A29</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E4:5_3"/><i><font color="Green" title="E11">A30</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E7:5"><i><font color="Green" title="E8">A14</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">r</font>
;<br/>

<b>thus </b><a NAME="E5:5_3"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E6:5"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="jordan_a.html#E3:5_3"><i><font color="Green" title="E10">A29</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:5_3"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:5_3"><i><font color="Green" title="E11">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:5"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E5:5"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="jordan_a.html#E8:5"><i><font color="Green" title="E9">A15</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>


</div>
