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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> implies <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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/>

<span class="kw">set </span><font color="Maroon" title="c2">i1</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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/>
<span class="kw">set </span><font color="Maroon" title="c3">i2</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E75">A1</font></span>: 
<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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E76">A2</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:83"/><span class="lab"><font color="Green" title="E77">A3</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`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#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T51" target="_self" title="SPRECT_2:th.51">Th55</a></span>;<br/>
<a NAME="E4:83"/>
<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#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="sprect_2.html#T30" target="_self" title="SPRECT_2:th.30">Th34</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E78">A4</font></span>: <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#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T39" title="PSCOMP_1:th.39">PSCOMP_1:39</a></span>
<br/>.= 
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E6:83"/>
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></span>;<br/>
<a NAME="E7:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A5</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K23" title="PSCOMP_1:func.23">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T24" title="PSCOMP_1:th.24">PSCOMP_1:24</a></span>;<br/>
<a NAME="E8:83"/><span class="lab"><font color="Green" title="E80">A6</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal5.html#T16" title="TOPREAL5:th.16">TOPREAL5:16</a></span>;<br/>
<a NAME="E9:83"/><span class="lab"><font color="Green" title="E81">A7</font></span>: 
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T42" target="_self" title="SPRECT_2:th.42">Th46</a></span>;<br/>
<a NAME="E10:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T20" title="FINSEQ_4:th.20">FINSEQ_4:20</a></span>;<br/>
<a NAME="E11:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A9</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E12:83"/><span class="lab"><font color="Green" title="E84">A10</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E85">A11</font></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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="ref" href="finseq_4.html#T19" title="FINSEQ_4:th.19">FINSEQ_4:19</a></span>
;<br/>
<a NAME="E14:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A12</font></span>: 
<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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E15:83"/><span class="lab"><font color="Green" title="E87">A13</font></span>: 
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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E16:83"/>
<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#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E17:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A14</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E84">A10</font></span></a>, <a class="txt" href="#E14:83"><span class="lab"><font color="Green" title="E86">A12</font></span></a>, <a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E80">A6</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">h</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E83">A9</font></span></a>, <a class="ref" href="sprect_2.html#T37" target="_self" title="SPRECT_2:th.37">Th41</a></span>;<br/>
<a NAME="E19:83"/><span class="lab"><font color="Green" title="E89">A15</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T60" title="NAT_D:th.60">NAT_D:60</a></span>;<br/>
<a NAME="E20:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A16</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:83"/><span class="kw">then </span>
<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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="ref" href="sprect_2.html#T8" target="_self" title="SPRECT_2:th.8">Th12</a></span>;<br/>
<a NAME="E22:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A17</font></span>: 
<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#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E23:83"/>
( <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> &amp; <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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="txt" href="#E20:83"><span class="lab"><font color="Green" title="E90">A16</font></span></a>, <a class="ref" href="sprect_2.html#T9" target="_self" title="SPRECT_2:th.9">Th13</a>, <a class="ref" href="sprect_2.html#T21" target="_self" title="SPRECT_2:th.21">Th25</a>, <a class="ref" href="sprect_2.html#T22" target="_self" title="SPRECT_2:th.22">Th26</a></span>;<br/>
<a NAME="E24:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A18</font></span>: 
(  <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 &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="txt" href="#E22:83"><span class="lab"><font color="Green" title="E91">A17</font></span></a>, <a class="ref" href="sprect_2.html#D3" target="_self" title="SPRECT_2:def.3">Def3</a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E25:83"/>
 <a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T11" title="SPRECT_1:th.11">SPRECT_1:11</a></span>;<br/>
<a NAME="E26:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A19</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T24" title="PSCOMP_1:th.24">PSCOMP_1:24</a></span>;<br/>
<a NAME="E27:83"/><span class="lab"><font color="Green" title="E94">A20</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T46" target="_self" title="SPRECT_2:th.46">Th50</a></span>;<br/>
<a NAME="E28:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A21</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T20" title="FINSEQ_4:th.20">FINSEQ_4:20</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E96">A22</font></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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E94">A20</font></span></a>, <a class="ref" href="finseq_4.html#T19" title="FINSEQ_4:th.19">FINSEQ_4:19</a></span>
;<br/>
<a NAME="E30:83"/><span class="lab"><font color="Green" title="E97">A23</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E31:83"/>
<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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>;<br/>
<a NAME="E32:83"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:83"><span class="lab"><font color="Green" title="E96">A22</font></span></a>, <a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E26:83"><span class="lab"><font color="Green" title="E93">A19</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E33:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A24</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:83"><span class="lab"><font color="Green" title="E97">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E34:83"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E35:83"/><span class="kw">then </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> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E36:83"/><span class="kw">then </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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/>
<a NAME="E37:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A25</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E38:83"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T53" target="_self" title="SPRECT_2:th.53">Th57</a></span>;<br/>
<a NAME="E39:83"/><span class="kw">then </span>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E40:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A26</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="txt" href="#E29:83"><span class="lab"><font color="Green" title="E96">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E41:83"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83"><span class="lab"><font color="Green" title="E87">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">M</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:83"><span class="lab"><font color="Green" title="E98">A24</font></span></a>, <a class="ref" href="sprect_2.html#T37" target="_self" title="SPRECT_2:th.37">Th41</a></span>;<br/>
<a NAME="E43:83"/><span class="lab"><font color="Green" title="E101">A27</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E44:83"/>
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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E45:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A28</font></span>: 
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:83"><span class="lab"><font color="Green" title="E97">A23</font></span></a>, <a class="ref" href="jordan4.html#T9" title="JORDAN4:th.9">JORDAN4:9</a></span>;<br/>
<a NAME="E46:83"/><span class="lab"><font color="Green" title="E103">A29</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E104">A30</font></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> = 
<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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="ref" href="sprect_2.html#T9" target="_self" title="SPRECT_2:th.9">Th13</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E94">A20</font></span></a>, <a class="ref" href="finseq_5.html#T38" title="FINSEQ_5:th.38">FINSEQ_5:38</a></span>
;<br/>
<a NAME="E48:83"/><span class="lab"><font color="Green" title="E105">A31</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:83"><span class="lab"><font color="Green" title="E97">A23</font></span></a>, <a class="txt" href="#E40:83"><span class="lab"><font color="Green" title="E100">A26</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E88">A14</font></span></a>, <a class="ref" href="sprect_2.html#T49" target="_self" title="SPRECT_2:th.49">Th53</a></span>;<br/>
<a NAME="E49:83"/><span class="lab"><font color="Green" title="E106">A32</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>;<br/>
<a NAME="E50:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A33</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="ref" href="sprect_2.html#T8" target="_self" title="SPRECT_2:th.8">Th12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_5"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_5"/>
(  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> or  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="83_5_1"><span class="kw">suppose </span></a><a NAME="E1:83_5_1"/><span class="lab"><font color="Green" title="E108">A34</font></span>: 
 <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:83_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="ref" href="sprect_2.html#T8" target="_self" title="SPRECT_2:th.8">Th12</a></span>;<br/><a NAME="E3:83_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A35</font></span>: 
<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#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="txt" href="#E49:83"><span class="lab"><font color="Green" title="E106">A32</font></span></a>, <a class="txt" href="#E1:83_5_1"><span class="lab"><font color="Green" title="E108">A34</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E4:83_5_1"/>
( <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> &amp; <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#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E47:83"><span class="lab"><font color="Green" title="E104">A30</font></span></a>, <a class="ref" href="sprect_2.html#T21" target="_self" title="SPRECT_2:th.21">Th25</a>, <a class="ref" href="sprect_2.html#T22" target="_self" title="SPRECT_2:th.22">Th26</a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E5:83_5_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_5_1"><span class="lab"><font color="Green" title="E109">A35</font></span></a>, <a class="ref" href="sprect_2.html#D2" target="_self" title="SPRECT_2:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:83_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:83"><span class="lab"><font color="Green" title="E92">A18</font></span></a>, <a class="txt" href="#E43:83"><span class="lab"><font color="Green" title="E101">A27</font></span></a>, <a class="txt" href="#E48:83"><span class="lab"><font color="Green" title="E105">A31</font></span></a>, <a class="ref" href="sprect_2.html#T29" target="_self" title="SPRECT_2:th.29">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_5_2"><span class="kw">suppose </span></a><a NAME="E1:83_5_2"/>
 <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">g</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="txt" href="#E49:83"><span class="lab"><font color="Green" title="E106">A32</font></span></a>, <a class="ref" href="sprect_2.html#T66" target="_self" title="SPRECT_2:th.66">Th70</a></span>;<br/><a NAME="E3:83_5_2"/><span class="lab"><font color="Green" title="E108">A36</font></span>: 
(  <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 &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T20" title="SPPOL_2:th.20">SPPOL_2:20</a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/><a NAME="E4:83_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#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></span>;<br/><a NAME="E5:83_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A37</font></span>: 
<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#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E6:83_5_2"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">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#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">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#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83"><span class="lab"><font color="Green" title="E87">A13</font></span></a>, <a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E83">A9</font></span></a>, <a class="txt" href="#E19:83"><span class="lab"><font color="Green" title="E89">A15</font></span></a>, <a class="ref" href="jordan4.html#T35" title="JORDAN4:th.35">JORDAN4:35</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><a NAME="E7:83_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A38</font></span>: 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">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#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K3" title="TOPREAL1:func.3">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="txt" href="#E50:83"><span class="lab"><font color="Green" title="E107">A33</font></span></a>, <a class="ref" href="pscomp_1.html#T43" title="PSCOMP_1:th.43">PSCOMP_1:43</a></span>;<br/><a NAME="E8:83_5_2"/>
(  <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> &amp;  <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="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>
<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#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="ref" href="sprect_2.html#T9" target="_self" title="SPRECT_2:th.9">Th13</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E94">A20</font></span></a>, <a class="ref" href="finseq_5.html#T38" title="FINSEQ_5:th.38">FINSEQ_5:38</a></span>
;<br/><a NAME="E10:83_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A39</font></span>: 
<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#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E11:83_5_2"/><span class="lab"><font color="Green" title="E112">A40</font></span>: 
<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#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:83"><span class="lab"><font color="Green" title="E102">A28</font></span></a>, <a class="txt" href="#E37:83"><span class="lab"><font color="Green" title="E99">A25</font></span></a>, <a class="ref" href="jordan3.html#T1" title="JORDAN3:th.1">JORDAN3:1</a></span>;<br/><a NAME="E12:83_5_2"/>
( <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> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E95">A21</font></span></a>, <a class="txt" href="#E46:83"><span class="lab"><font color="Green" title="E103">A29</font></span></a>, <a class="ref" href="sprect_2.html#T21" target="_self" title="SPRECT_2:th.21">Th25</a>, <a class="ref" href="sprect_2.html#T22" target="_self" title="SPRECT_2:th.22">Th26</a>, <a class="ref" href="sprect_2.html#T26" target="_self" title="SPRECT_2:th.26">Th30</a></span>;<br/><a NAME="E13:83_5_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T24" target="_self" title="SPRECT_2:th.24">Th28</a></span>;<br/><a NAME="E14:83_5_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_5_2"><span class="lab"><font color="Green" title="E109">A37</font></span></a>, <a class="txt" href="#E10:83_5_2"><span class="lab"><font color="Green" title="E111">A39</font></span></a>, <a class="ref" href="sprect_2.html#D2" target="_self" title="SPRECT_2:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:83_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:83"><span class="lab"><font color="Green" title="E92">A18</font></span></a>, <a class="txt" href="#E48:83"><span class="lab"><font color="Green" title="E105">A31</font></span></a>, <a class="txt" href="#E3:83_5_2"><span class="lab"><font color="Green" title="E108">A36</font></span></a>, <a class="txt" href="#E7:83_5_2"><span class="lab"><font color="Green" title="E110">A38</font></span></a>, <a class="txt" href="#E11:83_5_2"><span class="lab"><font color="Green" title="E112">A40</font></span></a>, <a class="ref" href="sprect_2.html#T29" target="_self" title="SPRECT_2:th.29">Th33</a>, <a class="ref" href="zfmisc_1.html#T125" title="ZFMISC_1:th.125">ZFMISC_1:125</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
