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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:56"/>
 <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="E5:56"/><b>then </b>
( 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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> &amp; 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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green">E36</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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> &amp;  <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> &amp;  <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> &amp;  <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:56"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a><br/>

<b>assume </b><a NAME="E7:56"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><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="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
 ;<br/>

<a NAME="E8:56"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><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="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E9:56"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="sprect_3.html#E7:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E39</font></i>: <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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></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/>
<a NAME="E11:56"/><i><font color="Green">E40</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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:56"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:56"/><i><font color="Green">E42</font></i>: 
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:56"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:56"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E15:56"/><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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E11:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="sprect_3.html#E12:56"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E16:56"/>
 <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E10:56"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>

<a NAME="E17:56"/><i><font color="Green">E43</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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E18:56"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E19:56"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <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="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E13:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E20:56"/><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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E17:56"><i><font color="Green">E43</font></i></a>, <a class="txt" href="sprect_3.html#E18:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E21:56"/>
<span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E10:56"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>

<i><font color="Green">E45</font></i>: <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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></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="E23:56"/><i><font color="Green">E46</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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E24:56"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E25:56"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E13:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E26:56"/><b>then </b>
 <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E23:56"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sprect_3.html#E24:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E27:56"/>
 <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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E22:56"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>

<a NAME="E28:56"/><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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E29:56"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E30:56"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:56"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E13:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E31:56"/><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">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E28:56"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E29:56"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E32:56"/>
<span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><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">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E22:56"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>


</div>
