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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; not <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <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"><font color="Maroon" title="c2">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c2">B</font></span>)</span> implies  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green" title="E40">A3</font></i>: 
not <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green" title="E41">A4</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"><font color="Maroon" title="c2">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green" title="E42">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c2">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>set </b><font color="Maroon" title="c3">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<a NAME="E6:54"/><i><font color="Green" title="E43">A6</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E7:54"/><i><font color="Green" title="E44">A7</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</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"><font color="Maroon" title="c2">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E5:54"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green" title="E45">A8</font></i>: 
not <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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="jordan21.html#E3:54"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan21.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green" title="E46">A9</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E10:54"/><b>then </b><i><font color="Green" title="E47">A10</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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c2">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green" title="E48">A11</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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="jordan21.html#E4:54"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan21.html#E5:54"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="jordan21.html#E9:54"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T4" title="RCOMP_1:th.4">RCOMP_1:4</a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E12:54"/><i><font color="Green" title="E49">A12</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E13:54"/><i><font color="Green" title="E50">A13</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan21.html#E5:54"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E14:54"/><i><font color="Green" title="E51">A14</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> <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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font>
 
<b>by </b><i><a class="txt" href="jordan21.html#E4:54"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan21.html#E5:54"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="jordan21.html#E7:54"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan21.html#E10:54"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E15:54"/>
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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s</font> 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> <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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">s</font> implies  ex <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 <br/>( <font color="Olive" title="b1">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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> ) )</span><br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E52">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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 <br/>( <font color="Olive" title="b1">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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> )</span><br/>

<b>take </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )</span><br/>

<a NAME="E2:54_1"/><i><font color="Green" title="E53">A16</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E3:54_1"/>
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E13:54"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="jordan6.html#T34" title="JORDAN6:th.34">JORDAN6:34</a></i>;<br/>
<a NAME="E4:54_1"/><b>then </b>
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E2:54"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan21.html#E2:54_1"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span><br/>

<a NAME="E6:54_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan21.html#E1:54_1"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E7:54_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:54"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan21.html#E6:54"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="jordan21.html#E8:54"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan21.html#E11:54"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="jordan21.html#E14:54"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:54"/>
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="jordan21.html#E12:54"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="jordan21.html#E13:54"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
