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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> )
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1c.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E3:22"/>
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T33">SPRECT_1:33</a></i>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E5:22"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>3</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T114">JORDAN2C:114</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:22"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan1c.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:22"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E2:22"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:22"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E9:22"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<a NAME="E10:22"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E11:22"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:22"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E13:22"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:22"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1c.html#E11:22"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E15:22"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:22"><i><font color="Green">E27</font></i></a>, <a class="ref" href="int_1.html#T51">INT_1:51</a></i>;<br/>


</div>
