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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#K31" title="PSCOMP_1:func.31">S-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="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> )</span><br/>

<b>set </b><font color="Maroon" title="c2">i1</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#K31" title="PSCOMP_1:func.31">S-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="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="E1:86"/><i><font color="Green" title="E78">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="E2:86"/><i><font color="Green" title="E79">A2</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#K31" title="PSCOMP_1:func.31">S-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="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:86"/><i><font color="Green" title="E80">A3</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#T50" target="_self" title="SPRECT_2:th.50">Th50</a></i>;<br/>
<a NAME="E4:86"/><i><font color="Green" title="E81">A4</font></i>: 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-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="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#T46" target="_self" title="SPRECT_2:th.46">Th46</a></i>;<br/>
<a NAME="E5:86"/><i><font color="Green" title="E82">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#E3:86"><i><font color="Green" title="E80">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="E6:86"/><b>then </b><i><font color="Green" title="E83">A6</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#K29" title="PSCOMP_1:func.29">E-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="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#K29" title="PSCOMP_1:func.29">E-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="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="E7:86"/><i><font color="Green" title="E84">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-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="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#E4:86"><i><font color="Green" title="E81">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="E8:86"/><b>then </b><i><font color="Green" title="E85">A8</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#K31" title="PSCOMP_1:func.31">S-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="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#K31" title="PSCOMP_1:func.31">S-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="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="E86">A9</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#K31" title="PSCOMP_1:func.31">S-max</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#K31" title="PSCOMP_1:func.31">S-max</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#E7:86"><i><font color="Green" title="E84">A7</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#K31" title="PSCOMP_1:func.31">S-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>
<b>by </b><i><a class="txt" href="sprect_2.html#E4:86"><i><font color="Green" title="E81">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="E10:86"/><i><font color="Green" title="E87">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"><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#E1:86"><i><font color="Green" title="E78">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="E88">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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</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#K29" title="PSCOMP_1:func.29">E-max</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#E5:86"><i><font color="Green" title="E82">A5</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#K29" title="PSCOMP_1:func.29">E-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>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:86"><i><font color="Green" title="E80">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="E12:86"/><i><font color="Green" title="E89">A12</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="euclid.html#K22" title="EUCLID:func.22">`2</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#K29" title="PSCOMP_1:func.29">E-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> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T57" target="_self" title="SPRECT_2:th.57">Th57</a></i>;<br/>
<a NAME="E13:86"/>
 <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="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#T16" title="SPRECT_1:th.16">SPRECT_1:16</a></i>;<br/>
<a NAME="E14:86"/><b>then </b>
 <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> <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="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="E15:86"/><b>then </b>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-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>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E12:86"><i><font color="Green" title="E89">A12</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:86"/><b>then </b><i><font color="Green" title="E90">A13</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#K31" title="PSCOMP_1:func.31">S-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="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#E2:86"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="sprect_2.html#E9:86"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="sprect_2.html#E11:86"><i><font color="Green" title="E88">A11</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="E17:86"/><b>then </b><i><font color="Green" title="E91">A14</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#E8:86"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:86"/><i><font color="Green" title="E92">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="E19:86"/><i><font color="Green" title="E93">A16</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> <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="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E20:86"/><i><font color="Green" title="E94">A17</font></i>: 
2 <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="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/>
<a NAME="E21:86"/><b>then </b><i><font color="Green" title="E95">A18</font></i>: 
2 <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_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:86"/>
<font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">N-most</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#E1:86"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="sprect_2.html#T34" target="_self" title="SPRECT_2:th.34">Th34</a></i>;<br/>
<b>then </b><i><font color="Green" title="E96">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</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#T98" title="PSCOMP_1:th.98">PSCOMP_1:98</a></i>
<br/>.= 
 <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></i>
;<br/>
<a NAME="E24:86"/><i><font color="Green" title="E97">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</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#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#E1:86"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E25:86"/><i><font color="Green" title="E98">A21</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-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="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:86"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:86"/><i><font color="Green" title="E99">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-max</a> <span class="p4">(<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>)</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="txt" href="sprect_2.html#E9:86"><i><font color="Green" title="E86">A9</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E27:86"/>
 <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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="topreal5.html#T22" title="TOPREAL5:th.22">TOPREAL5:22</a></i>;<br/>
<a NAME="E28:86"/><b>then </b><i><font color="Green" title="E100">A23</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">S-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="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> 2
 
<b>by </b><i><a class="txt" href="sprect_2.html#E23:86"><i><font color="Green" title="E96">A19</font></i></a>, <a class="txt" href="sprect_2.html#E24:86"><i><font color="Green" title="E97">A20</font></i></a>, <a class="txt" href="sprect_2.html#E25:86"><i><font color="Green" title="E98">A21</font></i></a>, <a class="txt" href="sprect_2.html#E26:86"><i><font color="Green" title="E99">A22</font></i></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">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#K31" title="PSCOMP_1:func.31">S-max</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>,2 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#E8:86"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="sprect_2.html#T41" target="_self" title="SPRECT_2:th.41">Th41</a></i>;<br/>
<a NAME="E30:86"/><i><font color="Green" title="E101">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</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/>
<a NAME="E31:86"/><i><font color="Green" title="E102">A25</font></i>: 
<font color="Maroon" title="c4">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#E7:86"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="sprect_2.html#E21:86"><i><font color="Green" title="E95">A18</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="E32:86"/>
<font color="Maroon" title="c4">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#K31" title="PSCOMP_1:func.31">S-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>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:86"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="sprect_2.html#E9:86"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="sprect_2.html#E21:86"><i><font color="Green" title="E95">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="E33:86"/><b>then </b><i><font color="Green" title="E103">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="E34:86"/>
<font color="Maroon" title="c4">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="c4">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> 2
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:86"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="sprect_2.html#E21:86"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self" title="SPRECT_2:th.13">Th13</a></i>;<br/>
<a NAME="E35:86"/><b>then </b><i><font color="Green" title="E104">A27</font></i>: 
<font color="Maroon" title="c4">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#E23:86"><i><font color="Green" title="E96">A19</font></i></a>, <a class="txt" href="sprect_2.html#E31:86"><i><font color="Green" title="E102">A25</font></i></a>, <a class="txt" href="sprect_2.html#E33:86"><i><font color="Green" title="E103">A26</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self" title="SPRECT_2:def.3">Def3</a></i>;<br/>
<a NAME="E36:86"/><i><font color="Green" title="E105">A28</font></i>: 
<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#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> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T55" target="_self" title="SPRECT_2:th.55">Th55</a></i>;<br/>
<a NAME="E37:86"/>
 <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="E38:86"/><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="E39:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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#E10:86"><i><font color="Green" title="E87">A10</font></i></a>, <a class="txt" href="sprect_2.html#E11:86"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="sprect_2.html#E36:86"><i><font color="Green" title="E105">A28</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E40:86"/><b>then </b><i><font color="Green" title="E106">A29</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-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="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#E6:86"><i><font color="Green" title="E83">A6</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="c5">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K29" title="PSCOMP_1:func.29">E-max</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#E17:86"><i><font color="Green" title="E91">A14</font></i></a>, <a class="ref" href="sprect_2.html#T41" target="_self" title="SPRECT_2:th.41">Th41</a></i>;<br/>
<a NAME="E42:86"/><i><font color="Green" title="E107">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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/>
<a NAME="E43:86"/><i><font color="Green" title="E108">A31</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">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="c4">h</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:86"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="sprect_2.html#E16:86"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="sprect_2.html#E28:86"><i><font color="Green" title="E100">A23</font></i></a>, <a class="ref" href="sprect_2.html#T53" target="_self" title="SPRECT_2:th.53">Th53</a></i>;<br/>
<i><font color="Green" title="E109">A32</font></i>: <font color="Maroon" title="c5">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="c5">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#K29" title="PSCOMP_1:func.29">E-max</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#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</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#K29" title="PSCOMP_1:func.29">E-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>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:86"><i><font color="Green" title="E80">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="E45:86"/><i><font color="Green" title="E110">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <span class="p4">(<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>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:86"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="jordan4.html#T21" title="JORDAN4:th.21">JORDAN4:21</a></i>;<br/>
<a NAME="E46:86"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="txt" href="sprect_2.html#E40:86"><i><font color="Green" title="E106">A29</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E47:86"/><b>then </b>
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E48:86"/><b>then </b>
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<a NAME="E49:86"/><b>then </b><i><font color="Green" title="E111">A34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <span class="p4">(<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>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E50:86"/><i><font color="Green" title="E112">A35</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="sprect_2.html#E19:86"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self" title="SPRECT_2:th.12">Th12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_5"/>
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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#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> or  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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#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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_5_1"><b>suppose </b></a><a NAME="E1:86_5_1"/><i><font color="Green" title="E113">A36</font></i>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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#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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:86_5_1"/><i><font color="Green" title="E114">A37</font></i>: 
<font color="Maroon" title="c5">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#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</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="E3:86_5_1"/>
<font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self" title="SPRECT_2:th.12">Th12</a></i>;<br/><a NAME="E4:86_5_1"/><b>then </b><i><font color="Green" title="E115">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#E1:86"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="sprect_2.html#E19:86"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="sprect_2.html#E1:86_5_1"><i><font color="Green" title="E113">A36</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:86_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">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#E44:86"><i><font color="Green" title="E109">A32</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E6:86_5_1"/><b>then </b>
<font color="Maroon" title="c5">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#E2:86_5_1"><i><font color="Green" title="E114">A37</font></i></a>, <a class="txt" href="sprect_2.html#E4:86_5_1"><i><font color="Green" title="E115">A38</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self" title="SPRECT_2:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E7:86_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E30:86"><i><font color="Green" title="E101">A24</font></i></a>, <a class="txt" href="sprect_2.html#E35:86"><i><font color="Green" title="E104">A27</font></i></a>, <a class="txt" href="sprect_2.html#E42:86"><i><font color="Green" title="E107">A30</font></i></a>, <a class="txt" href="sprect_2.html#E43:86"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self" title="SPRECT_2:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_5_2"><b>suppose </b></a><a NAME="E1:86_5_2"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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#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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">g</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">M</font> 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#E1:86"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="sprect_2.html#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="sprect_2.html#E19:86"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="sprect_2.html#T70" target="_self" title="SPRECT_2:th.70">Th70</a></i>;<br/><a NAME="E3:86_5_2"/><i><font color="Green" title="E113">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g</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/><a NAME="E4:86_5_2"/><i><font color="Green" title="E114">A40</font></i>: 
<font color="Maroon" title="c5">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#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</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="E5:86_5_2"/>
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <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="ref" href="sprect_2.html#T30" target="_self" title="SPRECT_2:th.30">Th30</a></i>;<br/><a NAME="E6:86_5_2"/><b>then </b><i><font color="Green" title="E115">A41</font></i>: 
<font color="Maroon" title="c6">g</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#E4:86_5_2"><i><font color="Green" title="E114">A40</font></i></a>, <a class="ref" href="sprect_2.html#T28" target="_self" title="SPRECT_2:th.28">Th28</a></i>;<br/><a NAME="E7:86_5_2"/>
<font color="Maroon" title="c6">g</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#K18" title="PSCOMP_1:func.18">NW-corner</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="finseq_5.html#T16" title="FINSEQ_5:th.16">FINSEQ_5:16</a></i>;<br/><a NAME="E8:86_5_2"/><b>then </b><i><font color="Green" title="E116">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</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="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E9:86_5_2"/><i><font color="Green" title="E117">A43</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</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="c5">M</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="E10:86_5_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">M</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <span class="p4">(<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></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>then </b><font color="Maroon" title="c6">g</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">g</font></span>)</span> = 
<font color="Maroon" title="c5">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="c5">M</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E9:86_5_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<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#K29" title="PSCOMP_1:func.29">E-max</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#E5:86"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sprect_2.html#E18:86"><i><font color="Green" title="E92">A15</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#K29" title="PSCOMP_1:func.29">E-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>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:86"><i><font color="Green" title="E80">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="E12:86_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</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">g</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="E13:86_5_2"/><b>then </b><i><font color="Green" title="E118">A44</font></i>: 
<font color="Maroon" title="c6">g</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#E6:86_5_2"><i><font color="Green" title="E115">A41</font></i></a>, <a class="txt" href="sprect_2.html#E8:86_5_2"><i><font color="Green" title="E116">A42</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self" title="SPRECT_2:def.2">Def2</a></i>;<br/><a NAME="E14:86_5_2"/><i><font color="Green" title="E119">A45</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T20" title="SPPOL_2:th.20">SPPOL_2:20</a></i>;<br/><a NAME="E15:86_5_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#E8:86"><i><font color="Green" title="E85">A8</font></i></a>, <a class="txt" href="sprect_2.html#E20:86"><i><font color="Green" title="E94">A17</font></i></a>, <a class="ref" href="jordan4.html#T47" title="JORDAN4:th.47">JORDAN4:47</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E16:86_5_2"/><b>then </b><i><font color="Green" title="E120">A46</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:86"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="sprect_2.html#E50:86"><i><font color="Green" title="E112">A35</font></i></a>, <a class="ref" href="pscomp_1.html#T102" title="PSCOMP_1:th.102">PSCOMP_1:102</a></i>;<br/><a NAME="E17:86_5_2"/>
<font color="Maroon" title="c5">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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="c5">M</font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E45:86"><i><font color="Green" title="E110">A33</font></i></a>, <a class="txt" href="sprect_2.html#E49:86"><i><font color="Green" title="E111">A34</font></i></a>, <a class="ref" href="jordan3.html#T34" title="JORDAN3:th.34">JORDAN3:34</a></i>;<br/><b>hence </b><a NAME="E18:86_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E30:86"><i><font color="Green" title="E101">A24</font></i></a>, <a class="txt" href="sprect_2.html#E35:86"><i><font color="Green" title="E104">A27</font></i></a>, <a class="txt" href="sprect_2.html#E43:86"><i><font color="Green" title="E108">A31</font></i></a>, <a class="txt" href="sprect_2.html#E3:86_5_2"><i><font color="Green" title="E113">A39</font></i></a>, <a class="txt" href="sprect_2.html#E13:86_5_2"><i><font color="Green" title="E118">A44</font></i></a>, <a class="txt" href="sprect_2.html#E14:86_5_2"><i><font color="Green" title="E119">A45</font></i></a>, <a class="txt" href="sprect_2.html#E16:86_5_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="sprect_2.html#T1" target="_self" title="SPRECT_2:th.1">Th1</a>, <a class="ref" href="sprect_2.html#T33" target="_self" title="SPRECT_2:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
