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

<b>let </b><font color="Maroon">c<sub>22</sub></font> be  <a href="compts_1.html#V6">compact</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>assume </b><a NAME="E1:128"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1">realize-max-dist-in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<a NAME="E2:128"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T71" target="_self">Th71</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/>
<a NAME="E3:128"/><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:128"><i><font color="Green">E153</font></i></a>, <a class="ref" href="jordan24.html#D1">JORDAN24:def 1</a></i>;<br/>
<i><font color="Green">E156</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan.html#E1:128"><i><font color="Green">E153</font></i></a>, <a class="ref" href="jordan.html#T75" target="_self">Th75</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<i><font color="Green">E157</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan.html#E1:128"><i><font color="Green">E153</font></i></a>, <a class="ref" href="jordan.html#T75" target="_self">Th75</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<b>thus </b><a NAME="E6:128"/>
 <a href="pscomp_1.html#K23">W-most</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:128_3"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<a NAME="E2:128_3"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <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> <b>by </b><i><a class="txt" href="jordan.html#E1:128_3"><i><font color="Green">E158</font></i></a></i>;<br/>
<a NAME="E4:128_3"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:128_3"><i><font color="Green">E158</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:128_3"/>
 <a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E6:128_3"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan.html#E4:128_3"><i><font color="Green">E160</font></i></a>, <a class="txt" href="jordan.html#E4:128"><i><font color="Green">E156</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E7:128_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E2:128"><i><font color="Green">E154</font></i></a>, <a class="txt" href="jordan.html#E2:128_3"><i><font color="Green">E159</font></i></a></i>;<br/>
<a NAME="E8:128_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <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> st <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 
;<br/>
<a NAME="E9:128_3"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E6:128_3"><i><font color="Green">E161</font></i></a>, <a class="txt" href="jordan.html#E140"><i><font color="Green">Lemma123</font></i></a></i>;<br/>
<a NAME="E10:128_3"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:128_3"><i><font color="Green">E159</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:128_3"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:128"><i><font color="Green">E153</font></i></a>, <a class="ref" href="jordan.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E12:128_3"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:128_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E6:128_3"><i><font color="Green">E161</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:128"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E8:128"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:128"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E10:128"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E11:128"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan.html#E9:128"><i><font color="Green">E159</font></i></a>, <a class="txt" href="jordan.html#E3:128"><i><font color="Green">E155</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E12:128"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:128"><i><font color="Green">E156</font></i></a>, <a class="txt" href="jordan.html#E5:128"><i><font color="Green">E157</font></i></a>, <a class="txt" href="jordan.html#E95"><i><font color="Green">Lemma82</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<b>hence </b><a NAME="E13:128"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E3:128"><i><font color="Green">E155</font></i></a>, <a class="txt" href="jordan.html#E8:128"><i><font color="Green">E158</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
