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

<b>set </b><font color="Maroon">P</font> =  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<a NAME="E1:3_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jgraph_6.html#D2">JGRAPH_6:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">w1</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>, <font color="Maroon">w2</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a><br/>

<b>assume </b><a NAME="E2:3_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</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="E3:3_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font>
 ;<br/>

<a NAME="E4:3_1_1"/><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">p3</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="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p3</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:3_1_1"/><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">p4</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="Olive">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:3_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span>;<br/>
<a NAME="E9:3_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T59">EUCLID:59</a></i>;<br/>
<a NAME="E10:3_1_1"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</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"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T61">EUCLID:61</a></i>;<br/>
<a NAME="E11:3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E12:3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">w2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E13:3_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T175">XREAL_1:175</a>, <a class="ref" href="xreal_1.html#T176">XREAL_1:176</a></i>;<br/>
<b>hence </b><a NAME="E14:3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div>
