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

<b>let </b><font color="Maroon">f</font> be  <a href="sprect_1.html#V1">rectangular</a> <a href="struct_0.html#NM7">FinSequence</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">p</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">q</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>assume </b><a NAME="E1:53"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green">E63</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E8:53"/><b>then </b><i><font color="Green">E65</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E5:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E9:53"/><i><font color="Green">E66</font></i>: 
(  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E5:53"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E67</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E12:53"/><b>then </b>
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sprect_3.html#T5" target="_self">Th5</a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E14:53"/>
(  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sprect_3.html#T5" target="_self">Th5</a>, <a class="ref" href="sprect_3.html#T6" target="_self">Th6</a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green">E70</font></i>: 
(  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<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="E16:53"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E17:53"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">p'</font> = <font color="Maroon">x</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/>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E20:53"/><i><font color="Green">E88</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E21:53"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E22:53"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p'</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">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<i><font color="Green">E93</font></i>: <font color="Maroon">p'</font> <a href="euclid.html#K21">`1</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">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</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"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</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">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
;<br/>
<i><font color="Green">E94</font></i>: <font color="Maroon">p'</font> <a href="euclid.html#K22">`2</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">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</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"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</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">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
;<br/>
<a NAME="E25:53"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_3"/>
( <font color="Maroon">p'</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> or <font color="Maroon">p'</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> or <font color="Maroon">p'</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> or <font color="Maroon">p'</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_1"><b>suppose </b></a><a NAME="E1:53_3_1"/>
<font color="Maroon">p'</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, <a class="ref" href="sprect_3.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xreal_1.html#T182">XREAL_1:182</a></i>;<br/><b>then </b><font color="Maroon">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_2"><b>suppose </b></a><a NAME="E1:53_3_2"/>
<font color="Maroon">p'</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, <a class="ref" href="sprect_3.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xreal_1.html#T181">XREAL_1:181</a></i>;<br/><b>then </b><font color="Maroon">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_3"><b>suppose </b></a><a NAME="E1:53_3_3"/>
<font color="Maroon">p'</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_3"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="sprect_3.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xreal_1.html#T182">XREAL_1:182</a></i>;<br/><b>then </b><font color="Maroon">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_4"><b>suppose </b></a><a NAME="E1:53_3_4"/>
<font color="Maroon">p'</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_4"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="sprect_3.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xreal_1.html#T181">XREAL_1:181</a></i>;<br/><b>then </b><font color="Maroon">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
