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

<b>let </b><font color="Maroon" title="c1">z</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/>

<b>set </b><font color="Maroon" title="c2">i1</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>;<br/>
<b>set </b><font color="Maroon" title="c3">i2</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>;<br/>
<b>set </b><font color="Maroon" title="c4">j</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:93"/><i><font color="Green" title="E84">A1</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:93"/><i><font color="Green" title="E85">A2</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>
 ;<br/>

<a NAME="E3:93"/><i><font color="Green" title="E86">A3</font></i>: 
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T49" target="_self" title="SPRECT_2:th.49">Th49</a></i>;<br/>
<a NAME="E4:93"/><i><font color="Green" title="E87">A4</font></i>: 
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T45" target="_self" title="SPRECT_2:th.45">Th45</a></i>;<br/>
<a NAME="E5:93"/><i><font color="Green" title="E88">A5</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T47" target="_self" title="SPRECT_2:th.47">Th47</a></i>;<br/>
<a NAME="E6:93"/><i><font color="Green" title="E89">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E3:93"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E7:93"/><b>then </b><i><font color="Green" title="E90">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:93"/><i><font color="Green" title="E91">A8</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E5:93"><i><font color="Green" title="E88">A5</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E9:93"/><b>then </b><i><font color="Green" title="E92">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E93">A10</font></i>: <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E8:93"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E5:93"><i><font color="Green" title="E88">A5</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E11:93"/><i><font color="Green" title="E94">A11</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:93"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/>
<i><font color="Green" title="E95">A12</font></i>: <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E6:93"><i><font color="Green" title="E89">A6</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:93"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E13:93"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E14:93"/><b>then </b>
<font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:93"><i><font color="Green" title="E93">A10</font></i></a>, <a class="txt" href="sprect_2.html#E12:93"><i><font color="Green" title="E95">A12</font></i></a>, <a class="ref" href="topreal5.html#T23" title="TOPREAL5:th.23">TOPREAL5:23</a></i>;<br/>
<a NAME="E15:93"/><b>then </b><i><font color="Green" title="E96">A13</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E2:93"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E16:93"/><b>then </b><i><font color="Green" title="E97">A14</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E9:93"><i><font color="Green" title="E92">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:93"/><i><font color="Green" title="E98">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E18:93"/><i><font color="Green" title="E99">A16</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E4:93"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E19:93"/><b>then </b><i><font color="Green" title="E100">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E101">A18</font></i>: <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E18:93"><i><font color="Green" title="E99">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E4:93"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E21:93"/>
(  <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">W-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="sprect_1.html#T15" title="SPRECT_1:th.15">SPRECT_1:15</a></i>;<br/>
<a NAME="E22:93"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E23:93"/><b>then </b>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T61" target="_self" title="SPRECT_2:th.61">Th61</a></i>;<br/>
<a NAME="E24:93"/><b>then </b><i><font color="Green" title="E102">A19</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:93"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="sprect_2.html#E9:93"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="sprect_2.html#E10:93"><i><font color="Green" title="E93">A10</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<i><font color="Green" title="E103">A20</font></i>: <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E18:93"><i><font color="Green" title="E99">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E4:93"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E26:93"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E27:93"/><b>then </b>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal5.html#T22" title="TOPREAL5:th.22">TOPREAL5:22</a></i>;<br/>
<a NAME="E28:93"/><b>then </b><i><font color="Green" title="E104">A21</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E11:93"><i><font color="Green" title="E94">A11</font></i></a>, <a class="txt" href="sprect_2.html#E19:93"><i><font color="Green" title="E100">A17</font></i></a>, <a class="txt" href="sprect_2.html#E25:93"><i><font color="Green" title="E103">A20</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E29:93"/><i><font color="Green" title="E105">A22</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:93"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="sprect_2.html#E90"><i><font color="Green" title="E81">Lm9</font></i></a></i>;<br/>
<a NAME="E30:93"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:93"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="sprect_2.html#E16:93"><i><font color="Green" title="E97">A14</font></i></a>, <a class="txt" href="sprect_2.html#E90"><i><font color="Green" title="E81">Lm9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">h</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">z</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font></span>)</span> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E28:93"><i><font color="Green" title="E104">A21</font></i></a>, <a class="ref" href="sprect_2.html#T42" target="_self" title="SPRECT_2:th.42">Th42</a></i>;<br/>
<a NAME="E32:93"/><i><font color="Green" title="E106">A23</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E17:93"><i><font color="Green" title="E98">A15</font></i></a>, <a class="txt" href="sprect_2.html#E18:93"><i><font color="Green" title="E99">A16</font></i></a>, <a class="ref" href="sprect_2.html#T25" target="_self" title="SPRECT_2:th.25">Th25</a>, <a class="ref" href="sprect_2.html#T26" target="_self" title="SPRECT_2:th.26">Th26</a></i>;<br/>
<a NAME="E33:93"/>
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E17:93"><i><font color="Green" title="E98">A15</font></i></a>, <a class="txt" href="sprect_2.html#E18:93"><i><font color="Green" title="E99">A16</font></i></a>, <a class="txt" href="sprect_2.html#E20:93"><i><font color="Green" title="E101">A18</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self" title="SPRECT_2:th.12">Th12</a></i>;<br/>
<a NAME="E34:93"/><b>then </b><i><font color="Green" title="E107">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E35:93"/>
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E17:93"><i><font color="Green" title="E98">A15</font></i></a>, <a class="txt" href="sprect_2.html#E18:93"><i><font color="Green" title="E99">A16</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self" title="SPRECT_2:th.13">Th13</a></i>;<br/>
<a NAME="E36:93"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E11:93"><i><font color="Green" title="E94">A11</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E37:93"/><b>then </b><i><font color="Green" title="E108">A25</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="sprect_2.html#R3" title="SPRECT_2:pred.3">is_a_v.c._for</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E32:93"><i><font color="Green" title="E106">A23</font></i></a>, <a class="txt" href="sprect_2.html#E34:93"><i><font color="Green" title="E107">A24</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self" title="SPRECT_2:def.3">Def3</a></i>;<br/>
<a NAME="E38:93"/>
 <a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T13" title="SPRECT_1:th.13">SPRECT_1:13</a></i>;<br/>
<a NAME="E39:93"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E40:93"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T55" target="_self" title="SPRECT_2:th.55">Th55</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:93"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E42:93"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:93"><i><font color="Green" title="E90">A7</font></i></a>, <a class="txt" href="sprect_2.html#E11:93"><i><font color="Green" title="E94">A11</font></i></a>, <a class="txt" href="sprect_2.html#E12:93"><i><font color="Green" title="E95">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">M</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">z</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E9:93"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="sprect_2.html#E15:93"><i><font color="Green" title="E96">A13</font></i></a>, <a class="ref" href="sprect_2.html#T42" target="_self" title="SPRECT_2:th.42">Th42</a></i>;<br/>
<a NAME="E44:93"/><i><font color="Green" title="E109">A26</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<i><font color="Green" title="E110">A27</font></i>: <font color="Maroon" title="c6">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font></span>)</span> = 
<font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E6:93"><i><font color="Green" title="E89">A6</font></i></a>, <a class="txt" href="sprect_2.html#E8:93"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self" title="SPRECT_2:th.13">Th13</a></i>
<br/>.= 
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:93"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="finseq_5.html#T41" title="FINSEQ_5:th.41">FINSEQ_5:41</a></i>
;<br/>
<a NAME="E46:93"/><i><font color="Green" title="E111">A28</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:93"><i><font color="Green" title="E89">A6</font></i></a>, <a class="txt" href="sprect_2.html#E8:93"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="sprect_2.html#E10:93"><i><font color="Green" title="E93">A10</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self" title="SPRECT_2:th.12">Th12</a></i>;<br/>
<a NAME="E47:93"/><i><font color="Green" title="E112">A29</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:93"><i><font color="Green" title="E89">A6</font></i></a>, <a class="txt" href="sprect_2.html#E8:93"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="sprect_2.html#T25" target="_self" title="SPRECT_2:th.25">Th25</a>, <a class="ref" href="sprect_2.html#T26" target="_self" title="SPRECT_2:th.26">Th26</a></i>;<br/>
<a NAME="E48:93"/><i><font color="Green" title="E113">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E46:93"><i><font color="Green" title="E111">A28</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E49:93"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E45:93"><i><font color="Green" title="E110">A27</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E50:93"/><b>then </b><i><font color="Green" title="E114">A31</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="sprect_2.html#R2" title="SPRECT_2:pred.2">is_a_h.c._for</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E47:93"><i><font color="Green" title="E112">A29</font></i></a>, <a class="txt" href="sprect_2.html#E48:93"><i><font color="Green" title="E113">A30</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self" title="SPRECT_2:def.2">Def2</a></i>;<br/>
<a NAME="E51:93"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">M</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E2:93"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="sprect_2.html#E19:93"><i><font color="Green" title="E100">A17</font></i></a>, <a class="txt" href="sprect_2.html#E24:93"><i><font color="Green" title="E102">A19</font></i></a>, <a class="txt" href="sprect_2.html#E29:93"><i><font color="Green" title="E105">A22</font></i></a>, <a class="ref" href="sprect_2.html#T51" target="_self" title="SPRECT_2:th.51">Th51</a></i>;<br/>
<b>hence </b><a NAME="E52:93"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E37:93"><i><font color="Green" title="E108">A25</font></i></a>, <a class="txt" href="sprect_2.html#E44:93"><i><font color="Green" title="E109">A26</font></i></a>, <a class="txt" href="sprect_2.html#E50:93"><i><font color="Green" title="E114">A31</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self" title="SPRECT_2:th.33">Th33</a></i>;<br/>


</div>
