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

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

<b>set </b><font color="Maroon" title="c2">X</font> = <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span>;<br/>
<a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="jordan18.html#T4" target="_self" title="JORDAN18:th.4">Th4</a></i>;<br/>
<div><i><font color="Green" title="E9">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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="E1:7_1"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c4">x</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> 2</span>)</span>
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font>
 <b>and </b><br/><a NAME="E5:7_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan18.html#E3:7_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/><a NAME="E7:7_1"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E5:7_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>;<br/><b>hence </b><a NAME="E8:7_1"/>
<font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E4:7_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="topreal1.html#D12" title="TOPREAL1:def.12">TOPREAL1:def 12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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="E1:7_2"/><i><font color="Green" title="E10">A6</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  as  <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>take </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">r</font>;<br/><a NAME="E3:7_2"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a>, <a class="ref" href="topreal1.html#T45" title="TOPREAL1:th.45">TOPREAL1:45</a></i>;<br/><b>hence </b><a NAME="E4:7_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E5:7_2"/>
<font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E1:7_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E6:7_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="jordan18.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>


</div>
